One of the most important aspects of QSym (the Hopf algebra of quasisymmetric functions) is a universal property discovered by Aguiar, Bergeron and Sottile in 2003 [2 ] ; among other applications, it gives a unifying framework for various quasisymmetric and symmetric functions constructed from combinatorial objects (e.g., the chromatic symmetric function of a graph).
On the other hand, let Λ k \Lambda_k Λ k k \mathbf{k} k H H H k \mathbf{k} k k \mathbf{k} k H → H ⊗ Λ k H \to H \otimes\Lambda_k H → H ⊗ Λ k Λ k \Lambda_k Λ k
In this note, I shall define an extended version of this homomorphism: a k \mathbf{k} k H → H ⊗ QSym k H \to H \otimes\operatorname{QSym}_k H → H ⊗ QSym k k \mathbf{k} k H H H Bernstein homomorphism , will generalize the second comultiplication of QSym k \operatorname{QSym}_k QSym k H H H H ⊗ Λ k H \otimes\Lambda_k H ⊗ Λ k
The Bernstein homomorphism H → H ⊗ QSym k H \to H \otimes\operatorname{QSym}_k H → H ⊗ QSym k H ′ ⊗ NSym k → H ′ H' \otimes\operatorname{NSym}_k \to H' H ′ ⊗ NSym k → H ′ H ′ H' H ′ NSym k \operatorname{NSym}_k NSym k H ′ H' H ′ [1 Propositions 84 and 88, and especially the Remark after Proposition 88] . What I believe to be new in this note is the way I will construct the Bernstein homomorphism: as a consequence of the Aguiar-Bergeron-Sottile universal property of QSym, but applied not to the k \mathbf{k} k QSym k \operatorname{QSym}_k QSym k H H H QSym H \operatorname{QSym}_H QSym H H H H H H H
Acknowledgments
Thanks to Marcelo Aguiar for enlightening discussions.
Definitions and conventions
For the rest of this note, we fix a commutative ring1 k k k ( ⊗ ) (\otimes) ( ⊗ ) ⊗ k \otimes_k ⊗ k k k k k k k k k k k k k k k k k k k 2 k k k C C C Δ C \Delta_C Δ C ε C \varepsilon_C ε C C C C Δ \Delta Δ ε \varepsilon ε
If V V V W W W k k k τ V , W \tau_{V,W} τ V , W k k k V ⊗ W → W ⊗ V V \otimes W \to W \otimes V V ⊗ W → W ⊗ V v ⊗ w ↦ w ⊗ v v \otimes w \mapsto w \otimes v v ⊗ w ↦ w ⊗ v k k k τ V , W \tau_{V,W} τ V , W twist map , and is a k k k
The next two definitions are taken from [GriRei14, §1.4]3
Definition 1.1. Let A A A k k k m A m_A m A k k k A ⊗ A → A A \otimes A \to A A ⊗ A → A a ⊗ b ↦ a b a \otimes b \mapsto ab a ⊗ b ↦ ab u A u_A u A k k k k → A k \to A k → A λ ↦ λ ⋅ 1 A \lambda\mapsto\lambda\cdot1_A λ ↦ λ ⋅ 1 A m A m_A m A u A u_A u A m m m u u u A A A k ∈ N k \in\mathbb{N} k ∈ N k k k m ( k − 1 ) : A ⊗ k → A m^{(k-1)}: A^{\otimes k} \to A m ( k − 1 ) : A ⊗ k → A m ( − 1 ) = u A m^{(-1)} = u_A m ( − 1 ) = u A m ( 0 ) = id A m^{(0)} = \operatorname{id}_A m ( 0 ) = id A
m ( k ) = m ∘ ( id A ⊗ m ( k − 1 ) ) for every k ≥ 1. m^{(k)} = m \circ\left(\operatorname{id}_A \otimes m^{(k-1)}\right) \qquad\text{for every } k \ge1. m ( k ) = m ∘ ( id A ⊗ m ( k − 1 ) ) for every k ≥ 1. The maps m ( k − 1 ) : A ⊗ k → A m^{(k-1)}: A^{\otimes k} \to A m ( k − 1 ) : A ⊗ k → A iterated multiplication maps of A A A
Notice that for every k ∈ N k \in\mathbb{N} k ∈ N m ( k − 1 ) m^{(k-1)} m ( k − 1 ) k k k A ⊗ k → A A^{\otimes k} \to A A ⊗ k → A a 1 ⊗ a 2 ⊗ ⋯ ⊗ a k ∈ A ⊗ k a_1 \otimes a_2 \otimes\cdots\otimes a_k \in A^{\otimes k} a 1 ⊗ a 2 ⊗ ⋯ ⊗ a k ∈ A ⊗ k a 1 a 2 ⋯ a k a_1a_2 \cdots a_k a 1 a 2 ⋯ a k
Definition 1.2. Let C C C k k k k ∈ N k \in\mathbb{N} k ∈ N k k k Δ ( k − 1 ) : C → C ⊗ k \Delta^{(k-1)}: C \to C^{\otimes k} Δ ( k − 1 ) : C → C ⊗ k Δ ( − 1 ) = ε C \Delta^{(-1)} = \varepsilon_C Δ ( − 1 ) = ε C Δ ( 0 ) = id C \Delta^{(0)} = \operatorname{id}_C Δ ( 0 ) = id C
Δ ( k ) = ( id C ⊗ Δ ( k − 1 ) ) ∘ Δ for every k ≥ 1. \Delta^{(k)} = \left(\operatorname{id}_C \otimes\Delta^{(k-1)}\right) \circ\Delta\qquad\text{for every } k \ge1. Δ ( k ) = ( id C ⊗ Δ ( k − 1 ) ) ∘ Δ for every k ≥ 1. The maps Δ ( k − 1 ) : C → C ⊗ k \Delta^{(k-1)}: C \to C^{\otimes k} Δ ( k − 1 ) : C → C ⊗ k iterated comultiplication maps of C C C
A composition shall mean a finite sequence of positive integers. The size of a composition α = ( α 1 , α 2 , … , α k ) \alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k) α = ( α 1 , α 2 , … , α k ) α 1 + α 2 + ⋯ + α k \alpha_1+\alpha_2+\cdots+\alpha_k α 1 + α 2 + ⋯ + α k ∣ α ∣ |\alpha| ∣ α ∣ Comp \operatorname{Comp} Comp N \mathbb{N} N { 0 , 1 , 2 , … } \{0,1,2,\ldots\} { 0 , 1 , 2 , … }
Definition 1.3. Let H H H k \mathbf{k} k n ∈ N n\in\mathbb{N} n ∈ N π n : H → H \pi_n:H\to H π n : H → H H H H n n n H n H_n H n H H H π n \pi_n π n H H H H H H H H H H n H_n H n H n H_n H n
For every composition α = ( a 1 , a 2 , … , a k ) \alpha=(a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k ) π α : H ⊗ k → H ⊗ k \pi_\alpha:H^{\otimes k}\to H^{\otimes k} π α : H ⊗ k → H ⊗ k π a 1 ⊗ π a 2 ⊗ ⋯ ⊗ π a k \pi_{a_1}\otimes\pi_{a_2}\otimes\cdots\otimes\pi_{a_k} π a 1 ⊗ π a 2 ⊗ ⋯ ⊗ π a k π a i : H → H \pi_{a_i}:H\to H π a i : H → H π α \pi_\alpha π α H a 1 ⊗ H a 2 ⊗ ⋯ ⊗ H a k H_{a_1}\otimes H_{a_2}\otimes\cdots\otimes H_{a_k} H a 1 ⊗ H a 2 ⊗ ⋯ ⊗ H a k
Let QSym k \operatorname{QSym}_{\mathbf{k}} QSym k k \mathbf{k} k k \mathbf{k} k QSym \operatorname{QSym} QSym [2 §3] ; it is also defined and denoted by QSym \operatorname{QSym} QSym QSym k \operatorname{QSym}_{\mathbf{k}} QSym k QSym k \operatorname{QSym}_{\mathbf{k}} QSym k k [ [ x 1 , x 2 , x 3 , … ] ] \mathbf{k}[[x_1,x_2,x_3,\ldots]] k [[ x 1 , x 2 , x 3 , … ]] x 1 , x 2 , x 3 , … x_1,x_2,x_3,\ldots x 1 , x 2 , x 3 , …
Let ε P \varepsilon_P ε P k \mathbf{k} k QSym k → k \operatorname{QSym}_{\mathbf{k}}\to\mathbf{k} QSym k → k f ∈ QSym k f\in\operatorname{QSym}_{\mathbf{k}} f ∈ QSym k f ( 1 , 0 , 0 , 0 , … ) ∈ k f(1,0,0,0,\ldots)\in\mathbf{k} f ( 1 , 0 , 0 , 0 , … ) ∈ k ε P \varepsilon_P ε P ζ Q \zeta_Q ζ Q [2 §4] and by ζ Q \zeta_Q ζ Q ε P \varepsilon_P ε P k \mathbf{k} k
Definition 1.4. For every composition α = ( α 1 , α 2 , … , α ℓ ) \alpha=(\alpha_1,\alpha_2,\ldots,\alpha_\ell) α = ( α 1 , α 2 , … , α ℓ ) M α ∈ k [ [ x 1 , x 2 , x 3 , … ] ] M_\alpha\in\mathbf{k}[[x_1,x_2,x_3,\ldots]] M α ∈ k [[ x 1 , x 2 , x 3 , … ]]
M α = ∑ 1 ≤ i 1 < i 2 < ⋯ < i ℓ x i 1 α 1 x i 2 α 2 ⋯ x i ℓ α ℓ M_\alpha=\sum_{1\leq i_1<i_2<\cdots<i_\ell}x_{i_1}^{\alpha_1}x_{i_2}^{\alpha_2}\cdots x_{i_\ell}^{\alpha_\ell} M α = 1 ≤ i 1 < i 2 < ⋯ < i ℓ ∑ x i 1 α 1 x i 2 α 2 ⋯ x i ℓ α ℓ (where the sum is over all strictly increasing ℓ \ell ℓ ( i 1 < i 2 < ⋯ < i ℓ ) (i_1<i_2<\cdots<i_\ell) ( i 1 < i 2 < ⋯ < i ℓ ) M α M_\alpha M α QSym k \operatorname{QSym}_{\mathbf{k}} QSym k M α M_\alpha M α monomial quasisymmetric function corresponding to α \alpha α ( M α ) α ∈ Comp (M_\alpha)_{\alpha\in\operatorname{Comp}} ( M α ) α ∈ Comp k \mathbf{k} k QSym k \operatorname{QSym}_{\mathbf{k}} QSym k monomial basis of QSym k \operatorname{QSym}_{\mathbf{k}} QSym k [2 §3] and [GriRei14, §5.1] for more about this basis.)
It is well-known that every ( b 1 , b 2 , … , b ℓ ) ∈ Comp (b_1,b_2,\ldots,b_\ell)\in\operatorname{Comp} ( b 1 , b 2 , … , b ℓ ) ∈ Comp
Δ ( M ( b 1 , b 2 , … , b ℓ ) ) = ∑ i = 0 ℓ M ( b 1 , b 2 , … , b i ) ⊗ M ( b i + 1 , b i + 2 , … , b ℓ ) (1) \Delta\left(M_{(b_1,b_2,\ldots,b_\ell)}\right)=\sum_{i=0}^{\ell}M_{(b_1,b_2,\ldots,b_i)}\otimes M_{(b_{i+1},b_{i+2},\ldots,b_\ell)} \tag*{(1)} Δ ( M ( b 1 , b 2 , … , b ℓ ) ) = i = 0 ∑ ℓ M ( b 1 , b 2 , … , b i ) ⊗ M ( b i + 1 , b i + 2 , … , b ℓ ) (1) and
ε ( M ( b 1 , b 2 , … , b ℓ ) ) = { 1 , if ℓ = 0 ; 0 , if ℓ ≠ 0. \varepsilon\left(M_{(b_1,b_2,\ldots,b_\ell)}\right)=
\begin{cases}
1, & \text{if }\ell=0;\\
0, & \text{if }\ell\ne0.
\end{cases} ε ( M ( b 1 , b 2 , … , b ℓ ) ) = { 1 , 0 , if ℓ = 0 ; if ℓ = 0. These two equalities can be used as a definition of the k \mathbf{k} k QSym k \operatorname{QSym}_{\mathbf{k}} QSym k ( M α ) α ∈ Comp (M_\alpha)_{\alpha\in\operatorname{Comp}} ( M α ) α ∈ Comp k \mathbf{k} k QSym k \operatorname{QSym}_{\mathbf{k}} QSym k k \mathbf{k} k Δ \Delta Δ ε \varepsilon ε M α M_\alpha M α
The Aguiar-Bergeron-Sottile theorem
The cornerstone of the Aguiar-Bergeron-Sottile paper [2 ] is the following result:
Theorem 2.1. Let k \mathbf{k} k H H H k \mathbf{k} k ζ : H → k \zeta:H\to\mathbf{k} ζ : H → k k \mathbf{k} k
(a) Then, there exists a unique graded k \mathbf{k} k Ψ : H → QSym k \Psi:H\to\operatorname{QSym}_{\mathbf{k}} Ψ : H → QSym k
H → Ψ QSym k ζ ↓ ↓ ε P k = k \begin{CD}
H @>{\Psi}>> \operatorname{QSym}_{\mathbf{k}} \\
@V{\zeta}VV @VV{\varepsilon_P}V \\
\mathbf{k} @= \mathbf{k}
\end{CD} H ζ ↓ ⏐ k Ψ QSym k ↓ ⏐ ε P k is commutative.
(b) This unique k \mathbf{k} k Ψ : H → QSym k \Psi:H\to\operatorname{QSym}_{\mathbf{k}} Ψ : H → QSym k k \mathbf{k} k
(c) For every composition α = ( a 1 , a 2 , … , a k ) \alpha=(a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k ) k \mathbf{k} k ζ α : H → k \zeta_\alpha:H\to\mathbf{k} ζ α : H → k
H → Δ ( k − 1 ) H ⊗ k → π α H ⊗ k → ζ ⊗ k k ⊗ k → ≅ k . H \xrightarrow{\Delta^{(k-1)}} H^{\otimes k} \xrightarrow{\pi_\alpha} H^{\otimes k} \xrightarrow{\zeta^{\otimes k}} \mathbf{k}^{\otimes k} \xrightarrow{\cong} \mathbf{k}. H Δ ( k − 1 ) H ⊗ k π α H ⊗ k ζ ⊗ k k ⊗ k ≅ k . (Here, the map k ⊗ k → ≅ k \mathbf{k}^{\otimes k}\xrightarrow{\cong}\mathbf{k} k ⊗ k ≅ k k \mathbf{k} k k ⊗ k \mathbf{k}^{\otimes k} k ⊗ k k \mathbf{k} k Δ ( k − 1 ) : H → H ⊗ k \Delta^{(k-1)}:H\to H^{\otimes k} Δ ( k − 1 ) : H → H ⊗ k π α : H ⊗ k → H ⊗ k \pi_\alpha:H^{\otimes k}\to H^{\otimes k} π α : H ⊗ k → H ⊗ k
Then, the unique k \mathbf{k} k Ψ \Psi Ψ Theorem 2.1 (a) is given by the formula
Ψ ( h ) = ∑ α ∈ Comp ∣ α ∣ = n ζ α ( h ) ⋅ M α whenever n ∈ N and h ∈ H n . \Psi(h)=\sum_{\substack{\alpha\in\operatorname{Comp}\\|\alpha|=n}}\zeta_\alpha(h)\cdot M_\alpha\qquad\text{whenever }n\in\mathbb{N}\text{ and }h\in H_n. Ψ ( h ) = α ∈ Comp ∣ α ∣ = n ∑ ζ α ( h ) ⋅ M α whenever n ∈ N and h ∈ H n . (Recall that H n H_n H n n n n
(d) The unique k \mathbf{k} k Ψ \Psi Ψ Theorem 2.1 (a) is also given by
Ψ ( h ) = ∑ α ∈ Comp ζ α ( h ) ⋅ M α for every h ∈ H \Psi(h)=\sum_{\alpha\in\operatorname{Comp}}\zeta_\alpha(h)\cdot M_\alpha\qquad\text{for every }h\in H Ψ ( h ) = α ∈ Comp ∑ ζ α ( h ) ⋅ M α for every h ∈ H (in particular, the sum on the right hand side of this equality has only finitely many nonzero addends).
(e) Assume that the k k k H H H k k k Ψ \Psi Ψ Theorem 2.1 (a) satisfies Ψ ( H ) ⊆ Λ k \Psi(H) \subseteq\Lambda_k Ψ ( H ) ⊆ Λ k Λ k \Lambda_k Λ k k k k k k k Λ k \Lambda_k Λ k Λ k \Lambda_k Λ k k k k QSym k \operatorname{QSym}_k QSym k
Parts (a), (b) and (c) of Theorem 2.1 are proven in [2 proof of Theorem 4.1] and [GriRei14, proof of Theorem 7.1.3] (although we are using different notations here4 [2 ] which needlessly require k k k H H H Theorem 2.1 (d) easily follows from Theorem 2.1 (c).5 Theorem 2.1 (e) appears in [GriRei14, Remark 7.1.4] (and something very close is proven in [2 Theorem 4.3] ). For the sake of completeness, let me give some details on the proof of Theorem 2.1 (e):
H ′ → α H ζ ′ ↓ ↓ ζ k = k \begin{CD}
H' @>\alpha>> H \\
@V\zeta'VV @VV\zeta V \\
k @= k
\end{CD} H ′ ζ ′ ↓ ⏐ k α H ↓ ⏐ ζ k is commutative. Theorem 2.1 (a) translates into this language as follows: There exists a unique morphism from the combinatorial coalgebra ( H , ζ ) (H,\zeta) ( H , ζ ) ( QSym k , ε P ) (\operatorname{QSym}_k,\varepsilon_P) ( QSym k , ε P ) [2 ] is also using the notations k k k H H H QSym \operatorname{QSym} QSym ζ Q \zeta_Q ζ Q k k k H H H QSym k \operatorname{QSym}_k QSym k ε P \varepsilon_P ε P
π α ( Δ ( k − 1 ) ( ∑ ( h ) h ) ) ∈ π α ( Δ ( k − 1 ) ( H n ) ) = 0 \pi_\alpha\left(\Delta^{(k-1)}\left(\sum_{(h)} h\right)\right) \in\pi_\alpha\left(\Delta^{(k-1)}(H_n)\right)=0 π α Δ ( k − 1 ) ( h ) ∑ h ∈ π α ( Δ ( k − 1 ) ( H n ) ) = 0 for reasons of gradedness). Hence, for every n ∈ N n \in\mathbb{N} n ∈ N h ∈ H n h \in H_n h ∈ H n
∑ α ∈ Comp ζ α ( h ) ⋅ M α = ∑ α ∈ Comp ∣ α ∣ = n ζ α ( h ) ⋅ M α + ∑ α ∈ Comp ∣ α ∣ ≠ n ζ α ( h ) ⋅ M α = ∑ α ∈ Comp ∣ α ∣ = n ζ α ( h ) ⋅ M α = Ψ ( h ) (by xref‘Theorem 2.1 (c)‘). \begin{aligned}
\sum_{\alpha\in\operatorname{Comp}} \zeta_\alpha(h)\cdot M_\alpha
=
\sum_{\substack{\alpha\in\operatorname{Comp}\\|\alpha|=n}} \zeta_\alpha(h)\cdot M_\alpha
+
\sum_{\substack{\alpha\in\operatorname{Comp}\\|\alpha|\ne n}} \zeta_\alpha(h)\cdot M_\alpha\\
&=
\sum_{\substack{\alpha\in\operatorname{Comp}\\|\alpha|=n}} \zeta_\alpha(h)\cdot M_\alpha
=
\Psi(h)
\qquad\text{(by {xref}`Theorem 2.1 (c)`).}
\end{aligned} α ∈ Comp ∑ ζ α ( h ) ⋅ M α = α ∈ Comp ∣ α ∣ = n ∑ ζ α ( h ) ⋅ M α + α ∈ Comp ∣ α ∣ = n ∑ ζ α ( h ) ⋅ M α = α ∈ Comp ∣ α ∣ = n ∑ ζ α ( h ) ⋅ M α = Ψ ( h ) (by xref ‘Theorem 2.1 (c)‘). Both sides of this equality are k k k h h h h ∈ H h \in H h ∈ H h h h Theorem 2.1 (d).
which the diagram
H → Ψ ′ Λ k ↘ ζ ↙ ε p k (2) \begin{array}{c}
H \xrightarrow{\Psi'} \Lambda_k \\
\searrow_{\zeta} \quad\swarrow_{\varepsilon_p} \\
\mathbf{k} \tag*{(2)}
\end{array} H Ψ ′ Λ k ↘ ζ ↙ ε p k (2) is commutative. Consider this Ψ ′ \Psi' Ψ ′ ι : Λ k → QSym k \iota:\Lambda_k\to\operatorname{QSym}_k ι : Λ k → QSym k k \mathbf{k} k ε p = ε P ∘ ι \varepsilon_p=\varepsilon_P\circ\iota ε p = ε P ∘ ι ε p \varepsilon_p ε p (2) yields ζ = ε p ⏟ = ε P ∘ ι ∘ Ψ ′ = ε P ∘ ι ∘ Ψ ′ \zeta=\underbrace{\varepsilon_p}_{=\varepsilon_P\circ\iota}\circ\Psi'=\varepsilon_P\circ\iota\circ\Psi' ζ = = ε P ∘ ι ε p ∘ Ψ ′ = ε P ∘ ι ∘ Ψ ′
Now, consider the unique k \mathbf{k} k Ψ \Psi Ψ Theorem 2.1 (a). Due to its uniqueness, it has the following property: If Ψ ~ \widetilde{\Psi} Ψ k \mathbf{k} k H → QSym k H\to\operatorname{QSym}_k H → QSym k
H → Ψ ~ QSym k ↘ ζ ↙ ε P k (3) \begin{array}{c}
H \xrightarrow{\widetilde{\Psi}} \operatorname{QSym}_k \\
\searrow_{\zeta} \quad\swarrow_{\varepsilon_P} \\
\mathbf{k} \tag*{(3)}
\end{array} H Ψ QSym k ↘ ζ ↙ ε P k (3) is commutative, then Ψ ~ = Ψ \widetilde{\Psi}=\Psi Ψ = Ψ Ψ ~ = ι ∘ Ψ ′ \widetilde{\Psi}=\iota\circ\Psi' Ψ = ι ∘ Ψ ′ ι ∘ Ψ ′ = Ψ \iota\circ\Psi'=\Psi ι ∘ Ψ ′ = Ψ (3) is commutative for Ψ ~ = ι ∘ Ψ ′ \widetilde{\Psi}=\iota\circ\Psi' Ψ = ι ∘ Ψ ′ ζ = ε P ∘ ι ∘ Ψ ′ \zeta=\varepsilon_P\circ\iota\circ\Psi' ζ = ε P ∘ ι ∘ Ψ ′
Hence,
Ψ ⏟ = ι ∘ Ψ ′ ( H ) = ( ι ∘ Ψ ′ ) ( H ) = ι ( Ψ ′ ( H ) ⏟ ⊆ Λ k ) ⊆ ι ( Λ k ) = Λ k . \underbrace{\Psi}_{=\iota\circ\Psi'}(H)=(\iota\circ\Psi')(H)=\iota\left(\underbrace{\Psi'(H)}_{\subseteq\Lambda_k}\right)\subseteq\iota(\Lambda_k)=\Lambda_k. = ι ∘ Ψ ′ Ψ ( H ) = ( ι ∘ Ψ ′ ) ( H ) = ι ⊆ Λ k Ψ ′ ( H ) ⊆ ι ( Λ k ) = Λ k . This proves Theorem 2.1 (e).
Extension of scalars and ( k , A ) (k,A) ( k , A )
Various applications of Theorem 2.1 can be found in [2 ] and [GriRei14, Chapter 7]. We are going to present another application, which we will obtain by “leveraging” Theorem 2.1 through an extension-of-scalars argument6
Definition 3.1. Let H H H k \mathbf{k} k H ‾ \underline{H} H k \mathbf{k} k H H H k \mathbf{k} k H H H H H H k \mathbf{k} k H H H H ‾ \underline{H} H H H H k \mathbf{k} k H H H k \mathbf{k} k
In other words, H ‾ \underline{H} H H H H H H H k \mathbf{k} k H ‾ \underline{H} H H H H H H H k \mathbf{k} k H H H
Definition 3.2. Let A A A k \mathbf{k} k
(a) If H H H k \mathbf{k} k A ⊗ H A \otimes H A ⊗ H A A A A ⊗ H A \otimes H A ⊗ H A A A
a ( b ⊗ h ) = a b ⊗ h for all a ∈ A , b ∈ A and h ∈ H . a(b \otimes h)=ab \otimes h \qquad\text{for all } a \in A, b \in A \text{ and } h \in H. a ( b ⊗ h ) = ab ⊗ h for all a ∈ A , b ∈ A and h ∈ H . This A A A A ⊗ H A \otimes H A ⊗ H k \mathbf{k} k H H H A A A
We can define a functor Mod k → Mod A \operatorname{Mod}_{\mathbf{k}} \to\operatorname{Mod}_{A} Mod k → Mod A Mod B \operatorname{Mod}_{B} Mod B B B B H ∈ Mod k H \in\operatorname{Mod}_{\mathbf{k}} H ∈ Mod k A ⊗ H A \otimes H A ⊗ H f ∈ Mod k ( H 1 , H 2 ) f \in\operatorname{Mod}_{\mathbf{k}}(H_1,H_2) f ∈ Mod k ( H 1 , H 2 ) id ⊗ f ∈ Mod A ( A ⊗ H 1 , A ⊗ H 2 ) \operatorname{id} \otimes f \in\operatorname{Mod}_{A}(A \otimes H_1,A \otimes H_2) id ⊗ f ∈ Mod A ( A ⊗ H 1 , A ⊗ H 2 ) extension of scalars (from k \mathbf{k} k A A A
(b) If H H H k \mathbf{k} k A A A A ⊗ H A \otimes H A ⊗ H A A A n n n A ⊗ H n A \otimes H_n A ⊗ H n H n H_n H n n n n H H H A A A A ⊗ H A \otimes H A ⊗ H A ⊗ H A \otimes H A ⊗ H A ‾ ⊗ H \underline{A} \otimes H A ⊗ H
As before, we can define a functor from the category of graded k \mathbf{k} k A A A H H H A ⊗ H A \otimes H A ⊗ H extension of scalars .
(c) If H H H k \mathbf{k} k A A A A ⊗ H A \otimes H A ⊗ H A A A
( a ⊗ h ) ( b ⊗ g ) = a b ⊗ h g for all a ∈ A , b ∈ A , h ∈ H and g ∈ H . (a \otimes h)(b \otimes g)=ab \otimes hg \qquad\text{for all } a \in A, b \in A, h \in H \text{ and } g \in H. ( a ⊗ h ) ( b ⊗ g ) = ab ⊗ h g for all a ∈ A , b ∈ A , h ∈ H and g ∈ H . (This is, of course, the same rule as used in the standard definition of the tensor product A ⊗ H A \otimes H A ⊗ H A ⊗ H A \otimes H A ⊗ H A A A k \mathbf{k} k A A A A ⊗ H A \otimes H A ⊗ H k \mathbf{k} k H H H scalars extended to A A A .
As before, we can define a functor from the category of k \mathbf{k} k A A A H H H A ⊗ H A \otimes H A ⊗ H extension of scalars .
(d) If H H H k \mathbf{k} k A A A A ⊗ H A \otimes H A ⊗ H A A A
id A ⊗ Δ H : A ⊗ H ⟶ A ⊗ ( H ⊗ H ) ≅ ( A ⊗ H ) ⊗ A ( A ⊗ H ) \operatorname{id}_{A} \otimes\Delta_{H}: A \otimes H \longrightarrow A \otimes(H \otimes H) \cong(A \otimes H) \otimes_{A} (A \otimes H) id A ⊗ Δ H : A ⊗ H ⟶ A ⊗ ( H ⊗ H ) ≅ ( A ⊗ H ) ⊗ A ( A ⊗ H ) and its counit is defined to be
id A ⊗ ε H : A ⊗ H ⟶ A ⊗ k ≅ A \operatorname{id}_{A} \otimes\varepsilon_{H}: A \otimes H \longrightarrow A \otimes\mathbf{k} \cong A id A ⊗ ε H : A ⊗ H ⟶ A ⊗ k ≅ A (recalling that Δ H \Delta_{H} Δ H ε H \varepsilon_{H} ε H H H H A ⊗ H A \otimes H A ⊗ H A A A A A A A ⊗ H A \otimes H A ⊗ H A A A A ⊗ H A \otimes H A ⊗ H k \mathbf{k} k H H H scalars extended to A A A .
As before, we can define a functor from the category of k \mathbf{k} k A A A H H H A ⊗ H A \otimes H A ⊗ H extension of scalars .
Notice that A ⊗ H A \otimes H A ⊗ H A A A k \mathbf{k} k A A A k \mathbf{k} k A A A A ⊗ H A \otimes H A ⊗ H k \mathbf{k} k A ⊗ H A \otimes H A ⊗ H k \mathbf{k} k A A A H H H
(e) If H H H k \mathbf{k} k A A A A ⊗ H A \otimes H A ⊗ H A A A A A A A A A A ⊗ H A \otimes H A ⊗ H A A A A A A A ⊗ H A \otimes H A ⊗ H k \mathbf{k} k H H H scalars extended to A A A .
As before, we can define a functor from the category of k \mathbf{k} k A A A H H H A ⊗ H A \otimes H A ⊗ H extension of scalars .
(f) Similarly, extension of scalars is defined for k \mathbf{k} k k \mathbf{k} k A A A k \mathbf{k} k
Definition 3.3. Let A A A k \mathbf{k} k
(a) Let H H H k \mathbf{k} k G G G A A A k \mathbf{k} k f : H → G f:H \to G f : H → G f ♯ f^{\sharp} f ♯ A A A
A ⊗ H ⟶ G , a ⊗ h ↦ a f ( h ) . A \otimes H \longrightarrow G,\qquad a \otimes h \mapsto af(h). A ⊗ H ⟶ G , a ⊗ h ↦ a f ( h ) . (It is easy to see that this latter map is indeed well-defined and A A A
any A A A g : A ⊗ H → G g : A \otimes H \to G g : A ⊗ H → G g ♭ g^{\flat} g ♭ k k k H → G H \to G H → G h ↦ g ( 1 ⊗ h ) h \mapsto g(1 \otimes h) h ↦ g ( 1 ⊗ h )
Sometimes we will use the notations f ♯ ( A , k ) f^{\sharp(A,k)} f ♯ ( A , k ) g ♭ ( A , k ) g^{\flat(A,k)} g ♭ ( A , k ) f ♯ f^\sharp f ♯ g ♭ g^\flat g ♭ A A A k k k ( f ♯ ) ♭ = f (f^\sharp)^\flat= f ( f ♯ ) ♭ = f k k k f : H → G f : H \to G f : H → G ( g ♭ ) ♯ = g (g^\flat)^\sharp= g ( g ♭ ) ♯ = g A A A g : A ⊗ H → G g : A \otimes H \to G g : A ⊗ H → G
{ k -linear maps H → G } → { A -linear maps A ⊗ H → G } , f ↦ f ♯ (4) \{k\text{-linear maps }H \to G\} \to\{A\text{-linear maps }A \otimes H \to G\}, \qquad f \mapsto f^\sharp \tag*{(4)} { k -linear maps H → G } → { A -linear maps A ⊗ H → G } , f ↦ f ♯ (4) and
{ A -linear maps A ⊗ H → G } → { k -linear maps H → G } , g ↦ g ♭ (5) \{A\text{-linear maps }A \otimes H \to G\} \to\{k\text{-linear maps }H \to G\}, \qquad g \mapsto g^\flat \tag*{(5)} { A -linear maps A ⊗ H → G } → { k -linear maps H → G } , g ↦ g ♭ (5) are mutually inverse.
This is a particular case of an adjunction between functors (namely, the Hom-tensor adjunction, with a slight simplification, also known as the induction-restriction adjunction); this is also the reason why we are using the ♯ \sharp ♯ ♭ \flat ♭ H H H G G G
(b) Let H H H k k k G G G A A A k k k f : H → G f : H \to G f : H → G ( k , A ) (k,A) ( k , A ) A A A f ♯ : A ⊗ H → G f^\sharp: A \otimes H \to G f ♯ : A ⊗ H → G A A A
Proposition 3.4. Let A A A k k k H H H k k k G G G A A A f : H → G f : H \to G f : H → G k k k f f f k k k f ♯ f^\sharp f ♯ A A A
Proof of Proposition 3.4. Straightforward and left to the reader. (The main step is to observe that f ♯ f^\sharp f ♯ A A A
We have f ♯ ( 1 ⊗ 1 ) = 1 f^\sharp(1 \otimes1) = 1 f ♯ ( 1 ⊗ 1 ) = 1
Every a , b ∈ A a,b \in A a , b ∈ A h , g ∈ H h,g \in H h , g ∈ H f ♯ ( ( a ⊗ h ) ( b ⊗ g ) ) = f ♯ ( a ⊗ h ) f ♯ ( b ⊗ g ) f^\sharp((a \otimes h)(b \otimes g)) = f^\sharp(a \otimes h)f^\sharp(b \otimes g) f ♯ (( a ⊗ h ) ( b ⊗ g )) = f ♯ ( a ⊗ h ) f ♯ ( b ⊗ g )
This is because the tensor product A ⊗ H A \otimes H A ⊗ H
Proposition 3.5. Let A A A k k k H H H k k k G G G A A A f : H → G f : H \to G f : H → G k k k k k k f f f k k k f ♯ f^\sharp f ♯
Proof of Proposition 3.5. Again, straightforward and therefore omitted.
Let us first prove some easily-checked properties of ( k , A ) (\mathbf{k},A) ( k , A )
Proposition 3.6. Let k \mathbf{k} k A A A k \mathbf{k} k H H H k \mathbf{k} k G G G I I I A A A f : H → G f:H\to G f : H → G ( k , A ) (\mathbf{k},A) ( k , A ) g : G → I g:G\to I g : G → I A A A g ∘ f g\circ f g ∘ f ( k , A ) (\mathbf{k},A) ( k , A )
Proof of Proposition 3.6. Since f f f ( k , A ) (\mathbf{k},A) ( k , A ) f ♯ : A ⊗ H → G f^\sharp:A\otimes H\to G f ♯ : A ⊗ H → G A A A f f f k \mathbf{k} k g g g A A A
( g ∘ f ) ♯ = g ∘ f ♯ . (6) (g\circ f)^\sharp=g\circ f^\sharp. \tag*{(6)} ( g ∘ f ) ♯ = g ∘ f ♯ . (6) Thus, ( g ∘ f ) ♯ (g\circ f)^\sharp ( g ∘ f ) ♯ A A A g g g f ♯ f^\sharp f ♯ A A A g ∘ f g\circ f g ∘ f ( k , A ) (\mathbf{k},A) ( k , A )
Proposition 3.7. Let k \mathbf{k} k A A A k \mathbf{k} k F F F H H H k \mathbf{k} k G G G A A A f : H → G f:H\to G f : H → G ( k , A ) (\mathbf{k},A) ( k , A ) e : F → H e:F\to H e : F → H k \mathbf{k} k f ∘ e f\circ e f ∘ e ( k , A ) (\mathbf{k},A) ( k , A )
Proof of Proposition 3.7. Since f f f ( k , A ) (\mathbf{k},A) ( k , A ) f ♯ : A ⊗ H → G f^\sharp:A\otimes H\to G f ♯ : A ⊗ H → G A A A id A ⊗ e : A ⊗ F → A ⊗ H \operatorname{id}_A\otimes e:A\otimes F\to A\otimes H id A ⊗ e : A ⊗ F → A ⊗ H A A A e : F → H e:F\to H e : F → H k \mathbf{k} k ( f ∘ e ) ♯ = f ♯ ∘ ( id A ⊗ e ) (f\circ e)^\sharp=f^\sharp\circ(\operatorname{id}_A\otimes e) ( f ∘ e ) ♯ = f ♯ ∘ ( id A ⊗ e ) ( f ∘ e ) ♯ (f\circ e)^\sharp ( f ∘ e ) ♯ A A A f ♯ f^\sharp f ♯ id A ⊗ e \operatorname{id}_A\otimes e id A ⊗ e A A A f ∘ e f\circ e f ∘ e ( k , A ) (\mathbf{k},A) ( k , A )
Proposition 3.8. Let k \mathbf{k} k A A A k \mathbf{k} k H H H k \mathbf{k} k G G G A A A B B B A A A p : A → B p:A\to B p : A → B A A A p p p A A A B B B p G : G → B ⊗ A G p_G:G\to B\otimes_A G p G : G → B ⊗ A G A A A
G → ≅ A ⊗ A G → p ⊗ id B ⊗ A G . G\xrightarrow{\cong}A\otimes_A G\xrightarrow{p\otimes\operatorname{id}}B\otimes_A G. G ≅ A ⊗ A G p ⊗ id B ⊗ A G . Let f : H → G f:H\to G f : H → G ( k , A ) (\mathbf{k},A) ( k , A ) p G ∘ f : H → B ⊗ A G p_G\circ f:H\to B\otimes_A G p G ∘ f : H → B ⊗ A G ( k , B ) (\mathbf{k},B) ( k , B )
Proof of Proposition 3.8. Since f f f ( k , A ) (\mathbf{k}, A) ( k , A ) f # = f # ( A , k ) : A ⊗ H → G f^{\#}=f^{\#(A,\mathbf{k})}: A\otimes H\to G f # = f # ( A , k ) : A ⊗ H → G A A A id B ⊗ A f # : B ⊗ A ( A ⊗ H ) → B ⊗ A G \operatorname{id}_{B}\otimes_{A}f^{\#}: B\otimes_{A}(A\otimes H)\to B\otimes_{A}G id B ⊗ A f # : B ⊗ A ( A ⊗ H ) → B ⊗ A G B B B
Let κ : B ⊗ H → B ⊗ A ( A ⊗ H ) \kappa:B\otimes H\to B\otimes_{A}(A\otimes H) κ : B ⊗ H → B ⊗ A ( A ⊗ H ) B B B b ⊗ h ∈ B ⊗ H b\otimes h\in B\otimes H b ⊗ h ∈ B ⊗ H b ⊗ A ( 1 ⊗ h b\otimes_{A}(1\otimes h b ⊗ A ( 1 ⊗ h κ \kappa κ B B B 7 ( id B ⊗ A f # ) ∘ κ (\operatorname{id}_{B}\otimes_{A}f^{\#})\circ\kappa ( id B ⊗ A f # ) ∘ κ B B B id B ⊗ A f # \operatorname{id}_{B}\otimes_{A}f^{\#} id B ⊗ A f # κ \kappa κ B B B
The definition of p G p_G p G
p G ( u ) = 1 ⊗ A u (7) p_G(u)=1\otimes_A u \tag*{(7)} p G ( u ) = 1 ⊗ A u (7) for every u ∈ G u\in G u ∈ G
The map p G ∘ f : H → B ⊗ A G p_G\circ f:H\to B\otimes_A G p G ∘ f : H → B ⊗ A G ( p G ∘ f ) # ( B , k ) : B ⊗ H → B ⊗ A G (p_G\circ f)^{\#(B,\mathbf{k})}:B\otimes H\to B\otimes_A G ( p G ∘ f ) # ( B , k ) : B ⊗ H → B ⊗ A G ( p G ∘ f ) # ( B , k ) = ( id B ⊗ A f # ) ∘ κ (p_G\circ f)^{\#(B,\mathbf{k})}=(\operatorname{id}_{B}\otimes_A f^{\#})\circ\kappa ( p G ∘ f ) # ( B , k ) = ( id B ⊗ A f # ) ∘ κ ( p G ∘ f ) # ( B , k ) (p_G\circ f)^{\#(B,\mathbf{k})} ( p G ∘ f ) # ( B , k ) b ⊗ h ∈ B ⊗ H b\otimes h\in B\otimes H b ⊗ h ∈ B ⊗ H b ( p G ∘ f ) ( h ) = b p G ( f ( h ) ) = b ( 1 ⊗ A f ( h ) ) = b ⊗ A f ( h ) b(p_G\circ f)(h)=b\,p_G(f(h))=b(1\otimes_A f(h))=b\otimes_A f(h) b ( p G ∘ f ) ( h ) = b p G ( f ( h )) = b ( 1 ⊗ A f ( h )) = b ⊗ A f ( h ) ( id B ⊗ A f # ) ∘ κ (\operatorname{id}_{B}\otimes_A f^{\#})\circ\kappa ( id B ⊗ A f # ) ∘ κ b ⊗ h ∈ B ⊗ H b\otimes h\in B\otimes H b ⊗ h ∈ B ⊗ H
( ( id B ⊗ A f # ) ∘ κ ) ( b ⊗ h ) = ( id B ⊗ A f # ) ( b ⊗ A ( 1 ⊗ h ) ) = b ⊗ A f # ( 1 ⊗ h ) = b ⊗ A f ( h ) ((\operatorname{id}_{B}\otimes_A f^{\#})\circ\kappa)(b\otimes h)=(\operatorname{id}_{B}\otimes_A f^{\#})(b\otimes_A(1\otimes h))=b\otimes_A f^{\#}(1\otimes h)=b\otimes_A f(h) (( id B ⊗ A f # ) ∘ κ ) ( b ⊗ h ) = ( id B ⊗ A f # ) ( b ⊗ A ( 1 ⊗ h )) = b ⊗ A f # ( 1 ⊗ h ) = b ⊗ A f ( h ) as well). Thus, ( p G ∘ f ) # ( B , k ) (p_G\circ f)^{\#(B,\mathbf{k})} ( p G ∘ f ) # ( B , k ) B B B ( id B ⊗ A f # ) ∘ κ (\operatorname{id}_{B}\otimes_A f^{\#})\circ\kappa ( id B ⊗ A f # ) ∘ κ B B B p G ∘ f p_G\circ f p G ∘ f ( k , B ) (\mathbf{k}, B) ( k , B )
Proposition 3.9. Let k \mathbf{k} k A A A B B B k \mathbf{k} k H H H G G G k \mathbf{k} k f : H → A ⊗ G f:H\to A\otimes G f : H → A ⊗ G ( k , A ) (\mathbf{k}, A) ( k , A ) p : A → B p:A\to B p : A → B k \mathbf{k} k ( p ⊗ id ) ∘ f : H → B ⊗ G (p\otimes\operatorname{id})\circ f:H\to B\otimes G ( p ⊗ id ) ∘ f : H → B ⊗ G ( k , B ) (\mathbf{k}, B) ( k , B )
Proof of Proposition 3.9. Consider B B B A A A k \mathbf{k} k p : A → B p:A\to B p : A → B p p p A A A A → B A\to B A → B A ⊗ G A\otimes G A ⊗ G A A A p A ⊗ G : A ⊗ G → B ⊗ A ( A ⊗ G ) p_{A\otimes G}:A\otimes G\to B\otimes_A(A\otimes G) p A ⊗ G : A ⊗ G → B ⊗ A ( A ⊗ G ) A A A
A ⊗ G → ≅ A ⊗ A ( A ⊗ G ) → p ⊗ A id B ⊗ A ( A ⊗ G ) . A\otimes G \xrightarrow{\cong} A\otimes_A(A\otimes G) \xrightarrow{p\otimes_A\operatorname{id}} B\otimes_A(A\otimes G). A ⊗ G ≅ A ⊗ A ( A ⊗ G ) p ⊗ A id B ⊗ A ( A ⊗ G ) . Proposition 3.8 (applied to A ‾ ⊗ G \underline{A} \otimes G A ⊗ G p A ‾ ⊗ G p_{\underline{A}\otimes G} p A ⊗ G G G G p G p_G p G p A ‾ ⊗ G ∘ f : H → B ‾ ⊗ A ‾ ( A ‾ ⊗ G ) p_{\underline{A}\otimes G} \circ f:H \to\underline{B} \otimes_{\underline{A}} (\underline{A} \otimes G) p A ⊗ G ∘ f : H → B ⊗ A ( A ⊗ G ) ( k , B ‾ ) (\mathbf{k},\underline{B}) ( k , B )
Now, let ϕ \phi ϕ B B B
B ‾ ⊗ A ‾ ( A ‾ ⊗ G ) → ( B ‾ ⊗ A ‾ A ‾ ) ⊗ G → B ‾ ⊗ G . \underline{B} \otimes_{\underline{A}} (\underline{A} \otimes G) \to(\underline{B} \otimes_{\underline{A}} \underline{A}) \otimes G \to\underline{B} \otimes G. B ⊗ A ( A ⊗ G ) → ( B ⊗ A A ) ⊗ G → B ⊗ G . Then, ϕ \phi ϕ B B B p ⊗ id = ϕ ∘ p A ‾ ⊗ G p \otimes\operatorname{id} = \phi\circ p_{\underline{A}\otimes G} p ⊗ id = ϕ ∘ p A ⊗ G A ‾ ⊗ G → B ‾ ⊗ G \underline{A} \otimes G \to\underline{B} \otimes G A ⊗ G → B ⊗ G
( p ⊗ id ) ∘ f = ϕ ∘ p A ‾ ⊗ G ∘ f = ϕ ∘ ( p A ‾ ⊗ G ∘ f ) (p \otimes\operatorname{id}) \circ f = \phi\circ p_{\underline{A}\otimes G} \circ f = \phi\circ(p_{\underline{A}\otimes G} \circ f) ( p ⊗ id ) ∘ f = ϕ ∘ p A ⊗ G ∘ f = ϕ ∘ ( p A ⊗ G ∘ f ) must be a ( k , B ‾ ) (\mathbf{k},\underline{B}) ( k , B ) p A ‾ ⊗ G ∘ f p_{\underline{A}\otimes G} \circ f p A ⊗ G ∘ f ( k , B ‾ ) (\mathbf{k},\underline{B}) ( k , B ) ϕ \phi ϕ B B B
Proposition 3.10. Let k \mathbf{k} k A A A B B B k \mathbf{k} k H H H k \mathbf{k} k G G G A A A f : H → G f:H \to G f : H → G ( k , A ) (\mathbf{k},A) ( k , A ) id ⊗ f : B ⊗ H → B ⊗ G \operatorname{id} \otimes f:B \otimes H \to B \otimes G id ⊗ f : B ⊗ H → B ⊗ G ( B , B ⊗ A ) (B,B \otimes A) ( B , B ⊗ A )
Proof of Proposition 3.10. Since f f f ( k , A ) (\mathbf{k},A) ( k , A ) f ♯ = f ♯ ( A , k ) : A ⊗ H → G f^{\sharp}=f^{\sharp(A,\mathbf{k})}:A \otimes H \to G f ♯ = f ♯ ( A , k ) : A ⊗ H → G A A A id B ⊗ f ♯ : B ⊗ ( A ⊗ H ) → B ⊗ G \operatorname{id}_B \otimes f^{\sharp}:B \otimes(A \otimes H) \to B \otimes G id B ⊗ f ♯ : B ⊗ ( A ⊗ H ) → B ⊗ G B ⊗ A B \otimes A B ⊗ A
But the B B B id ⊗ f : B ⊗ H → B ⊗ G \operatorname{id} \otimes f:B \otimes H \to B \otimes G id ⊗ f : B ⊗ H → B ⊗ G B ⊗ A B \otimes A B ⊗ A
( id ⊗ f ) ♯ ( B ⊗ A , B ) : ( B ⊗ A ) ⊗ B ( B ⊗ H ) → B ⊗ G . (\operatorname{id} \otimes f)^{\sharp(B\otimes A,B)}:(B \otimes A) \otimes_B (B \otimes H) \to B \otimes G. ( id ⊗ f ) ♯ ( B ⊗ A , B ) : ( B ⊗ A ) ⊗ B ( B ⊗ H ) → B ⊗ G . Now, let γ \gamma γ B ⊗ A B \otimes A B ⊗ A ( B ⊗ A ) ⊗ B ( B ⊗ H ) → B ⊗ ( A ⊗ H ) (B \otimes A) \otimes_B (B \otimes H) \to B \otimes(A \otimes H) ( B ⊗ A ) ⊗ B ( B ⊗ H ) → B ⊗ ( A ⊗ H ) ( b ⊗ a ) ⊗ B ( b ′ ⊗ h ) ∈ ( B ⊗ A ) ⊗ B ( B ⊗ H ) (b \otimes a) \otimes_B (b' \otimes h) \in(B \otimes A) \otimes_B (B \otimes H) ( b ⊗ a ) ⊗ B ( b ′ ⊗ h ) ∈ ( B ⊗ A ) ⊗ B ( B ⊗ H ) b b ′ ⊗ ( a ⊗ h ) bb' \otimes(a \otimes h) b b ′ ⊗ ( a ⊗ h ) γ \gamma γ B ⊗ A B \otimes A B ⊗ A ( id B ⊗ f ♯ ) ∘ γ (\operatorname{id}_B \otimes f^{\sharp}) \circ\gamma ( id B ⊗ f ♯ ) ∘ γ B ⊗ A B \otimes A B ⊗ A id B ⊗ f ♯ \operatorname{id}_B \otimes f^{\sharp} id B ⊗ f ♯ γ \gamma γ B ⊗ A B \otimes A B ⊗ A
Now, it is straightforward to see that ( id ⊗ f ) ♯ ( B ⊗ A , B ) = ( id B ⊗ f ♯ ) ∘ γ (\operatorname{id} \otimes f)^{\sharp(B\otimes A,B)}=(\operatorname{id}_B \otimes f^{\sharp}) \circ\gamma ( id ⊗ f ) ♯ ( B ⊗ A , B ) = ( id B ⊗ f ♯ ) ∘ γ 8 ( id ⊗ f ) ♯ ( B ⊗ A , B ) (\operatorname{id} \otimes f)^{\sharp(B\otimes A,B)} ( id ⊗ f ) ♯ ( B ⊗ A , B ) B ⊗ A B \otimes A B ⊗ A ( id B ⊗ f ♯ ) ∘ γ (\operatorname{id}_B \otimes f^{\sharp}) \circ\gamma ( id B ⊗ f ♯ ) ∘ γ B ⊗ A B \otimes A B ⊗ A id ⊗ f : B ⊗ H → B ⊗ G \operatorname{id} \otimes f:B \otimes H \to B \otimes G id ⊗ f : B ⊗ H → B ⊗ G ( B , B ⊗ A ) (B,B \otimes A) ( B , B ⊗ A )
( id ⊗ f ) ♯ ( B ⊗ A , B ) ( ( b ⊗ a ) ⊗ B ( b ′ ⊗ h ) ) = ( ( id B ⊗ f ♯ ) ∘ γ ) ( ( b ⊗ a ) ⊗ B ( b ′ ⊗ h ) ) (\operatorname{id} \otimes f)^{\sharp(B\otimes A,B)}\left((b \otimes a) \otimes_B (b' \otimes h)\right)=\left((\operatorname{id}_B \otimes f^{\sharp}) \circ\gamma\right)\left((b \otimes a) \otimes_B (b' \otimes h)\right) ( id ⊗ f ) ♯ ( B ⊗ A , B ) ( ( b ⊗ a ) ⊗ B ( b ′ ⊗ h ) ) = ( ( id B ⊗ f ♯ ) ∘ γ ) ( ( b ⊗ a ) ⊗ B ( b ′ ⊗ h ) ) for each b ∈ B b \in B b ∈ B a ∈ A a \in A a ∈ A b ′ ∈ B b' \in B b ′ ∈ B h ∈ H h \in H h ∈ H b b ′ ⊗ a f ( h ) bb' \otimes af(h) b b ′ ⊗ a f ( h )
Proposition 3.11. Let k k k A A A k k k B B B A A A H H H k k k G G G A A A I I I B B B f : H → G f:H\to G f : H → G ( k , A ) (k,A) ( k , A ) g : G → I g:G\to I g : G → I ( A , B ) (A,B) ( A , B ) g ∘ f : H → I g\circ f:H\to I g ∘ f : H → I ( k , B ) (k,B) ( k , B )
Proof of Proposition 3.11. Since f f f ( k , A ) (k,A) ( k , A ) f ♯ ( A , k ) : A ⊗ H → G f^{\sharp(A,k)}:A\otimes H\to G f ♯ ( A , k ) : A ⊗ H → G A A A id B ⊗ A f ♯ ( A , k ) : B ⊗ A ( A ⊗ H ) → B ⊗ A G \operatorname{id}_B\otimes_A f^{\sharp(A,k)}:B\otimes_A(A\otimes H)\to B\otimes_A G id B ⊗ A f ♯ ( A , k ) : B ⊗ A ( A ⊗ H ) → B ⊗ A G B B B
Since g : G → I g:G\to I g : G → I ( A , B ) (A,B) ( A , B ) g ♯ ( B , A ) : B ⊗ A G → I g^{\sharp(B,A)}:B\otimes_A G\to I g ♯ ( B , A ) : B ⊗ A G → I B B B
Let δ : B ⊗ H → B ⊗ A ( A ⊗ H ) \delta:B\otimes H\to B\otimes_A(A\otimes H) δ : B ⊗ H → B ⊗ A ( A ⊗ H ) B B B b ⊗ h b\otimes h b ⊗ h b ⊗ A ( 1 ⊗ h b\otimes_A(1\otimes h b ⊗ A ( 1 ⊗ h δ \delta δ B B B
Straightforward elementwise computation shows that ( g ∘ f ) ♯ ( B , k ) = g ♯ ( B , A ) ∘ ( id B ⊗ A f ♯ ( A , k ) ) ∘ δ (g\circ f)^{\sharp(B,k)}=g^{\sharp(B,A)}\circ(\operatorname{id}_B\otimes_A f^{\sharp(A,k)})\circ\delta ( g ∘ f ) ♯ ( B , k ) = g ♯ ( B , A ) ∘ ( id B ⊗ A f ♯ ( A , k ) ) ∘ δ ( g ∘ f ) ♯ ( B , k ) (g\circ f)^{\sharp(B,k)} ( g ∘ f ) ♯ ( B , k ) B B B g ♯ ( B , A ) g^{\sharp(B,A)} g ♯ ( B , A ) id B ⊗ A f ♯ ( A , k ) \operatorname{id}_B\otimes_A f^{\sharp(A,k)} id B ⊗ A f ♯ ( A , k ) δ \delta δ B B B g ∘ f : H → I g\circ f:H\to I g ∘ f : H → I ( k , B ) (k,B) ( k , B ) □ \square □
With these basics in place, we can now “escalate” Theorem 2.1 to the following setting:
Corollary 3.12. Let k k k H H H k k k A A A k k k ξ : H → A \xi:H\to A ξ : H → A k k k
(a) Then, there exists a unique graded ( k , A ) (k,A) ( k , A ) Ξ : H → A ⊗ QSym k \Xi:H\to A\otimes\operatorname{QSym}_k Ξ : H → A ⊗ QSym k
H → Ξ A ⊗ QSym k ↘ ξ ↙ id A ⊗ ε P A (8) \begin{aligned}
H&\xrightarrow{\ \Xi\ } A\otimes\operatorname{QSym}_k\\
&\searrow_{\xi}\qquad\quad\swarrow_{\operatorname{id}_A\otimes\varepsilon_P}\\
&\qquad A \tag*{(8)}
\end{aligned} H Ξ A ⊗ QSym k ↘ ξ ↙ id A ⊗ ε P A (8) is commutative (where we regard id A ⊗ ε P : A ⊗ QSym k → A ⊗ k \operatorname{id}_A\otimes\varepsilon_P:A\otimes\operatorname{QSym}_k\to A\otimes k id A ⊗ ε P : A ⊗ QSym k → A ⊗ k A ⊗ QSym k A\otimes\operatorname{QSym}_k A ⊗ QSym k A A A A ⊗ k A\otimes k A ⊗ k A A A
(b) This unique ( k , A ) (k,A) ( k , A ) Ξ : H → A ⊗ QSym k \Xi:H\to A\otimes\operatorname{QSym}_k Ξ : H → A ⊗ QSym k k k k
(c) For every composition α = ( a 1 , a 2 , … , a k ) \alpha=(a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k ) k k k ξ α : H → A \xi_\alpha:H\to A ξ α : H → A k k k
H → Δ ( k − 1 ) H ⊗ k → π α H ⊗ k → ξ ⊗ k A ⊗ k → m ( k − 1 ) A . H\xrightarrow{\Delta^{(k-1)}}H^{\otimes k}\xrightarrow{\pi_\alpha}H^{\otimes k}\xrightarrow{\xi^{\otimes k}}A^{\otimes k}\xrightarrow{m^{(k-1)}}A. H Δ ( k − 1 ) H ⊗ k π α H ⊗ k ξ ⊗ k A ⊗ k m ( k − 1 ) A . (Recall that Δ ( k − 1 ) : H → H ⊗ k \Delta^{(k-1)}:H\to H^{\otimes k} Δ ( k − 1 ) : H → H ⊗ k m ( k − 1 ) : A ⊗ k → A m^{(k-1)}:A^{\otimes k}\to A m ( k − 1 ) : A ⊗ k → A π α : H ⊗ k → H ⊗ k \pi_\alpha:H^{\otimes k}\to H^{\otimes k} π α : H ⊗ k → H ⊗ k Definition 1.3 .)
Then, the unique ( k , A ) (\mathbf{k}, A) ( k , A ) Ξ \Xi Ξ Corollary 3.12 (a) is given by
Ξ ( h ) = ∑ α ∈ Comp ξ α ( h ) ⊗ M α for every h ∈ H \Xi(h)=\sum_{\alpha\in\operatorname{Comp}}\xi_{\alpha}(h)\otimes M_{\alpha}\qquad\text{for every }h\in H Ξ ( h ) = α ∈ Comp ∑ ξ α ( h ) ⊗ M α for every h ∈ H (in particular, the sum on the right hand side of this equality has only finitely many nonzero addends).
(d) If the k \mathbf{k} k H H H Ξ ( H ) \Xi(H) Ξ ( H ) A ⊗ Λ k A\otimes\Lambda_{\mathbf{k}} A ⊗ Λ k A ⊗ QSym k A\otimes\operatorname{QSym}_{\mathbf{k}} A ⊗ QSym k Λ k \Lambda_{\mathbf{k}} Λ k k \mathbf{k} k k \mathbf{k} k
Proof of Corollary 3.12. We have A ⊗ QSym k ≅ QSym A A\otimes\operatorname{QSym}_{\mathbf{k}}\cong\operatorname{QSym}_{A} A ⊗ QSym k ≅ QSym A A A A QSym k \operatorname{QSym}_{\mathbf{k}} QSym k k \mathbf{k} k k \mathbf{k} k
Recall that we have defined a k \mathbf{k} k ε P : QSym k → k \varepsilon_{P}:\operatorname{QSym}_{\mathbf{k}}\to\mathbf{k} ε P : QSym k → k ε P \varepsilon_{P} ε P ε P , k \varepsilon_{P,\mathbf{k}} ε P , k k \mathbf{k} k m \mathbf{m} m ε P , m : QSym m → m \varepsilon_{P,\mathbf{m}}:\operatorname{QSym}_{\mathbf{m}}\to\mathbf{m} ε P , m : QSym m → m m \mathbf{m} m A A A ε P , A : QSym A → A \varepsilon_{P,A}:\operatorname{QSym}_{A}\to A ε P , A : QSym A → A ε P , m \varepsilon_{P,\mathbf{m}} ε P , m m \mathbf{m} m ε P , A : QSym A → A \varepsilon_{P,A}:\operatorname{QSym}_{A}\to A ε P , A : QSym A → A id A ⊗ ε P , k : A ⊗ QSym k → A ⊗ k \operatorname{id}_{A}\otimes\varepsilon_{P,\mathbf{k}}:A\otimes\operatorname{QSym}_{\mathbf{k}}\to A\otimes\mathbf{k} id A ⊗ ε P , k : A ⊗ QSym k → A ⊗ k A ⊗ QSym k A\otimes\operatorname{QSym}_{\mathbf{k}} A ⊗ QSym k QSym A \operatorname{QSym}_{A} QSym A A ⊗ k A\otimes\mathbf{k} A ⊗ k A A A
The k \mathbf{k} k ξ : H → A \xi:H\to A ξ : H → A A A A ξ # : A ⊗ H → A \xi^{\#}:A\otimes H\to A ξ # : A ⊗ H → A ξ # ( a ⊗ h ) = a ξ ( h ) \xi^{\#}(a\otimes h)=a\xi(h) ξ # ( a ⊗ h ) = a ξ ( h ) a ∈ A a\in A a ∈ A h ∈ H h\in H h ∈ H G = A G=A G = A f = ξ f=\xi f = ξ ξ # \xi^{\#} ξ # A A A ξ \xi ξ k \mathbf{k} k
Theorem 2.1 (a) (applied to A A A A ⊗ H A\otimes H A ⊗ H ξ # \xi^{\#} ξ # k \mathbf{k} k H H H ζ \zeta ζ A A A Ψ : A ⊗ H → QSym A \Psi:A\otimes H\to\operatorname{QSym}_{A} Ψ : A ⊗ H → QSym A
A ⊗ H → Ψ QSym A ξ # ↓ ↓ ε P , A A = A (9) \begin{CD}
A\otimes H @>{\Psi}>> \operatorname{QSym}_{A}\\
@V{\xi^{\#}}VV @VV{\varepsilon_{P,A}}V\\
A @= A \tag*{(9)}
\end{CD} A ⊗ H ξ # ↓ ⏐ A Ψ QSym A ↓ ⏐ ε P , A A (9) is commutative. Since we are identifying the map ε P , A : QSym A → A \varepsilon_{P,A}:\operatorname{QSym}_{A}\to A ε P , A : QSym A → A id A ⊗ ε P , k : A ⊗ QSym k → A ⊗ k = A \operatorname{id}_{A}\otimes\varepsilon_{P,\mathbf{k}}:A\otimes\operatorname{QSym}_{\mathbf{k}}\to A\otimes\mathbf{k}=A id A ⊗ ε P , k : A ⊗ QSym k → A ⊗ k = A A A A Ψ : A ⊗ H → A ⊗ QSym k \Psi:A\otimes H\to A\otimes\operatorname{QSym}_{\mathbf{k}} Ψ : A ⊗ H → A ⊗ QSym k
A ⊗ H → Ψ A ⊗ QSym k ξ # ↓ ↓ id A ⊗ ε P , k A = A \begin{CD}
A\otimes H @>{\Psi}>> A\otimes\operatorname{QSym}_{\mathbf{k}}\\
@V{\xi^{\#}}VV @VV{\operatorname{id}_{A}\otimes\varepsilon_{P,\mathbf{k}}}V\\
A @= A
\end{CD} A ⊗ H ξ # ↓ ⏐ A Ψ A ⊗ QSym k ↓ ⏐ id A ⊗ ε P , k A is commutative. In other words, there exists a unique graded A ‾ \underline{A} A Ψ : A ‾ ⊗ H → A ‾ ⊗ Q S y m k \Psi: \underline{A} \otimes H \to\underline{A} \otimes\mathrm{QSym}_{k} Ψ : A ⊗ H → A ⊗ QSym k ( id A ⊗ ε P , k ) ∘ Ψ = ξ ♯ (\operatorname{id}_{A} \otimes\varepsilon_{P,k}) \circ\Psi= \xi^{\sharp} ( id A ⊗ ε P , k ) ∘ Ψ = ξ ♯ intermediate universal property .
The ( k , A ‾ ) (k,\underline{A}) ( k , A ) H → A ‾ ⊗ Q S y m k H \to\underline{A} \otimes\mathrm{QSym}_{k} H → A ⊗ QSym k A ‾ \underline{A} A A ‾ ⊗ H → A ‾ ⊗ Q S y m k \underline{A} \otimes H \to\underline{A} \otimes\mathrm{QSym}_{k} A ⊗ H → A ⊗ QSym k A ‾ \underline{A} A A ‾ ⊗ H → Q S y m A ‾ \underline{A} \otimes H \to\mathrm{QSym}_{\underline{A}} A ⊗ H → QSym A A ‾ ⊗ Q S y m k ≅ Q S y m A ‾ \underline{A} \otimes\mathrm{QSym}_{k} \cong\mathrm{QSym}_{\underline{A}} A ⊗ QSym k ≅ QSym A ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k A ‾ \underline{A} A Ξ ♯ : A ‾ ⊗ H → A ‾ ⊗ Q S y m k \Xi^{\sharp} : \underline{A} \otimes H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ ♯ : A ⊗ H → A ⊗ QSym k Ξ \Xi Ξ Ξ ♯ \Xi^{\sharp} Ξ ♯ ( k , A ‾ ) (k,\underline{A}) ( k , A ) H → A ‾ ⊗ Q S y m k H \to\underline{A} \otimes\mathrm{QSym}_{k} H → A ⊗ QSym k A ‾ \underline{A} A A ‾ ⊗ H → A ‾ ⊗ Q S y m k \underline{A} \otimes H \to\underline{A} \otimes\mathrm{QSym}_{k} A ⊗ H → A ⊗ QSym k ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k ( id A ⊗ ε P , k ) ∘ Ξ ♯ = ξ ♯ (\operatorname{id}_{A} \otimes\varepsilon_{P,k}) \circ\Xi^{\sharp} = \xi^{\sharp} ( id A ⊗ ε P , k ) ∘ Ξ ♯ = ξ ♯ ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k ( ( id A ⊗ ε P , k ) ∘ Ξ ) ♯ = ξ ♯ ((\operatorname{id}_{A} \otimes\varepsilon_{P,k}) \circ\Xi)^{\sharp} = \xi^{\sharp} (( id A ⊗ ε P , k ) ∘ Ξ ) ♯ = ξ ♯ ( ( id A ⊗ ε P , k ) ∘ Ξ ) ♯ = ( id A ⊗ ε P , k ) ∘ Ξ ♯ ((\operatorname{id}_{A} \otimes\varepsilon_{P,k}) \circ\Xi)^{\sharp} = (\operatorname{id}_{A} \otimes\varepsilon_{P,k}) \circ\Xi^{\sharp} (( id A ⊗ ε P , k ) ∘ Ξ ) ♯ = ( id A ⊗ ε P , k ) ∘ Ξ ♯ ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k ( id A ⊗ ε P , k ) ∘ Ξ = ξ (\operatorname{id}_{A} \otimes\varepsilon_{P,k}) \circ\Xi= \xi ( id A ⊗ ε P , k ) ∘ Ξ = ξ ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k
By tracing back the above argument, we see that it yields an explicit construction of the unique graded ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k Ξ ♯ = Ψ \Xi^{\sharp} = \Psi Ξ ♯ = Ψ Ψ \Psi Ψ A ‾ \underline{A} A Ψ : A ‾ ⊗ H → Q S y m A ‾ \Psi: \underline{A} \otimes H \to\mathrm{QSym}_{\underline{A}} Ψ : A ⊗ H → QSym A Ξ \Xi Ξ Ψ \Psi Ψ
Theorem 2.1 (b) (applied to A ‾ \underline{A} A A ‾ ⊗ H \underline{A} \otimes H A ⊗ H ξ ♯ \xi^{\sharp} ξ ♯ k k k H H H ζ \zeta ζ Ψ : A ‾ ⊗ H → Q S y m A ‾ \Psi: \underline{A} \otimes H \to\mathrm{QSym}_{\underline{A}} Ψ : A ⊗ H → QSym A A ‾ \underline{A} A A ‾ \underline{A} A Ξ ♯ : A ‾ ⊗ H → A ‾ ⊗ Q S y m k \Xi^{\sharp} : \underline{A} \otimes H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ ♯ : A ⊗ H → A ⊗ QSym k A ‾ \underline{A} A Ξ ♯ : A ‾ ⊗ H → A ‾ ⊗ Q S y m k \Xi^{\sharp} : \underline{A} \otimes H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ ♯ : A ⊗ H → A ⊗ QSym k Ψ : A ‾ ⊗ H → Q S y m A ‾ \Psi: \underline{A} \otimes H \to\mathrm{QSym}_{\underline{A}} Ψ : A ⊗ H → QSym A Ξ : H → A ‾ ⊗ Q S y m k \Xi: H \to\underline{A} \otimes\mathrm{QSym}_{k} Ξ : H → A ⊗ QSym k k k k A ‾ \underline{A} A A ‾ ⊗ Q S y m k \underline{A} \otimes\mathrm{QSym}_{k} A ⊗ QSym k Ξ \Xi Ξ A ‾ \underline{A} A G G G f f f
(c) Theorem 2.1 (d) (applied to A ‾ \underline{A} A A ‾ ⊗ H \underline{A} \otimes H A ⊗ H ξ ♯ \xi^{\sharp} ξ ♯ k k k H H H ζ \zeta ζ Ψ \Psi Ψ
Ψ ( h ) = ∑ α ∈ Comp ( ξ ♯ ) α ( h ) ⋅ M α for every h ∈ A ‾ ⊗ H , (10) \Psi(h)=\sum_{\alpha\in\operatorname{Comp}}\left(\xi^{\sharp}\right)_{\alpha}(h)\cdot M_{\alpha}\qquad\text{for every }h\in\underline{A}\otimes H, \tag*{(10)} Ψ ( h ) = α ∈ Comp ∑ ( ξ ♯ ) α ( h ) ⋅ M α for every h ∈ A ⊗ H , (10) where the map ( ξ ♯ ) α : A ‾ ⊗ H → A ‾ (\xi^{\sharp})_{\alpha}:\underline{A}\otimes H\to\underline{A} ( ξ ♯ ) α : A ⊗ H → A ζ α : H → k \zeta_{\alpha}:H\to k ζ α : H → k k k k H H H ζ \zeta ζ A ‾ \underline{A} A A ‾ ⊗ H \underline{A}\otimes H A ⊗ H ξ ♯ \xi^{\sharp} ξ ♯ Q S y m A ‾ \mathrm{QSym}_{\underline{A}} QSym A QSym A \operatorname{QSym}_{A} QSym A A ⊗ QSym k A \otimes\operatorname{QSym}_{k} A ⊗ QSym k A ⊗ QSym k A \otimes\operatorname{QSym}_{k} A ⊗ QSym k
Ψ ( h ) = ∑ α ∈ Comp ( ξ ♯ ) α ( h ) ⊗ M α for every h ∈ A ⊗ H . (11) \Psi(h)=\sum_{\alpha\in\operatorname{Comp}}\left(\xi^{\sharp}\right)_{\alpha}(h)\otimes M_{\alpha}\qquad\text{for every }h\in A\otimes H. \tag*{(11)} Ψ ( h ) = α ∈ Comp ∑ ( ξ ♯ ) α ( h ) ⊗ M α for every h ∈ A ⊗ H . (11) Let ι H \iota_H ι H k \mathbf{k} k
H ⟶ A ⊗ H , h ⟼ 1 ⊗ h . H\longrightarrow A\otimes H,\qquad h\longmapsto1\otimes h. H ⟶ A ⊗ H , h ⟼ 1 ⊗ h . Also, for every k ∈ N k\in\mathbb{N} k ∈ N ι k \iota_k ι k k \mathbf{k} k
H ⊗ k ⟶ ( A ⊗ H ) ⊗ A k , g ⟼ 1 ⊗ g ∈ A ⊗ H ⊗ k ≅ ( A ⊗ H ) ⊗ A k H^{\otimes k}\longrightarrow(A\otimes H)^{\otimes_A k},\qquad g\longmapsto1\otimes g\in A\otimes H^{\otimes k}\cong(A\otimes H)^{\otimes_A k} H ⊗ k ⟶ ( A ⊗ H ) ⊗ A k , g ⟼ 1 ⊗ g ∈ A ⊗ H ⊗ k ≅ ( A ⊗ H ) ⊗ A k (where U ⊗ A k U^{\otimes_A k} U ⊗ A k k k k A A A U U U h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ∈ H ⊗ k h_1\otimes h_2\otimes\cdots\otimes h_k\in H^{\otimes k} h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ∈ H ⊗ k ( 1 ⊗ h 1 ) ⊗ A ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ( 1 ⊗ h k ) (1\otimes h_1)\otimes_A(1\otimes h_2)\otimes_A\cdots\otimes_A(1\otimes h_k) ( 1 ⊗ h 1 ) ⊗ A ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ( 1 ⊗ h k )
On the other hand, fix some α ∈ Comp \alpha\in\operatorname{Comp} α ∈ Comp α \alpha α α = ( a 1 , a 2 , … , a k ) \alpha=(a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k )
H → Δ ( k − 1 ) H ⊗ k → π α H ⊗ k → ζ ⊗ k A ⊗ k → m ( k − 1 ) A ι H ↓ ι k ↓ ι k ↓ ( ξ ♯ ) α ⊗ A k ↓ id ↓ A ⊗ H → Δ ( k − 1 ) ( A ⊗ H ) ⊗ A k → π α ( A ⊗ H ) ⊗ A k → ( ξ ♯ ) α ⊗ A k A ⊗ A k → ≅ A \begin{CD}
H @>{\Delta^{(k-1)}}>> H^{\otimes k} @>{\pi_\alpha}>> H^{\otimes k} @>{\zeta^{\otimes k}}>> A^{\otimes k} @>{m^{(k-1)}}>> A\\
@V{\iota_H}VV @V{\iota_k}VV @V{\iota_k}VV @V{\left(\xi^\sharp\right)_{\alpha}^{\otimes_A k}}VV @V{\operatorname{id}}VV\\
A\otimes H @>{\Delta^{(k-1)}}>> (A\otimes H)^{\otimes_A k} @>{\pi_\alpha}>> (A\otimes H)^{\otimes_A k} @>{\left(\xi^\sharp\right)_{\alpha}^{\otimes_A k}}>> A^{\otimes_A k} @>{\cong}>> A
\end{CD} H ι H ↓ ⏐ A ⊗ H Δ ( k − 1 ) Δ ( k − 1 ) H ⊗ k ι k ↓ ⏐ ( A ⊗ H ) ⊗ A k π α π α H ⊗ k ι k ↓ ⏐ ( A ⊗ H ) ⊗ A k ζ ⊗ k ( ξ ♯ ) α ⊗ A k A ⊗ k ( ξ ♯ ) α ⊗ A k ↓ ⏐ A ⊗ A k m ( k − 1 ) ≅ A id ↓ ⏐ A is commutative9 ( ξ ♯ ) α ∘ ι H = id ∘ ζ α = ζ α \left(\xi^\sharp\right)_{\alpha}\circ\iota_H=\operatorname{id}\circ\zeta_\alpha=\zeta_\alpha ( ξ ♯ ) α ∘ ι H = id ∘ ζ α = ζ α
Now, forget that we fixed α \alpha α
( ξ ♯ ) α ∘ ι H = ζ α for every α ∈ Comp . (12) \left(\xi^\sharp\right)_{\alpha}\circ\iota_H=\zeta_\alpha\qquad\text{for every }\alpha\in\operatorname{Comp}. \tag*{(12)} ( ξ ♯ ) α ∘ ι H = ζ α for every α ∈ Comp . (12) Its upper pentagon is commutative (by the definition of ζ α \zeta_\alpha ζ α
Its lower pentagon is commutative (by the definition of ( ξ ♯ ) α \left(\xi^\sharp\right)_\alpha ( ξ ♯ ) α
Its left square is commutative (since the operation Δ ( k − 1 ) \Delta^{(k-1)} Δ ( k − 1 ) k \mathbf{k} k
Its middle square is commutative (since the operation π α \pi_\alpha π α k \mathbf{k} k
Now, every h ∈ H h \in H h ∈ H
Ξ ( h ) = Ξ ♯ ⏟ = Ψ ( 1 ⊗ h ) = Ψ ( 1 ⊗ h ) = ∑ α ∈ Comp ( ξ ♯ ) α ( 1 ⊗ h ) ⏟ = ι H ( h ) ⊗ M α (by (11), applied to 1 ⊗ h instead of h ) = ∑ α ∈ Comp ( ξ ♯ ) α ⏟ = ( ( ξ ♯ ) α ∘ ι H ) ( h ) ( ι H ( h ) ) ⊗ M α = ∑ α ∈ Comp ( ( ξ ♯ ) α ∘ ι H ) ⏟ = ξ ~ α (by (12)) ( h ) ⊗ M α = ∑ α ∈ Comp ξ ~ α ( h ) ⊗ M α . \begin{aligned}
\Xi(h)=\underbrace{\Xi^{\sharp}}_{=\Psi}(1\otimes h)=\Psi(1\otimes h) \\
&=\sum_{\alpha\in\operatorname{Comp}}\underbrace{\left(\xi^{\sharp}\right)_{\alpha}(1\otimes h)}_{=\iota_H(h)}\otimes M_{\alpha}\qquad\text{(by (11), applied to $1\otimes h$ instead of $h$)} \\
&=\sum_{\alpha\in\operatorname{Comp}}\underbrace{\left(\xi^{\sharp}\right)_{\alpha}}_{=((\xi^{\sharp})_{\alpha}\circ\iota_H)(h)}(\iota_H(h))\otimes M_{\alpha}
=\sum_{\alpha\in\operatorname{Comp}}\underbrace{\left((\xi^{\sharp})_{\alpha}\circ\iota_H\right)}_{=\widetilde{\xi}_{\alpha}\ \text{(by (12))}}(h)\otimes M_{\alpha} \\
&=\sum_{\alpha\in\operatorname{Comp}}\widetilde{\xi}_{\alpha}(h)\otimes M_{\alpha}.
\end{aligned} Ξ ( h ) = = Ψ Ξ ♯ ( 1 ⊗ h ) = Ψ ( 1 ⊗ h ) = α ∈ Comp ∑ = ι H ( h ) ( ξ ♯ ) α ( 1 ⊗ h ) ⊗ M α (by (11), applied to 1 ⊗ h instead of h ) = α ∈ Comp ∑ = (( ξ ♯ ) α ∘ ι H ) ( h ) ( ξ ♯ ) α ( ι H ( h )) ⊗ M α = α ∈ Comp ∑ = ξ α (by (12)) ( ( ξ ♯ ) α ∘ ι H ) ( h ) ⊗ M α = α ∈ Comp ∑ ξ α ( h ) ⊗ M α . This proves Corollary 3.12 (c).
(d) Assume that the k \mathbf{k} k H H H A A A A ⊗ H A\otimes H A ⊗ H
Let us first see why A ⊗ Λ k A\otimes\Lambda_{\mathbf{k}} A ⊗ Λ k A ⊗ QSym k A\otimes\operatorname{QSym}_{\mathbf{k}} A ⊗ QSym k A A A A ⊗ QSym k → QSym A A\otimes\operatorname{QSym}_{\mathbf{k}}\to\operatorname{QSym}_{A} A ⊗ QSym k → QSym A QSym A \operatorname{QSym}_{A} QSym A A ⊗ QSym k A\otimes\operatorname{QSym}_{\mathbf{k}} A ⊗ QSym k A A A
Its right rectangle is commutative. (Indeed, every h 1 , h 2 , … , h k ∈ H h_1,h_2,\ldots,h_k\in H h 1 , h 2 , … , h k ∈ H
( id ∘ m ( k − 1 ) ∘ ξ ⊗ k ) ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) = m ( k − 1 ) ( ξ ⊗ k ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) ⏟ = ξ ( h 1 ) ⊗ ξ ( h 2 ) ⊗ ⋯ ⊗ ξ ( h k ) ) = m ( k − 1 ) ( ξ ( h 1 ) ⊗ ξ ( h 2 ) ⊗ ⋯ ⊗ ξ ( h k ) ) = ξ ( h 1 ) ξ ( h 2 ) ⋯ ξ ( h k ) \begin{aligned}
\left(\operatorname{id}\circ m^{(k-1)}\circ\xi^{\otimes k}\right)(h_1\otimes h_2\otimes\cdots\otimes h_k) \\
&=m^{(k-1)}\left(\underbrace{\xi^{\otimes k}(h_1\otimes h_2\otimes\cdots\otimes h_k)}_{=\xi(h_1)\otimes\xi(h_2)\otimes\cdots\otimes\xi(h_k)}\right)
=m^{(k-1)}\left(\xi(h_1)\otimes\xi(h_2)\otimes\cdots\otimes\xi(h_k)\right) \\
&=\xi(h_1)\xi(h_2)\cdots\xi(h_k)
\end{aligned} ( id ∘ m ( k − 1 ) ∘ ξ ⊗ k ) ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) = m ( k − 1 ) = ξ ( h 1 ) ⊗ ξ ( h 2 ) ⊗ ⋯ ⊗ ξ ( h k ) ξ ⊗ k ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) = m ( k − 1 ) ( ξ ( h 1 ) ⊗ ξ ( h 2 ) ⊗ ⋯ ⊗ ξ ( h k ) ) = ξ ( h 1 ) ξ ( h 2 ) ⋯ ξ ( h k ) and thus
( ( ξ ♯ ) ⊗ A k ∘ ι k ) ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) = ( ξ ♯ ) ⊗ A k ( ι k ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) ) ⏟ = ( 1 ⊗ h 1 ) ⊗ A ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ( 1 ⊗ h k ) = ( ξ ♯ ) ⊗ A k ( ( 1 ⊗ h 1 ) ⊗ A ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ( 1 ⊗ h k ) ) = ξ ♯ ( 1 ⊗ h 1 ) ⊗ A ξ ♯ ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ξ ♯ ( 1 ⊗ h k ) = ξ ( h 1 ) ⊗ A ξ ( h 2 ) ⊗ A ⋯ ⊗ A ξ ( h k ) (since ξ ♯ ( 1 ⊗ y ) = ξ ( y ) for every y ∈ H ) = ξ ( h 1 ) ξ ( h 2 ) ⋯ ξ ( h k ) (since A ⊗ A k ≅ A ) = ( id ∘ m ( k − 1 ) ∘ ξ ⊗ k ) ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) . \begin{aligned}
\left((\xi^{\sharp})^{\otimes_A k}\circ\iota_k\right)(h_1\otimes h_2\otimes\cdots\otimes h_k) \\
&=(\xi^{\sharp})^{\otimes_A k}\underbrace{\left(\iota_k(h_1\otimes h_2\otimes\cdots\otimes h_k)\right)}_{=(1\otimes h_1)\otimes_A(1\otimes h_2)\otimes_A\cdots\otimes_A(1\otimes h_k)} \\
&=(\xi^{\sharp})^{\otimes_A k}\left((1\otimes h_1)\otimes_A(1\otimes h_2)\otimes_A\cdots\otimes_A(1\otimes h_k)\right) \\
&=\xi^{\sharp}(1\otimes h_1)\otimes_A\xi^{\sharp}(1\otimes h_2)\otimes_A\cdots\otimes_A\xi^{\sharp}(1\otimes h_k) \\
&=\xi(h_1)\otimes_A\xi(h_2)\otimes_A\cdots\otimes_A\xi(h_k)\qquad\text{(since $\xi^{\sharp}(1\otimes y)=\xi(y)$ for every $y\in H$)} \\
&=\xi(h_1)\xi(h_2)\cdots\xi(h_k)\qquad\text{(since $A^{\otimes_A k}\cong A$)} \\
&=\left(\operatorname{id}\circ m^{(k-1)}\circ\xi^{\otimes k}\right)(h_1\otimes h_2\otimes\cdots\otimes h_k).
\end{aligned} ( ( ξ ♯ ) ⊗ A k ∘ ι k ) ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) = ( ξ ♯ ) ⊗ A k = ( 1 ⊗ h 1 ) ⊗ A ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ( 1 ⊗ h k ) ( ι k ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) ) = ( ξ ♯ ) ⊗ A k ( ( 1 ⊗ h 1 ) ⊗ A ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ( 1 ⊗ h k ) ) = ξ ♯ ( 1 ⊗ h 1 ) ⊗ A ξ ♯ ( 1 ⊗ h 2 ) ⊗ A ⋯ ⊗ A ξ ♯ ( 1 ⊗ h k ) = ξ ( h 1 ) ⊗ A ξ ( h 2 ) ⊗ A ⋯ ⊗ A ξ ( h k ) (since ξ ♯ ( 1 ⊗ y ) = ξ ( y ) for every y ∈ H ) = ξ ( h 1 ) ξ ( h 2 ) ⋯ ξ ( h k ) (since A ⊗ A k ≅ A ) = ( id ∘ m ( k − 1 ) ∘ ξ ⊗ k ) ( h 1 ⊗ h 2 ⊗ ⋯ ⊗ h k ) . Hence, ( ξ ♯ ) ⊗ A k ∘ ι k = id ∘ m ( k − 1 ) ∘ ξ ⊗ k (\xi^{\sharp})^{\otimes_A k}\circ\iota_k=\operatorname{id}\circ m^{(k-1)}\circ\xi^{\otimes k} ( ξ ♯ ) ⊗ A k ∘ ι k = id ∘ m ( k − 1 ) ∘ ξ ⊗ k A ⊗ Λ k → Λ A A \otimes\Lambda_k \to\Lambda_A A ⊗ Λ k → Λ A Λ A \Lambda_A Λ A A ⊗ Λ k A \otimes\Lambda_k A ⊗ Λ k A ⊗ Λ k = Λ A ⊆ QSym A = A ⊗ QSym k A \otimes\Lambda_k = \Lambda_A \subseteq\operatorname{QSym}_A = A \otimes\operatorname{QSym}_k A ⊗ Λ k = Λ A ⊆ QSym A = A ⊗ QSym k
Theorem 2.1 (e) (applied to A A A A ⊗ H A \otimes H A ⊗ H Ξ # \Xi^\# Ξ # k k k H H H ζ \zeta ζ Ψ ( A ⊗ H ) ⊆ Λ A = A ⊗ Λ k \Psi(A \otimes H) \subseteq\Lambda_A = A \otimes\Lambda_k Ψ ( A ⊗ H ) ⊆ Λ A = A ⊗ Λ k Ψ = Ξ # \Psi= \Xi^\# Ψ = Ξ # Ξ # ( A ⊗ H ) ⊆ A ⊗ Λ k \Xi^\#(A \otimes H) \subseteq A \otimes\Lambda_k Ξ # ( A ⊗ H ) ⊆ A ⊗ Λ k Ξ ( H ) ⊆ Ξ # ( A ⊗ H ) \Xi(H) \subseteq\Xi^\#(A \otimes H) Ξ ( H ) ⊆ Ξ # ( A ⊗ H ) h ∈ H h \in H h ∈ H Ξ ( h ) = Ξ # ( 1 ⊗ h ) ∈ Ξ # ( A ⊗ H ) \Xi(h) = \Xi^\#(1 \otimes h) \in\Xi^\#(A \otimes H) Ξ ( h ) = Ξ # ( 1 ⊗ h ) ∈ Ξ # ( A ⊗ H ) Ξ ( H ) ⊆ Ξ # ( A ⊗ H ) ⊆ A ⊗ Λ k \Xi(H) \subseteq\Xi^\#(A \otimes H) \subseteq A \otimes\Lambda_k Ξ ( H ) ⊆ Ξ # ( A ⊗ H ) ⊆ A ⊗ Λ k
The second comultiplication on QSym k \operatorname{QSym}_k QSym k
We now recall the definition of the second comultiplication (a.k.a. internal comultiplication ) of QSym k \operatorname{QSym}_k QSym k 10
Definition 4.2. (a) Given a u × v u \times v u × v A = ( a i , j ) 1 ≤ i ≤ u , 1 ≤ j ≤ v ∈ N u × v A = (a_{i,j})_{1 \leq i \leq u,\ 1 \leq j \leq v} \in\mathbb{N}^{u \times v} A = ( a i , j ) 1 ≤ i ≤ u , 1 ≤ j ≤ v ∈ N u × v u , v ∈ N u,v \in\mathbb{N} u , v ∈ N
The v v v A ∈ N v A \in\mathbb{N}^{v} A ∈ N v v v v j j j ∑ i = 1 u a i , j \sum_{i=1}^{u}a_{i,j} ∑ i = 1 u a i , j j j j A A A j j j A A A A A A
The u u u A ∈ N u A \in\mathbb{N}^{u} A ∈ N u u u u i i i ∑ j = 1 v a i , j \sum_{j=1}^{v}a_{i,j} ∑ j = 1 v a i , j i i i A A A i i i A A A A A A
The u v uv uv A ∈ N u v A \in\mathbb{N}^{uv} A ∈ N uv u v uv uv ( v ( i − 1 ) + j ) (v(i-1)+j) ( v ( i − 1 ) + j ) a i , j a_{i,j} a i , j i ∈ { 1 , 2 , … , u } i \in\{1,2,\ldots,u\} i ∈ { 1 , 2 , … , u } j ∈ { 1 , 2 , … , v } j \in\{1,2,\ldots,v\} j ∈ { 1 , 2 , … , v }
read A = ( a 1 , 1 , a 1 , 2 , … , a 1 , v , a 2 , 1 , a 2 , 2 , … , a 2 , v , … , a u , 1 , a u , 2 , … , a u , v ) . \operatorname{read} A = (a_{1,1},a_{1,2},\ldots,a_{1,v},a_{2,1},a_{2,2},\ldots,a_{2,v},\ldots,a_{u,1},a_{u,2},\ldots,a_{u,v}). read A = ( a 1 , 1 , a 1 , 2 , … , a 1 , v , a 2 , 1 , a 2 , 2 , … , a 2 , v , … , a u , 1 , a u , 2 , … , a u , v ) . We say that the matrix A A A column-reduced if column A A A A A A A A A 0 0 0
We say that the matrix A A A row-reduced if row A A A A A A A A A 0 0 0
We say that the matrix A A A reduced if A A A
(b) If w ∈ N k w \in\mathbb{N}^{k} w ∈ N k k k k k ∈ N k \in\mathbb{N} k ∈ N w r e d w^{\mathrm{red}} w red w w w 0 0 0 ( 3 , 1 , 0 , 1 , 0 , 0 , 2 ) r e d = ( 3 , 1 , 1 , 2 ) (3,1,0,1,0,0,2)^{\mathrm{red}}=(3,1,1,2) ( 3 , 1 , 0 , 1 , 0 , 0 , 2 ) red = ( 3 , 1 , 1 , 2 )
(c) Let N r e d ∙ , ∙ \mathbb{N}^{\bullet,\bullet}_{\mathrm{red}} N red ∙ , ∙ N u × v \mathbb{N}^{u\times v} N u × v u u u v v v N \mathbb{N} N
N r e d ∙ , ∙ = ⋃ ( u , v ) ∈ N 2 { A ∈ N u × v ∣ A is reduced } . \mathbb{N}^{\bullet,\bullet}_{\mathrm{red}}=\bigcup_{(u,v)\in\mathbb{N}^{2}}\{A\in\mathbb{N}^{u\times v}\mid A\text{ is reduced}\}. N red ∙ , ∙ = ( u , v ) ∈ N 2 ⋃ { A ∈ N u × v ∣ A is reduced } . (d) Let Δ P : QSym k → QSym k ⊗ QSym k \Delta_{P}:\operatorname{QSym}_{k}\to\operatorname{QSym}_{k}\otimes\operatorname{QSym}_{k} Δ P : QSym k → QSym k ⊗ QSym k k k k
Δ P ( M α ) = ∑ A ∈ N r e d ∙ , ∙ ( read A ) r e d = α M row A ⊗ M column A \Delta_{P}(M_{\alpha})=\sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}}\\(\operatorname{read}A)^{\mathrm{red}}=\alpha}}M_{\operatorname{row}A}\otimes M_{\operatorname{column}A} Δ P ( M α ) = A ∈ N red ∙ , ∙ ( read A ) red = α ∑ M row A ⊗ M column A for each α ∈ Comp \alpha\in\operatorname{Comp} α ∈ Comp
This map Δ P \Delta_{P} Δ P second comultiplication (or internal comultiplication ) of QSym k \operatorname{QSym}_{k} QSym k
(e) Let τ \tau τ τ QSym k , QSym k : QSym k ⊗ QSym k → QSym k ⊗ QSym k \tau_{\operatorname{QSym}_{k},\operatorname{QSym}_{k}}:\operatorname{QSym}_{k}\otimes\operatorname{QSym}_{k}\to\operatorname{QSym}_{k}\otimes\operatorname{QSym}_{k} τ QSym k , QSym k : QSym k ⊗ QSym k → QSym k ⊗ QSym k Δ P ′ = τ ∘ Δ P : QSym k → QSym k ⊗ QSym k \Delta'_{P}=\tau\circ\Delta_{P}:\operatorname{QSym}_{k}\to\operatorname{QSym}_{k}\otimes\operatorname{QSym}_{k} Δ P ′ = τ ∘ Δ P : QSym k → QSym k ⊗ QSym k
Example 4.3. The matrix ( 1 0 2 0 2 0 0 5 0 0 3 1 ) ∈ N 3 × 4 \begin{pmatrix} 1 & 0 & 2 & 0 \\ 2 & 0 & 0 & 5 \\ 0 & 0 & 3 & 1 \end{pmatrix} \in\mathbb{N}^{3 \times4} 1 2 0 0 0 0 2 0 3 0 5 1 ∈ N 3 × 4 A A A A = ( 3 , 7 , 4 ) A = (3,7,4) A = ( 3 , 7 , 4 ) A = ( 3 , 0 , 5 , 6 ) A = (3,0,5,6) A = ( 3 , 0 , 5 , 6 ) A = ( 1 , 0 , 2 , 0 , 2 , 0 , 0 , 5 , 0 , 0 , 3 , 1 ) A = (1,0,2,0,2,0,0,5,0,0,3,1) A = ( 1 , 0 , 2 , 0 , 2 , 0 , 0 , 5 , 0 , 0 , 3 , 1 )
Proposition 4.4. The k \mathit{k} k QSym k \operatorname{QSym}_{\mathit{k}} QSym k Δ P \Delta_P Δ P ε P \varepsilon_P ε P k \mathit{k} k
Proposition 4.4 is a well-known fact (appearing, for example, in [MalReu95, first paragraph of §3]), but we shall actually derive it further below using our results.
The (generalized) Bernstein homomorphism
Let us now define the Bernstein homomorphism of a commutative connected graded k \mathit{k} k
Definition 5.1. Let k \mathit{k} k H H H k \mathit{k} k α = ( a 1 , a 2 , … , a k ) \alpha= (a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k ) k \mathit{k} k ζ ~ α : H → H \widetilde{\zeta}_{\alpha}: H \to H ζ α : H → H k \mathit{k} k
H → Δ ( k − 1 ) H ⊗ k → π α H ⊗ k → m ( k − 1 ) H . H \xrightarrow{\Delta^{(k-1)}} H^{\otimes k} \xrightarrow{\pi_{\alpha}} H^{\otimes k} \xrightarrow{m^{(k-1)}} H . H Δ ( k − 1 ) H ⊗ k π α H ⊗ k m ( k − 1 ) H . (Recall that Δ ( k − 1 ) : H → H ⊗ k \Delta^{(k-1)} : H \to H^{\otimes k} Δ ( k − 1 ) : H → H ⊗ k m ( k − 1 ) : H ⊗ k → H m^{(k-1)} : H^{\otimes k} \to H m ( k − 1 ) : H ⊗ k → H π α : H ⊗ k → H ⊗ k \pi_{\alpha} : H^{\otimes k} \to H^{\otimes k} π α : H ⊗ k → H ⊗ k β H : H → H ⊗ QSym k \beta_H : H \to H \otimes\operatorname{QSym}_{\mathit{k}} β H : H → H ⊗ QSym k
β H ( h ) = ∑ α ∈ Comp ζ ~ α ( h ) ⊗ M α \beta_H(h) = \sum_{\alpha\in\operatorname{Comp}} \widetilde{\zeta}_{\alpha}(h) \otimes M_{\alpha} β H ( h ) = α ∈ Comp ∑ ζ α ( h ) ⊗ M α for every h ∈ H h \in H h ∈ H
It is easy to see that this map β H \beta_H β H 11 k \mathit{k} k
The graded k k k QSym k \operatorname{QSym}_k QSym k H = QSym k H=\operatorname{QSym}_k H = QSym k k k k β QSym k : QSym k → QSym k ⊗ QSym k \beta_{\operatorname{QSym}_k}:\operatorname{QSym}_k\to\operatorname{QSym}_k\otimes\operatorname{QSym}_k β QSym k : QSym k → QSym k ⊗ QSym k Δ p ′ \Delta'_p Δ p ′
Proposition 5.3. We have β QSym k = Δ p ′ \beta_{\operatorname{QSym}_k}=\Delta'_p β QSym k = Δ p ′
Before we prove this, let us recall a basic formula for multiplication of monomial quasisymmetric functions:
Proposition 5.4. Let k ∈ N k\in\mathbb{N} k ∈ N α 1 , α 2 , … , α k \alpha_1,\alpha_2,\ldots,\alpha_k α 1 , α 2 , … , α k k k k N C r e d k , ∙ \mathbb{N}^{k,\bullet}_{\mathrm{Cred}} N Cred k , ∙ N k × v \mathbb{N}^{k\times v} N k × v v v v N \mathbb{N} N
N C r e d k , ∙ = ⋃ v ∈ N { A ∈ N k × v ∣ A is column-reduced } . \mathbb{N}^{k,\bullet}_{\mathrm{Cred}}=\bigcup_{v\in\mathbb{N}}\left\{A\in\mathbb{N}^{k\times v}\mid A\text{ is column-reduced}\right\}. N Cred k , ∙ = v ∈ N ⋃ { A ∈ N k × v ∣ A is column-reduced } . Then,
M α 1 M α 2 ⋯ M α k = ∑ A ∈ N C r e d k , ∙ ; ( A g , ∙ ) = α g for each g M column A . M_{\alpha_1}M_{\alpha_2}\cdots M_{\alpha_k}
=
\sum_{\substack{A\in\mathbb{N}^{k,\bullet}_{\mathrm{Cred}};\\ (A_{g,\bullet})=\alpha_g\text{ for each }g}}
M_{\operatorname{column} A}. M α 1 M α 2 ⋯ M α k = A ∈ N Cred k , ∙ ; ( A g , ∙ ) = α g for each g ∑ M column A . Here, A i , ∙ A_{i,\bullet} A i , ∙ i i i A A A
Notice that the k = 2 k=2 k = 2 12
Proof of Proposition 5.4. We begin by introducing notations:
Let N k , ∞ \mathbb{N}^{k,\infty} N k , ∞ k k k 1 , 2 , … , k 1,2,\ldots,k 1 , 2 , … , k 1 , 2 , 3 , … 1,2,3,\ldots 1 , 2 , 3 , … N \mathbb{N} N
Let N f i n k , ∞ \mathbb{N}^{k,\infty}_{\mathrm{fin}} N fin k , ∞ N k , ∞ \mathbb{N}^{k,\infty} N k , ∞
Let N ∞ \mathbb{N}^{\infty} N ∞ ( a 1 , a 2 , a 3 , … ) (a_1,a_2,a_3,\ldots) ( a 1 , a 2 , a 3 , … ) N \mathbb{N} N
For every B ∈ N f i n k , ∞ B \in\mathbb{N}^{k,\infty}_{\mathrm{fin}} B ∈ N fin k , ∞ i ∈ { 1 , 2 , … , k } i \in\{1,2,\ldots,k\} i ∈ { 1 , 2 , … , k } B i , ⋅ ∈ N f i n ∞ B_{i,\mathbin{\cdot}} \in\mathbb{N}^{\infty}_{\mathrm{fin}} B i , ⋅ ∈ N fin ∞ i i i B B B
For every B = ( b i , j ) 1 ≤ i ≤ k , 1 ≤ j ∈ N f i n k , ∞ B = (b_{i,j})_{1\leq i\leq k,\ 1\leq j} \in\mathbb{N}^{k,\infty}_{\mathrm{fin}} B = ( b i , j ) 1 ≤ i ≤ k , 1 ≤ j ∈ N fin k , ∞ column B ∈ N f i n ∞ \operatorname{column} B \in\mathbb{N}^{\infty}_{\mathrm{fin}} column B ∈ N fin ∞ j j j ∑ i = 1 k b i , j \sum_{i=1}^{k} b_{i,j} ∑ i = 1 k b i , j j j j B B B j j j column B \operatorname{column} B column B B B B
For every β = ( b 1 , b 2 , b 3 , … ) ∈ N f i n ∞ \beta=(b_1,b_2,b_3,\ldots)\in\mathbb{N}^{\infty}_{\mathrm{fin}} β = ( b 1 , b 2 , b 3 , … ) ∈ N fin ∞ x β x^\beta x β x 1 , x 2 , x 3 , … x_1,x_2,x_3,\ldots x 1 , x 2 , x 3 , …
x β = x 1 b 1 x 2 b 2 x 3 b 3 ⋯ . x^\beta=x_1^{b_1}x_2^{b_2}x_3^{b_3}\cdots. x β = x 1 b 1 x 2 b 2 x 3 b 3 ⋯ . Then, it is easy to see that
M α = ∑ β ∈ N f i n ∞ β r e d = α x β for every composition α . (13) M_\alpha=\sum_{\substack{\beta\in\mathbb{N}^{\infty}_{\mathrm{fin}}\\ \beta^{\mathrm{red}}=\alpha}}x^\beta
\qquad\text{for every composition }\alpha. \tag*{(13)} M α = β ∈ N fin ∞ β red = α ∑ x β for every composition α . (13) Now,
M α 1 M α 2 ⋯ M α k = ∏ g = 1 k M α g ⏟ = ∑ β ∈ N f i n ∞ β r e d = α g x β (by (14)) = ∏ g = 1 k ∑ β ∈ N f i n ∞ β r e d = α g x β = ∑ ( β 1 , β 2 , … , β k ) ∈ ( N f i n ∞ ) k ( β g ) r e d = α g for each g x β 1 x β 2 ⋯ x β k (by the product rule) = ∑ B ∈ N f i n k , ∞ ( B g , ∙ ) r e d = α g for each g x B 1 , ∙ x B 2 , ∙ ⋯ x B k , ∙ ⏟ = x column B ( since column B is the sum of the rows of B (as vectors) ) ( here, we have substituted ( B 1 , ∙ , B 2 , ∙ , … , B k , ∙ ) for ( β 1 , β 2 , … , β k ) in the sum, since the map N f i n k , ∞ → ( N f i n ∞ ) k , B ↦ ( B 1 , ∙ , B 2 , ∙ , … , B k , ∙ ) is a bijection ) = ∑ B ∈ N f i n k , ∞ ( B g , ∙ ) r e d = α g for each g x column B . (14) \begin{aligned}
M_{\alpha_1}M_{\alpha_2}\cdots M_{\alpha_k}
&= \prod_{g=1}^{k} \underbrace{M_{\alpha_g}}_{=\sum_{\substack{\beta\in\mathbb{N}_{\mathrm{fin}}^\infty\\ \beta^{\mathrm{red}=\alpha_g}}} x^\beta\text{ (by (14))}} \\
&= \prod_{g=1}^{k}\sum_{\substack{\beta\in\mathbb{N}_{\mathrm{fin}}^\infty\\ \beta^{\mathrm{red}=\alpha_g}}} x^\beta\\
&= \sum_{\substack{(\beta_1,\beta_2,\ldots,\beta_k)\in(\mathbb{N}_{\mathrm{fin}}^\infty)^k\\(\beta_g)^{\mathrm{red}}=\alpha_g\text{ for each }g}} x^{\beta_1}x^{\beta_2}\cdots x^{\beta_k}
\qquad\text{(by the product rule)} \\
&= \sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\(B_{g,\bullet})^{\mathrm{red}}=\alpha_g\text{ for each }g}} \underbrace{x^{B_{1,\bullet}}x^{B_{2,\bullet}}\cdots x^{B_{k,\bullet}}}_{=x^{\text{column }B}} \\
&\qquad\left(\text{since column }B\text{ is the sum of the rows of }B\text{ (as vectors)}\right) \\
&\qquad\left(\text{here, we have substituted }(B_{1,\bullet},B_{2,\bullet},\ldots,B_{k,\bullet})\text{ for }(\beta_1,\beta_2,\ldots,\beta_k)\text{ in the sum, since the map }\right.\\
&\qquad\left.\mathbb{N}_{\mathrm{fin}}^{k,\infty}\to(\mathbb{N}_{\mathrm{fin}}^\infty)^k,\ B\mapsto(B_{1,\bullet},B_{2,\bullet},\ldots,B_{k,\bullet})\text{ is a bijection}\right)\\
&= \sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\(B_{g,\bullet})^{\mathrm{red}}=\alpha_g\text{ for each }g}}x^{\text{column }B}. \tag*{(14)}
\end{aligned} M α 1 M α 2 ⋯ M α k = g = 1 ∏ k = ∑ β ∈ N fin ∞ β red = α g x β (by (14)) M α g = g = 1 ∏ k β ∈ N fin ∞ β red = α g ∑ x β = ( β 1 , β 2 , … , β k ) ∈ ( N fin ∞ ) k ( β g ) red = α g for each g ∑ x β 1 x β 2 ⋯ x β k (by the product rule) = B ∈ N fin k , ∞ ( B g , ∙ ) red = α g for each g ∑ = x column B x B 1 , ∙ x B 2 , ∙ ⋯ x B k , ∙ ( since column B is the sum of the rows of B (as vectors) ) ( here, we have substituted ( B 1 , ∙ , B 2 , ∙ , … , B k , ∙ ) for ( β 1 , β 2 , … , β k ) in the sum, since the map N fin k , ∞ → ( N fin ∞ ) k , B ↦ ( B 1 , ∙ , B 2 , ∙ , … , B k , ∙ ) is a bijection ) = B ∈ N fin k , ∞ ( B g , ∙ ) red = α g for each g ∑ x column B . (14) Now, let us introduce one more notation: For every matrix B ∈ N f i n k , ∞ B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞ B C r e d B^{\mathrm{Cred}} B Cred B B B 14 B C r e d ∈ N C r e d k , ∙ B^{\mathrm{Cred}}\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet} B Cred ∈ N Cred k , ∙ B ∈ N f i n k , ∞ B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞ B ∈ N f i n k , ∞ B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞ A = B C r e d A=B^{\mathrm{Cred}} A = B Cred
( B g , ∙ ) r e d = ( A g , ∙ ) r e d for each g (15) (B_{g,\bullet})^{\mathrm{red}}=(A_{g,\bullet})^{\mathrm{red}}\quad\text{for each }g \tag*{(15)} ( B g , ∙ ) red = ( A g , ∙ ) red for each g (15) (indeed, A g , ∙ A_{g,\bullet} A g , ∙ B g , ∙ B_{g,\bullet} B g , ∙
Now, (15) becomes
M α 1 M α 2 ⋯ M α k = ∑ B ∈ N f i n k , ∞ ( B g , ∙ ) r e d = α g for each g x column B = ∑ A ∈ N C r e d k , ∙ ∑ B ∈ N f i n k , ∞ ( B g , ∙ ) r e d = α g for each g B C r e d = A x column B = ∑ B ∈ N f i n k , ∞ B C r e d = A ( B g , ∙ ) r e d = α g for each g x column B = ∑ B ∈ N f i n k , ∞ B C r e d = A ( A g , ∙ ) r e d = α g for each g x column B (because if B C r e d = A , then ( B g , ∙ ) r e d = ( A g , ∙ ) r e d for each g (because of (16))) (since B C r e d ∈ N C r e d k , ∙ for each B ∈ N f i n k , ∞ ) = ∑ A ∈ N C r e d k , ∙ ∑ B ∈ N f i n k , ∞ B C r e d = A ( A g , ∙ ) r e d = α g for each g x column B = ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = α g for each g ∑ B ∈ N f i n k , ∞ B C r e d = A x column B = ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = α g for each g ∑ B ∈ N f i n k , ∞ B C r e d = A x column B . (16) \begin{aligned}
M_{\alpha_1}M_{\alpha_2}\cdots M_{\alpha_k}
&=\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\(B_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}x^{\operatorname{column} B}\\
&=\sum_{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}}
\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\(B_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g\\B^{\mathrm{Cred}}=A}}x^{\operatorname{column} B}\\
&=\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\B^{\mathrm{Cred}}=A\\(B_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
x^{\operatorname{column} B}
=\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\B^{\mathrm{Cred}}=A\\(A_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
x^{\operatorname{column} B}\\
&\qquad\text{(because if }B^{\mathrm{Cred}}=A,\text{ then }(B_{g,\bullet})^{\mathrm{red}}=(A_{g,\bullet})^{\mathrm{red}}\text{ for each }g\text{ (because of (16)))}\\
&\qquad\text{(since }B^{\mathrm{Cred}}\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\text{ for each }B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\text{)}\\
&=\sum_{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}}
\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\B^{\mathrm{Cred}}=A\\(A_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
x^{\operatorname{column} B}\\
&=\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\(A_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\B^{\mathrm{Cred}}=A}}x^{\operatorname{column} B}\\
&=\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\(A_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\B^{\mathrm{Cred}}=A}}x^{\operatorname{column} B}. \tag*{(16)}
\end{aligned} M α 1 M α 2 ⋯ M α k = B ∈ N fin k , ∞ ( B g , ∙ ) red = α g for each g ∑ x column B = A ∈ N Cred k , ∙ ∑ B ∈ N fin k , ∞ ( B g , ∙ ) red = α g for each g B Cred = A ∑ x column B = B ∈ N fin k , ∞ B Cred = A ( B g , ∙ ) red = α g for each g ∑ x column B = B ∈ N fin k , ∞ B Cred = A ( A g , ∙ ) red = α g for each g ∑ x column B (because if B Cred = A , then ( B g , ∙ ) red = ( A g , ∙ ) red for each g (because of (16))) (since B Cred ∈ N Cred k , ∙ for each B ∈ N fin k , ∞ ) = A ∈ N Cred k , ∙ ∑ B ∈ N fin k , ∞ B Cred = A ( A g , ∙ ) red = α g for each g ∑ x column B = A ∈ N Cred k , ∙ ( A g , ∙ ) red = α g for each g ∑ B ∈ N fin k , ∞ B Cred = A ∑ x column B = A ∈ N Cred k , ∙ ( A g , ∙ ) red = α g for each g ∑ B ∈ N fin k , ∞ B Cred = A ∑ x column B . (16) However, for every matrix A ∈ N C r e d k , ∙ A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet} A ∈ N Cred k , ∙
∑ B ∈ N f i n k , ∞ B C r e d = A x column B = M column A . (17) \sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty}\\B^{\mathrm{Cred}}=A}}x^{\operatorname{column} B}=M_{\operatorname{column} A}. \tag*{(17)} B ∈ N fin k , ∞ B Cred = A ∑ x column B = M column A . (17) Proof of (18): Let A ∈ N C r e d k , ∙ A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet} A ∈ N Cred k , ∙ B ∈ N f i n k , ∞ B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞ ( column B ) r e d = column ( B C r e d ) (\operatorname{column} B)^{\mathrm{red}}=\operatorname{column}(B^{\mathrm{Cred}}) ( column B ) red = column ( B Cred ) B B B B B B B ∈ N f i n k , ∞ B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞ B C r e d = A B^{\mathrm{Cred}}=A B Cred = A column B ∈ N f i n ∞ \operatorname{column} B\in\mathbb{N}_{\mathrm{fin}}^{\infty} column B ∈ N fin ∞ ( column B ) r e d = column ( B C r e d ) ⏟ = A (\operatorname{column} B)^{\mathrm{red}}=\operatorname{column}\underbrace{(B^{\mathrm{Cred}})}_{=A} ( column B ) red = column = A ( B Cred )
{ B ∈ N f i n k , ∞ ∣ B C r e d = A } → { β ∈ N f i n ∞ ∣ β r e d = column A } , B ↦ column B (18) \left\{B \in\mathbb{N}_{\mathrm{fin}}^{k,\infty} \mid B^{\mathrm{Cred}}=A\right\}\to\left\{\beta\in\mathbb{N}_{\mathrm{fin}}^{\infty} \mid\beta^{\mathrm{red}}=\operatorname{column} A\right\},
\qquad B\mapsto\operatorname{column} B \tag*{(18)} { B ∈ N fin k , ∞ ∣ B Cred = A } → { β ∈ N fin ∞ ∣ β red = column A } , B ↦ column B (18) is well-defined.
On the other hand, if β ∈ N f i n ∞ \beta\in\mathbb{N}_{\mathrm{fin}}^{\infty} β ∈ N fin ∞ β r e d = column A \beta^{\mathrm{red}}=\operatorname{column} A β red = column A B ∈ N f i n k , ∞ B \in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞ B C r e d = A B^{\mathrm{Cred}}=A B Cred = A column B = β \operatorname{column} B=\beta column B = β 15
Let us briefly sketch a proof of the fact that this B B B N f i n k , ∞ \mathbb{N}_{\mathrm{fin}}^{k,\infty} N fin k , ∞ B C r e d = A B^{\mathrm{Cred}}=A B Cred = A column B = β \operatorname{column} B=\beta column B = β
Indeed, it is clear that B ∈ N f i n k , ∞ B \in\mathbb{N}_{\mathrm{fin}}^{k,\infty} B ∈ N fin k , ∞
(the j -th entry of column B ) = β j (19) \text{(the $j$-th entry of column $B$)}=\beta_j \tag*{(19)} (the j -th entry of column B ) = β j (19) for every j ∈ { 1 , 2 , 3 , … } j\in\{1,2,3,\ldots\} j ∈ { 1 , 2 , 3 , … }
Proof of (20): Let j ∈ { 1 , 2 , 3 , … } j\in\{1,2,3,\ldots\} j ∈ { 1 , 2 , 3 , … } j ∈ { i 1 , i 2 , … , i h } j\in\{i_1,i_2,\ldots,i_h\} j ∈ { i 1 , i 2 , … , i h } j ∉ { i 1 , i 2 , … , i h } j\notin\{i_1,i_2,\ldots,i_h\} j ∈ / { i 1 , i 2 , … , i h } j ∈ { i 1 , i 2 , … , i h } j\in\{i_1,i_2,\ldots,i_h\} j ∈ { i 1 , i 2 , … , i h } g ∈ { 1 , 2 , … , h } g\in\{1,2,\ldots,h\} g ∈ { 1 , 2 , … , h } j = i g j=i_g j = i g g g g
(the j -th entry of column B ) = ( the sum of the entries of the j -th column of B ) j = i g = ( the sum of the entries of the i g -th column of B ) = (the g -th column of A ) (by the definition of B ) = (the sum of the entries of the g -th column of A ) = ( the g -th entry of column A ) = β r e d = ( β i 1 , β i 2 , … , β i h ) (by the definition of β r e d ) = (the g -th entry of ( β i 1 , β i 2 , … , β i h ) ) = β i g = β j (since i g = j ). \begin{aligned}
\text{(the $j$-th entry of column $B$)}
&=\left(\text{the sum of the entries of the $j$-th column of $B$}\right)_{\!j=i_g}\\
&=\left(\text{the sum of the entries of the $i_g$-th column of $B$}\right)_{\!=\text{(the $g$-th column of $A$)\\(by the definition of $B$)}}\\
&=\text{(the sum of the entries of the $g$-th column of $A$)}\\
&=\left(\text{the $g$-th entry of column $A$}\right)_{\!=\beta^{\mathrm{red}}=(\beta_{i_1},\beta_{i_2},\ldots,\beta_{i_h})\\\text{(by the definition of $\beta^{\mathrm{red}}$)}}\\
&=\text{(the $g$-th entry of $(\beta_{i_1},\beta_{i_2},\ldots,\beta_{i_h})$)}=\beta_{i_g}=\beta_j
\qquad\text{(since $i_g=j$).}
\end{aligned} (the j -th entry of column B ) = ( the sum of the entries of the j -th column of B ) j = i g = ( the sum of the entries of the i g -th column of B ) = (the g -th column of A ) (by the definition of B ) = (the sum of the entries of the g -th column of A ) = ( the g -th entry of column A ) = β red = ( β i 1 , β i 2 , … , β i h ) (by the definition of β red ) = (the g -th entry of ( β i 1 , β i 2 , … , β i h ) ) = β i g = β j (since i g = j ). Thus, (20) is proven in Case 1. Let us now consider Case 2. In this case, we have j ∉ { i 1 , i 2 , … , i h } j\notin\{i_1,i_2,\ldots,i_h\} j ∈ / { i 1 , i 2 , … , i h } j j j (18) is bijective. Thus, we can substitute β \beta β B B B
∑ B ∈ N f i n k , ∞ ; B C r e d = A x column B \sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty};\\ B^{\mathrm{Cred}}=A}} x^{\operatorname{column} B} B ∈ N fin k , ∞ ; B Cred = A ∑ x column B and obtain
∑ B ∈ N f i n k , ∞ ; B C r e d = A x column B = ∑ β ∈ N f i n ∞ ; β r e d = column A x β = M column A \sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty};\\ B^{\mathrm{Cred}}=A}} x^{\operatorname{column} B}
=
\sum_{\substack{\beta\in\mathbb{N}_{\mathrm{fin}}^{\infty};\\ \beta^{\mathrm{red}}=\operatorname{column} A}} x^\beta
=
M_{\operatorname{column} A} B ∈ N fin k , ∞ ; B Cred = A ∑ x column B = β ∈ N fin ∞ ; β red = column A ∑ x β = M column A (by (13) , applied to α = column A \alpha=\operatorname{column} A α = column A (17) .
Now, (16) becomes
M α 1 M α 2 ⋯ M α k = ∑ A ∈ N C r e d ′ k , ∙ ; ( A g , ∙ ) r e d = α g for each g ∑ B ∈ N f i n k , ∞ ; B C r e d = A x column B = ∑ A ∈ N C r e d ′ k , ∙ ; ( A g , ∙ ) r e d = α g for each g M column A . \begin{aligned}
M_{\alpha_1}M_{\alpha_2}\cdots M_{\alpha_k}
=
\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}^{\prime}}^{k,\bullet};\\ (A_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
\sum_{\substack{B\in\mathbb{N}_{\mathrm{fin}}^{k,\infty};\\ B^{\mathrm{Cred}}=A}}x^{\operatorname{column} B} \\
&=
\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}^{\prime}}^{k,\bullet};\\ (A_{g,\bullet})^{\mathrm{red}}=\alpha_g\ \text{for each }g}}
M_{\operatorname{column} A}.
\end{aligned} M α 1 M α 2 ⋯ M α k = A ∈ N Cred ′ k , ∙ ; ( A g , ∙ ) red = α g for each g ∑ B ∈ N fin k , ∞ ; B Cred = A ∑ x column B = A ∈ N Cred ′ k , ∙ ; ( A g , ∙ ) red = α g for each g ∑ M column A . belong to the list ( i 1 < i 2 < ⋯ < i h ) (i_1<i_2<\cdots<i_h) ( i 1 < i 2 < ⋯ < i h ) j j j c ∈ { 1 , 2 , 3 , … } c\in\{1,2,3,\ldots\} c ∈ { 1 , 2 , 3 , … } β c ≠ 0 \beta_c\ne0 β c = 0 ( i 1 < i 2 < ⋯ < i h ) (i_1<i_2<\cdots<i_h) ( i 1 < i 2 < ⋯ < i h ) β j = 0 \beta_j=0 β j = 0
Recall that j ∉ { i 1 , i 2 , … , i h } j\notin\{i_1,i_2,\ldots,i_h\} j ∈ / { i 1 , i 2 , … , i h } j j j B B B 0 0 0 B B B
(the j -th entry of column B ) = ( the sum of the entries of the j -th column of B ) = (the sum of the entries of the 0 vector) = 0 = β j . \begin{aligned}
\text{(the $j$-th entry of column $B$)}
&=\left(\text{the sum of the entries of the $j$-th column of $B$}\right)\\
&=\text{(the sum of the entries of the $0$ vector)}\\
&=0=\beta_j.
\end{aligned} (the j -th entry of column B ) = ( the sum of the entries of the j -th column of B ) = (the sum of the entries of the 0 vector) = 0 = β j . Thus, (19) is proven in Case 2.
Hence, (19) is proven in both Cases 1 and 2. Thus, the proof of (19) is complete.
Now, from (19) , we immediately obtain column B = ( β 1 , β 2 , β 3 , … ) = β B=(\beta_1,\beta_2,\beta_3,\ldots)=\beta B = ( β 1 , β 2 , β 3 , … ) = β
It remains to prove that B C r e d = A B^{\mathrm{Cred}}=A B Cred = A A ∈ N C r e d ′ k , ∙ A\in\mathbb{N}_{\mathrm{Cred}^{\prime}}^{k,\bullet} A ∈ N Cred ′ k , ∙ A A A A A A i 1 i_1 i 1 i 2 i_2 i 2 … \ldots … i h i_h i h B B B A A A B B B B B B j j j j j j B B B { i 1 , i 2 , … , i h } \{i_1,i_2,\ldots,i_h\} { i 1 , i 2 , … , i h } ( i 1 < i 2 < ⋯ < i h ) (i_1<i_2<\cdots<i_h) ( i 1 < i 2 < ⋯ < i h ) B C r e d B^{\mathrm{Cred}} B Cred B C r e d B^{\mathrm{Cred}} B Cred k × h k\times h k × h i 1 i_1 i 1 B B B i 2 i_2 i 2 B B B … \ldots … i h i_h i h B B B A A A B C r e d B^{\mathrm{Cred}} B Cred A A A B C r e d = A B^{\mathrm{Cred}}=A B Cred = A
Thus, we have proven that B B B N f i n k , ∞ \mathbb{N}_{\mathrm{fin}}^{k,\infty} N fin k , ∞ B C r e d = A B^{\mathrm{Cred}}=A B Cred = A B = β B=\beta B = β
It is fairly easy to see that it is the only such element (because the condition column B = β B=\beta B = β B B B B C r e d = A B^{\mathrm{Cred}}=A B Cred = A
This proves Proposition 5.4.
We need one more piece of notation:
Definition 5.5. We define a (multiplicative) monoid structure on the set Comp \operatorname{Comp} Comp α = ( a 1 , a 2 , … , a n ) \alpha=(a_1,a_2,\ldots,a_n) α = ( a 1 , a 2 , … , a n ) β = ( b 1 , b 2 , … , b m ) \beta=(b_1,b_2,\ldots,b_m) β = ( b 1 , b 2 , … , b m ) α β = ( a 1 , a 2 , … , a n , b 1 , b 2 , … , b m ) \alpha\beta=(a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_m) α β = ( a 1 , a 2 , … , a n , b 1 , b 2 , … , b m ) Comp \operatorname{Comp} Comp ∅ = ( ) \varnothing=() ∅ = ( ) { 1 , 2 , 3 , … } \{1,2,3,\ldots\} { 1 , 2 , 3 , … }
Proposition 5.6. Let γ ∈ Comp \gamma\in\operatorname{Comp} γ ∈ Comp k ∈ N k\in\mathbb{N} k ∈ N
Δ ( k − 1 ) M γ = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k γ 1 γ 2 ⋯ γ k = γ M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k . \Delta^{(k-1)}M_\gamma=
\sum_{\substack{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\operatorname{Comp}^k\\
\gamma_1\gamma_2\cdots\gamma_k=\gamma}}
M_{\gamma_1}\otimes M_{\gamma_2}\otimes\cdots\otimes M_{\gamma_k}. Δ ( k − 1 ) M γ = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k γ 1 γ 2 ⋯ γ k = γ ∑ M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k . Proof of Proposition 5.6 (sketched). We can rewrite (1) as follows:
Δ M β = ∑ ( σ , τ ) ∈ Comp × Comp σ τ = β M σ ⊗ M τ for every β ∈ Comp . (20) \Delta M_\beta=
\sum_{\substack{(\sigma,\tau)\in\operatorname{Comp}\times\operatorname{Comp}\\
\sigma\tau=\beta}}
M_\sigma\otimes M_\tau
\qquad\text{for every }\beta\in\operatorname{Comp}. \tag*{(20)} Δ M β = ( σ , τ ) ∈ Comp × Comp σ τ = β ∑ M σ ⊗ M τ for every β ∈ Comp . (20) Proposition 5.6 can easily be proven by induction using (20) .
Proof of Proposition 5.3. Fix α ∈ Comp \alpha\in\operatorname{Comp} α ∈ Comp γ ∈ Comp \gamma\in\operatorname{Comp} γ ∈ Comp α \alpha α α = ( a 1 , a 2 , … , a k ) \alpha=(a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k ) k \mathbf{k} k ξ α : QSym k → QSym k \xi_\alpha:\operatorname{QSym}_{\mathbf{k}}\to\operatorname{QSym}_{\mathbf{k}} ξ α : QSym k → QSym k H = QSym k H=\operatorname{QSym}_{\mathbf{k}} H = QSym k
We shall prove that
ξ α ( M γ ) = ∑ A ∈ N r e d ∙ , ∙ ( read A ) r e d = γ row A = α M column A . (21) \xi_\alpha(M_\gamma)=
\sum_{\substack{A\in\mathbb{N}_{\mathrm{red}}^{\bullet,\bullet}\\
(\operatorname{read} A)_{\mathrm{red}}=\gamma\\
\operatorname{row} A=\alpha}}
M_{\operatorname{column} A}. \tag*{(21)} ξ α ( M γ ) = A ∈ N red ∙ , ∙ ( read A ) red = γ row A = α ∑ M column A . (21) Proof of (22): The definition of ξ ~ α \widetilde{\xi}_{\alpha} ξ α ξ ~ α = m ( k − 1 ) ∘ π α ∘ Δ ( k − 1 ) \widetilde{\xi}_{\alpha}=m^{(k-1)}\circ\pi_{\alpha}\circ\Delta^{(k-1)} ξ α = m ( k − 1 ) ∘ π α ∘ Δ ( k − 1 )
ξ ~ α ( M γ ) = ( m ( k − 1 ) ∘ π α ∘ Δ ( k − 1 ) ) ( M γ ) = ( m ( k − 1 ) ∘ π α ) ( Δ ( k − 1 ) ( M γ ) ) ⏟ ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k = ( m ( k − 1 ) ∘ π α ) ( ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k ) = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ m ( k − 1 ) ( π α ( M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k ) ) = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ m ( k − 1 ) ( π α 1 ( M γ 1 ) ⊗ π α 2 ( M γ 2 ) ⊗ ⋯ ⊗ π α k ( M γ k ) ) = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ m ( k − 1 ) ( π α 1 ( M γ 1 ) ⊗ π α 2 ( M γ 2 ) ⊗ ⋯ ⊗ π α k ( M γ k ) ) ⏟ = π α 1 ( M γ 1 ) ⋅ π α 2 ( M γ 2 ) ⋯ π α k ( M γ k ) = ∏ g = 1 k π α g ( M γ g ) = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∏ g = 1 k π α g ( M γ g ) ⏟ = { M γ g , if ∣ γ g ∣ = a g ; 0 , if ∣ γ g ∣ ≠ a g (since the power series M γ g is homogeneous of degree ∣ γ g ∣ ) \begin{aligned}
\widetilde{\xi}_{\alpha}(M_{\gamma})
&=\left(m^{(k-1)}\circ\pi_{\alpha}\circ\Delta^{(k-1)}\right)(M_{\gamma})\\
&=\left(m^{(k-1)}\circ\pi_{\alpha}\right)
\underbrace{\left(\Delta^{(k-1)}(M_{\gamma})\right)}_{ \sum_{\substack{(\gamma_{1},\gamma_{2},\ldots,\gamma_{k})\in\operatorname{Comp}^{k};\\ \gamma_{1}\gamma_{2}\cdots\gamma_{k}=\gamma}}M_{\gamma_{1}}\otimes M_{\gamma_{2}}\otimes\cdots\otimes M_{\gamma_{k}}}\\
&=\left(m^{(k-1)}\circ\pi_{\alpha}\right)\left(\sum_{\substack{(\gamma_{1},\gamma_{2},\ldots,\gamma_{k})\in\operatorname{Comp}^{k};\\ \gamma_{1}\gamma_{2}\cdots\gamma_{k}=\gamma}}M_{\gamma_{1}}\otimes M_{\gamma_{2}}\otimes\cdots\otimes M_{\gamma_{k}}\right)\\
&=\sum_{\substack{(\gamma_{1},\gamma_{2},\ldots,\gamma_{k})\in\operatorname{Comp}^{k};\\ \gamma_{1}\gamma_{2}\cdots\gamma_{k}=\gamma}}m^{(k-1)}\left(\pi_{\alpha}(M_{\gamma_{1}}\otimes M_{\gamma_{2}}\otimes\cdots\otimes M_{\gamma_{k}})\right)\\
&=\sum_{\substack{(\gamma_{1},\gamma_{2},\ldots,\gamma_{k})\in\operatorname{Comp}^{k};\\ \gamma_{1}\gamma_{2}\cdots\gamma_{k}=\gamma}}m^{(k-1)}\left(\pi_{\alpha_{1}}(M_{\gamma_{1}})\otimes\pi_{\alpha_{2}}(M_{\gamma_{2}})\otimes\cdots\otimes\pi_{\alpha_{k}}(M_{\gamma_{k}})\right)\\
&=\sum_{\substack{(\gamma_{1},\gamma_{2},\ldots,\gamma_{k})\in\operatorname{Comp}^{k};\\ \gamma_{1}\gamma_{2}\cdots\gamma_{k}=\gamma}}\underbrace{m^{(k-1)}\left(\pi_{\alpha_{1}}(M_{\gamma_{1}})\otimes\pi_{\alpha_{2}}(M_{\gamma_{2}})\otimes\cdots\otimes\pi_{\alpha_{k}}(M_{\gamma_{k}})\right)}_{ =\pi_{\alpha_{1}}(M_{\gamma_{1}})\cdot\pi_{\alpha_{2}}(M_{\gamma_{2}})\cdots\pi_{\alpha_{k}}(M_{\gamma_{k}})=\prod_{g=1}^{k}\pi_{\alpha_{g}}(M_{\gamma_{g}})}\\
&=\sum_{\substack{(\gamma_{1},\gamma_{2},\ldots,\gamma_{k})\in\operatorname{Comp}^{k};\\ \gamma_{1}\gamma_{2}\cdots\gamma_{k}=\gamma}}\prod_{g=1}^{k}\underbrace{\pi_{\alpha_{g}}(M_{\gamma_{g}})}_{ =\begin{cases}M_{\gamma_{g}},&\text{if }|\gamma_{g}|=a_{g};\\0,&\text{if }|\gamma_{g}|\ne a_{g}\end{cases}}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\text{(since the power series $M_{\gamma_{g}}$ is homogeneous of degree $|\gamma_{g}|$)}
\end{aligned} ξ α ( M γ ) = ( m ( k − 1 ) ∘ π α ∘ Δ ( k − 1 ) ) ( M γ ) = ( m ( k − 1 ) ∘ π α ) ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k ( Δ ( k − 1 ) ( M γ ) ) = ( m ( k − 1 ) ∘ π α ) ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ m ( k − 1 ) ( π α ( M γ 1 ⊗ M γ 2 ⊗ ⋯ ⊗ M γ k ) ) = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ m ( k − 1 ) ( π α 1 ( M γ 1 ) ⊗ π α 2 ( M γ 2 ) ⊗ ⋯ ⊗ π α k ( M γ k ) ) = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ = π α 1 ( M γ 1 ) ⋅ π α 2 ( M γ 2 ) ⋯ π α k ( M γ k ) = ∏ g = 1 k π α g ( M γ g ) m ( k − 1 ) ( π α 1 ( M γ 1 ) ⊗ π α 2 ( M γ 2 ) ⊗ ⋯ ⊗ π α k ( M γ k ) ) = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ g = 1 ∏ k = { M γ g , 0 , if ∣ γ g ∣ = a g ; if ∣ γ g ∣ = a g π α g ( M γ g ) (since the power series M γ g is homogeneous of degree ∣ γ g ∣) = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ C o m p k ; γ 1 γ 2 ⋯ γ k = γ ∏ g = 1 k { M γ g , if ∣ γ g ∣ = a g ; 0 , if ∣ γ g ∣ ≠ a g = { ∏ g = 1 k M γ g , if ∣ γ g ∣ = a g for all g ; 0 , otherwise = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ C o m p k ; γ 1 γ 2 ⋯ γ k = γ { ∏ g = 1 k M γ g , if ∣ γ g ∣ = a g for all g ; 0 , otherwise = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ C o m p k ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∏ g = 1 k M γ g = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ C o m p k ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g M γ 1 M γ 2 ⋯ M γ k = ∑ A ∈ N C r e d k , ∗ ( A g , ∙ ) r e d = γ g for each g (by Proposition 5.4, applied to α i = γ i ) \begin{aligned}
&= \sum_{\substack{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\mathrm{Comp}^k;\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma}}
\prod_{g=1}^{k}
\begin{cases}
M_{\gamma_g}, & \text{if } \lvert\gamma_g\rvert=a_g;\\
0, & \text{if } \lvert\gamma_g\rvert\ne a_g
\end{cases}\\
&= \begin{cases}
\prod_{g=1}^{k}M_{\gamma_g}, & \text{if } \lvert\gamma_g\rvert=a_g\text{ for all }g;\\
0, & \text{otherwise}
\end{cases}\\
&= \sum_{\substack{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\mathrm{Comp}^k;\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma}}
\begin{cases}
\prod_{g=1}^{k}M_{\gamma_g}, & \text{if } \lvert\gamma_g\rvert=a_g\text{ for all }g;\\
0, & \text{otherwise}
\end{cases}\\
&= \sum_{\substack{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\mathrm{Comp}^k;\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma;\\ \lvert\gamma_g\rvert=a_g\text{ for all }g}}
\prod_{g=1}^{k}M_{\gamma_g}\\
&= \sum_{\substack{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\mathrm{Comp}^k;\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma;\\ \lvert\gamma_g\rvert=a_g\text{ for all }g}}
M_{\gamma_1}M_{\gamma_2}\cdots M_{\gamma_k}\\
&= \sum_{A\in\mathbb{N}_{\mathrm{Cred}}^{k,*}}(A_g,\bullet)^{\mathrm{red}}=\gamma_g\text{ for each }g\\
&\qquad\text{(by Proposition 5.4, applied to }\alpha_i=\gamma_i\text{)}
\end{aligned} = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ g = 1 ∏ k { M γ g , 0 , if ∣ γ g ∣ = a g ; if ∣ γ g ∣ = a g = { ∏ g = 1 k M γ g , 0 , if ∣ γ g ∣ = a g for all g ; otherwise = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ∑ { ∏ g = 1 k M γ g , 0 , if ∣ γ g ∣ = a g for all g ; otherwise = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∑ g = 1 ∏ k M γ g = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∑ M γ 1 M γ 2 ⋯ M γ k = A ∈ N Cred k , ∗ ∑ ( A g , ∙ ) red = γ g for each g (by Proposition 5.4, applied to α i = γ i ) (here, we have filtered out the zero addends)
= ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = γ g for each g M c o l u m n A = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = γ g for each g ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g M c o l u m n A = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = γ g for each g ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g M c o l u m n A = ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = γ g for each g ; ( A 1 , ∙ ) r e d ( A 2 , ∙ ) r e d ⋯ ( A k , ∙ ) r e d = γ ; ∣ ( A g , ∙ ) r e d ∣ = a g for all g M c o l u m n A (here, we have replaced every γ g in the conditions ( γ 1 γ 2 ⋯ γ k = γ ) and ( ∣ γ g ∣ = a g for all g ) by the corresponding ( A g , ∙ ) r e d , because of the condition that ( A g , ∙ ) r e d = γ g ) = ∑ ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ∑ A ∈ N C r e d k , ∙ ( A g , ∙ ) r e d = γ g for each g ; ( A 1 , ∙ ) r e d ( A 2 , ∙ ) r e d ⋯ ( A k , ∙ ) r e d = γ ; ∣ ( A g , ∙ ) r e d ∣ = a g for all g M c o l u m n A = ∑ A ∈ N C r e d k , ∙ ( A 1 , ∙ ) r e d ( A 2 , ∙ ) r e d ⋯ ( A k , ∙ ) r e d = γ ; ∣ ( A g , ∙ ) r e d ∣ = a g for all g M c o l u m n A . (22) \begin{aligned}
&= \sum_{\substack{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\operatorname{Comp}^{k}\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma;\\ |\gamma_g|=a_g\ \text{for all }g}}
\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\ (A_{g,\bullet})^{\mathrm{red}}=\gamma_g\ \text{for each }g}}
M_{\mathrm{column}\ A}\\
&= \sum_{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\operatorname{Comp}^{k}}
\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\ (A_{g,\bullet})^{\mathrm{red}}=\gamma_g\ \text{for each }g;\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma;\\ |\gamma_g|=a_g\ \text{for all }g}}
M_{\mathrm{column}\ A}\\
&= \sum_{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\operatorname{Comp}^{k}}
\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\ (A_{g,\bullet})^{\mathrm{red}}=\gamma_g\ \text{for each }g;\\ \gamma_1\gamma_2\cdots\gamma_k=\gamma;\\ |\gamma_g|=a_g\ \text{for all }g}}
M_{\mathrm{column}\ A}\\
&= \sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\ (A_{g,\bullet})^{\mathrm{red}}=\gamma_g\ \text{for each }g;\\ (A_{1,\bullet})^{\mathrm{red}}(A_{2,\bullet})^{\mathrm{red}}\cdots(A_{k,\bullet})^{\mathrm{red}}=\gamma;\\ \left|(A_{g,\bullet})^{\mathrm{red}}\right|=a_g\ \text{for all }g}}
M_{\mathrm{column}\ A}\\
&\qquad\text{(here, we have replaced every }\gamma_g\text{ in the conditions }(\gamma_1\gamma_2\cdots\gamma_k=\gamma)\text{ and }(|\gamma_g|=a_g\text{ for all }g)\text{ by the corresponding }(A_{g,\bullet})^{\mathrm{red}},\text{ because}\\
&\qquad\text{of the condition that }(A_{g,\bullet})^{\mathrm{red}}=\gamma_g\text{)}\\
&= \sum_{(\gamma_1,\gamma_2,\ldots,\gamma_k)\in\operatorname{Comp}^{k}}
\sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\ (A_{g,\bullet})^{\mathrm{red}}=\gamma_g\ \text{for each }g;\\ (A_{1,\bullet})^{\mathrm{red}}(A_{2,\bullet})^{\mathrm{red}}\cdots(A_{k,\bullet})^{\mathrm{red}}=\gamma;\\ \left|(A_{g,\bullet})^{\mathrm{red}}\right|=a_g\ \text{for all }g}}
M_{\mathrm{column}\ A}\\
&= \sum_{\substack{A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet}\\ (A_{1,\bullet})^{\mathrm{red}}(A_{2,\bullet})^{\mathrm{red}}\cdots(A_{k,\bullet})^{\mathrm{red}}=\gamma;\\ \left|(A_{g,\bullet})^{\mathrm{red}}\right|=a_g\ \text{for all }g}}
M_{\mathrm{column}\ A}. \tag*{(22)}
\end{aligned} = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∑ A ∈ N Cred k , ∙ ( A g , ∙ ) red = γ g for each g ∑ M column A = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ∑ A ∈ N Cred k , ∙ ( A g , ∙ ) red = γ g for each g ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∑ M column A = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ∑ A ∈ N Cred k , ∙ ( A g , ∙ ) red = γ g for each g ; γ 1 γ 2 ⋯ γ k = γ ; ∣ γ g ∣ = a g for all g ∑ M column A = A ∈ N Cred k , ∙ ( A g , ∙ ) red = γ g for each g ; ( A 1 , ∙ ) red ( A 2 , ∙ ) red ⋯ ( A k , ∙ ) red = γ ; ∣ ( A g , ∙ ) red ∣ = a g for all g ∑ M column A (here, we have replaced every γ g in the conditions ( γ 1 γ 2 ⋯ γ k = γ ) and ( ∣ γ g ∣ = a g for all g ) by the corresponding ( A g , ∙ ) red , because of the condition that ( A g , ∙ ) red = γ g ) = ( γ 1 , γ 2 , … , γ k ) ∈ Comp k ∑ A ∈ N Cred k , ∙ ( A g , ∙ ) red = γ g for each g ; ( A 1 , ∙ ) red ( A 2 , ∙ ) red ⋯ ( A k , ∙ ) red = γ ; ∣ ( A g , ∙ ) red ∣ = a g for all g ∑ M column A = A ∈ N Cred k , ∙ ( A 1 , ∙ ) red ( A 2 , ∙ ) red ⋯ ( A k , ∙ ) red = γ ; ∣ ( A g , ∙ ) red ∣ = a g for all g ∑ M column A . (22) Now, we observe that every A ∈ N C r e d k , ∙ A\in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet} A ∈ N Cred k , ∙
( A 1 , ∙ ) r e d ( A 2 , ∙ ) r e d ⋯ ( A k , ∙ ) r e d = ( read A ) r e d (23) (A_{1,\bullet})^{\mathrm{red}}(A_{2,\bullet})^{\mathrm{red}}\cdots(A_{k,\bullet})^{\mathrm{red}}=(\operatorname{read} A)^{\mathrm{red}} \tag*{(23)} ( A 1 , ∙ ) red ( A 2 , ∙ ) red ⋯ ( A k , ∙ ) red = ( read A ) red (23) Also, for every A ∈ N C r e d k , ∙ A \in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet} A ∈ N Cred k , ∙
( ∣ ( A g , ∙ ) r e d ∣ = a g for all g ) ⟺ ( row A = α ) (24) \left(\left|\left(A_{g,\bullet}\right)^{\mathrm{red}}\right|=a_g\text{ for all }g\right)\Longleftrightarrow(\text{row }A=\alpha) \tag*{(24)} ( ( A g , ∙ ) red = a g for all g ) ⟺ ( row A = α ) (24) Also, every A ∈ N C r e d k , ∙ A \in\mathbb{N}_{\mathrm{Cred}}^{k,\bullet} A ∈ N Cred k , ∙ A = α A=\alpha A = α N r e d ∙ , ∙ \mathbb{N}_{\mathrm{red}}^{\bullet,\bullet} N red ∙ , ∙ 16 A ∈ N r e d ∙ , ∙ A \in\mathbb{N}_{\mathrm{red}}^{\bullet,\bullet} A ∈ N red ∙ , ∙ A = α A=\alpha A = α N C r e d k , ∙ \mathbb{N}_{\mathrm{Cred}}^{k,\bullet} N Cred k , ∙ 17
Let N ∙ \mathbb{N}^{\bullet} N ∙ Comp ⊆ N ∙ \operatorname{Comp} \subseteq\mathbb{N}^{\bullet} Comp ⊆ N ∙ Comp \operatorname{Comp} Comp N ∙ \mathbb{N}^{\bullet} N ∙ α = ( a 1 , a 2 , … , a n ) \alpha=(a_1,a_2,\ldots,a_n) α = ( a 1 , a 2 , … , a n ) β = ( b 1 , b 2 , … , b m ) \beta=(b_1,b_2,\ldots,b_m) β = ( b 1 , b 2 , … , b m ) α β = ( a 1 , a 2 , … , a n , b 1 , b 2 , … , b m ) \alpha\beta=(a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_m) α β = ( a 1 , a 2 , … , a n , b 1 , b 2 , … , b m ) N ∙ \mathbb{N}^{\bullet} N ∙ N \mathbb{N} N A A A
read A = A 1 , ∙ A 2 , ∙ ⋯ A k , ∙ . \operatorname{read} A=A_{1,\bullet}A_{2,\bullet}\cdots A_{k,\bullet}. read A = A 1 , ∙ A 2 , ∙ ⋯ A k , ∙ . Clearly, the map N ∙ → Comp \mathbb{N}^{\bullet}\to\operatorname{Comp} N ∙ → Comp β ↦ β r e d \beta\mapsto\beta^{\mathrm{red}} β ↦ β red
( A 1 , ∙ ) r e d ( A 2 , ∙ ) r e d ⋯ ( A k , ∙ ) r e d = ( A 1 , ∙ A 2 , ∙ ⋯ A k , ∙ ⏟ = read A ) r e d = ( read A ) r e d . \left(A_{1,\bullet}\right)^{\mathrm{red}}\left(A_{2,\bullet}\right)^{\mathrm{red}}\cdots\left(A_{k,\bullet}\right)^{\mathrm{red}}
=
\left(\underbrace{A_{1,\bullet}A_{2,\bullet}\cdots A_{k,\bullet}}_{=\operatorname{read} A}\right)^{\mathrm{red}}
=(\operatorname{read} A)^{\mathrm{red}}. ( A 1 , ∙ ) red ( A 2 , ∙ ) red ⋯ ( A k , ∙ ) red = = read A A 1 , ∙ A 2 , ∙ ⋯ A k , ∙ red = ( read A ) red . This proves (24).
∣ ( A g , ∙ ) r e d ∣ = ( sum of all entries of ( A g , ∙ ) r e d ) = ( sum of all nonzero entries in A g , ∙ ) \left|\left(A_{g,\bullet}\right)^{\mathrm{red}}\right|
=
\left(\text{sum of all entries of }\left(A_{g,\bullet}\right)^{\mathrm{red}}\right)
=
\left(\text{sum of all nonzero entries in }A_{g,\bullet}\right) ( A g , ∙ ) red = ( sum of all entries of ( A g , ∙ ) red ) = ( sum of all nonzero entries in A g , ∙ ) ( by the definition of ( A g , ∙ ) r e d ) = ( sum of all nonzero entries in the g -th row of A ) = ( sum of all entries in the g -th row of A ) = ( the g -th entry of row A ) . \begin{aligned}
\left(\text{by the definition of }\left(A_{g,\bullet}\right)^{\mathrm{red}}\right) \\
&=
\left(\text{sum of all nonzero entries in the }g\text{-th row of }A\right)
=
\left(\text{sum of all entries in the }g\text{-th row of }A\right)
=
\left(\text{the }g\text{-th entry of row }A\right).
\end{aligned} ( by the definition of ( A g , ∙ ) red ) = ( sum of all nonzero entries in the g -th row of A ) = ( sum of all entries in the g -th row of A ) = ( the g -th entry of row A ) . Hence, we have the following chain of equivalences:
( ∣ ( A g , ∙ ) r e d ∣ ⏟ = the g -th entry of row A = a g for all g ) \left(\underbrace{\left|\left(A_{g,\bullet}\right)^{\mathrm{red}}\right|}_{=\text{the }g\text{-th entry of row }A}=a_g\text{ for all }g\right) = the g -th entry of row A ( A g , ∙ ) red = a g for all g ⟺ ( ( the g -th entry of row A ) = a g for all g ) \Longleftrightarrow
\left((\text{the }g\text{-th entry of row }A)=a_g\text{ for all }g\right) ⟺ ( ( the g -th entry of row A ) = a g for all g ) ⟺ ( row A = ( a 1 , a 2 , … , a k ) ⏟ = α ) = ( row A = α ) . \Longleftrightarrow
\left(\text{row }A=\underbrace{(a_1,a_2,\ldots,a_k)}_{=\alpha}\right)
=
(\text{row }A=\alpha). ⟺ ( row A = = α ( a 1 , a 2 , … , a k ) ) = ( row A = α ) . This proves (25). two observations, we see that
( the matrices A ∈ N C r e d k , ∙ satisfying row A = α are precisely the matrices A ∈ N r e d ∙ , ∙ satisfying row A = α ) . (25) \left(
\begin{gathered}
\text{the matrices } A \in\mathbb{N}^{k,\bullet}_{\mathrm{Cred}} \text{ satisfying row } A=\alpha\\
\text{are precisely the matrices } A \in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}} \text{ satisfying row } A=\alpha
\end{gathered}
\right). \tag*{(25)} ( the matrices A ∈ N Cred k , ∙ satisfying row A = α are precisely the matrices A ∈ N red ∙ , ∙ satisfying row A = α ) . (25) Now, (23) becomes
ξ α ( M γ ) = ∑ A ∈ N C r e d k , ∙ ; ( A 1 , ∙ ) r e d ( A 2 , ∙ ) r e d ⋯ ( A k , ∙ ) r e d = γ ; ∣ ( A g , ∙ ) r e d ∣ = a g for all g M column A = ∑ A ∈ N C r e d k , ∙ ; ( read A ) r e d = γ ; row A = α M column A = ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ ; row A = α M column A . \begin{aligned}
\xi_{\alpha}(M_{\gamma})
&=
\sum_{\substack{A\in\mathbb{N}^{k,\bullet}_{\mathrm{Cred}};\\
(A_{1,\bullet})^{\mathrm{red}}(A_{2,\bullet})^{\mathrm{red}}\cdots(A_{k,\bullet})^{\mathrm{red}}=\gamma;\\
\left|(A_{g,\bullet})^{\mathrm{red}}\right|=a_g\text{ for all }g}}
M_{\operatorname{column} A}\\
&=
\sum_{\substack{A\in\mathbb{N}^{k,\bullet}_{\mathrm{Cred}};\\
(\operatorname{read} A)^{\mathrm{red}}=\gamma;\\
\operatorname{row} A=\alpha}}
M_{\operatorname{column} A}
=
\sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\
(\operatorname{read} A)^{\mathrm{red}}=\gamma;\\
\operatorname{row} A=\alpha}}
M_{\operatorname{column} A}.
\end{aligned} ξ α ( M γ ) = A ∈ N Cred k , ∙ ; ( A 1 , ∙ ) red ( A 2 , ∙ ) red ⋯ ( A k , ∙ ) red = γ ; ∣ ( A g , ∙ ) red ∣ = a g for all g ∑ M column A = A ∈ N Cred k , ∙ ; ( read A ) red = γ ; row A = α ∑ M column A = A ∈ N red ∙ , ∙ ; ( read A ) red = γ ; row A = α ∑ M column A . This proves (22).
rows of A A A A A A k k k row A = α = ( a 1 , a 2 , … , a k ) \operatorname{row} A=\alpha=(a_1,a_2,\ldots,a_k) row A = α = ( a 1 , a 2 , … , a k ) A A A k k k A A A A ∈ N r e d ∙ , ∙ A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}} A ∈ N red ∙ , ∙ A ∈ N C r e d k , ∙ A\in\mathbb{N}^{k,\bullet}_{\mathrm{Cred}} A ∈ N Cred k , ∙ A A A A A A k k k
Now, forget that we fixed α \alpha α γ \gamma γ γ ∈ C o m p \gamma\in\mathrm{Comp} γ ∈ Comp
β Q S y m k ( M γ ) = ∑ α ∈ C o m p ξ α ( M γ ) ⊗ M α (by the definition of β Q S y m k ) = ∑ α ∈ C o m p ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ ; row A = α M column A ⊗ M α (by (22)) = ∑ α ∈ C o m p ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ ; row A = α M column A ⊗ M row A = ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ ; row A ∈ C o m p M column A ⊗ M row A . (26) \begin{aligned}
\beta_{\mathrm{QSym}_k}(M_\gamma)
&= \sum_{\alpha\in\mathrm{Comp}} \xi_\alpha(M_\gamma)\otimes M_\alpha
&&\text{(by the definition of $\beta_{\mathrm{QSym}_k}$)}\\
&= \sum_{\alpha\in\mathrm{Comp}}
\sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma;\\\operatorname{row} A=\alpha}}
M_{\operatorname{column} A}\otimes M_\alpha
&&\text{(by (22))}\\
&= \sum_{\alpha\in\mathrm{Comp}}
\sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma;\\\operatorname{row} A=\alpha}}
M_{\operatorname{column} A}\otimes M_{\operatorname{row} A}\\
&= \sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma;\\\operatorname{row} A\in\mathrm{Comp}}}
M_{\operatorname{column} A}\otimes M_{\operatorname{row} A}. \tag*{(26)}
\end{aligned} β QSym k ( M γ ) = α ∈ Comp ∑ ξ α ( M γ ) ⊗ M α = α ∈ Comp ∑ A ∈ N red ∙ , ∙ ; ( read A ) red = γ ; row A = α ∑ M column A ⊗ M α = α ∈ Comp ∑ A ∈ N red ∙ , ∙ ; ( read A ) red = γ ; row A = α ∑ M column A ⊗ M row A = A ∈ N red ∙ , ∙ ; ( read A ) red = γ ; row A ∈ Comp ∑ M column A ⊗ M row A . (by the definition of β QSym k ) (by (22)) (26) But every A ∈ N r e d ∙ , ∙ A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}} A ∈ N red ∙ , ∙ row A ∈ C o m p \operatorname{row} A\in\mathrm{Comp} row A ∈ Comp 20 (26) can be replaced by ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ \sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma}} ∑ A ∈ N red ∙ , ∙ ; ( read A ) red = γ
β Q S y m k ( M γ ) = ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ ; row A ∈ C o m p M column A ⊗ M row A = ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ M column A ⊗ M row A . (27) \begin{aligned}
\beta_{\mathrm{QSym}_k}(M_\gamma)
&= \sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma;\\\operatorname{row} A\in\mathrm{Comp}}}
M_{\operatorname{column} A}\otimes M_{\operatorname{row} A}\\
&= \sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma}}
M_{\operatorname{column} A}\otimes M_{\operatorname{row} A}. \tag*{(27)}
\end{aligned} β QSym k ( M γ ) = A ∈ N red ∙ , ∙ ; ( read A ) red = γ ; row A ∈ Comp ∑ M column A ⊗ M row A = A ∈ N red ∙ , ∙ ; ( read A ) red = γ ∑ M column A ⊗ M row A . (27) On the other hand, every γ ∈ Comp \gamma\in\operatorname{Comp} γ ∈ Comp
Δ p ′ ( M γ ) = ( τ ∘ Δ p ) ( M γ ) = τ ( ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ M r o w A ⊗ M c o l u m n A ) = τ ( ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ M r o w A ⊗ M c o l u m n A ) = ∑ A ∈ N r e d ∙ , ∙ ; ( read A ) r e d = γ M c o l u m n A ⊗ M r o w A (by the definition of τ ) = β Q S y m k ( M γ ) (by xref‘(28)‘). \begin{aligned}
\Delta'_{p}(M_{\gamma}) &= (\tau\circ\Delta_{p})(M_{\gamma}) = \tau\left(\sum_{\substack{A \in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma}} M_{\mathrm{row}\,A} \otimes M_{\mathrm{column}\,A}\right) \\
&= \tau\left(\sum_{\substack{A \in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma}} M_{\mathrm{row}\,A} \otimes M_{\mathrm{column}\,A}\right) \\
&= \sum_{\substack{A \in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\(\operatorname{read} A)^{\mathrm{red}}=\gamma}} M_{\mathrm{column}\,A} \otimes M_{\mathrm{row}\,A} \qquad\text{(by the definition of $\tau$)} \\
&= \beta_{\mathrm{QSym}_{k}}(M_{\gamma}) \qquad\text{(by {xref}`(28)`).}
\end{aligned} Δ p ′ ( M γ ) = ( τ ∘ Δ p ) ( M γ ) = τ A ∈ N red ∙ , ∙ ; ( read A ) red = γ ∑ M row A ⊗ M column A = τ A ∈ N red ∙ , ∙ ; ( read A ) red = γ ∑ M row A ⊗ M column A = A ∈ N red ∙ , ∙ ; ( read A ) red = γ ∑ M column A ⊗ M row A (by the definition of τ ) = β QSym k ( M γ ) (by xref ‘(28)‘). Since both maps Δ p ′ \Delta'_{p} Δ p ′ β Q S y m k \beta_{\mathrm{QSym}_{k}} β QSym k k k k Δ p ′ = β Q S y m k \Delta'_{p}=\beta_{\mathrm{QSym}_{k}} Δ p ′ = β QSym k ( M γ ) γ ∈ Comp (M_{\gamma})_{\gamma\in\operatorname{Comp}} ( M γ ) γ ∈ Comp k k k Q S y m k \mathrm{QSym}_{k} QSym k Proposition 5.3 . □
The next theorem is an analogue for Q S y m \mathrm{QSym} QSym
Theorem 5.7. Let k k k H H H k k k α \alpha α k k k ζ α : H ⟶ H \zeta_{\alpha}: H \longrightarrow H ζ α : H ⟶ H Definition 5.1 . Define a map β H : H ⟶ H ⊗ Q S y m k \beta_{H}: H \longrightarrow H \otimes\mathrm{QSym}_{k} β H : H ⟶ H ⊗ QSym k Definition 5.1 .
(a) The map β H \beta_{H} β H k k k H ⟶ H ⊗ Q S y m k H \longrightarrow H \otimes\mathrm{QSym}_{k} H ⟶ H ⊗ QSym k ( k , H ) (k,H) ( k , H )
(b) We have ( id ⊗ ε P ) ∘ β H = id (\operatorname{id} \otimes\varepsilon_{P}) \circ\beta_{H}=\operatorname{id} ( id ⊗ ε P ) ∘ β H = id id ⊗ ε P : H ⊗ Q S y m k ⟶ H ⊗ k \operatorname{id} \otimes\varepsilon_{P}: H \otimes\mathrm{QSym}_{k} \longrightarrow H \otimes k id ⊗ ε P : H ⊗ QSym k ⟶ H ⊗ k H ⊗ Q S y m k H \otimes\mathrm{QSym}_{k} H ⊗ QSym k H H H H ⊗ k H \otimes k H ⊗ k H H H
(c) Define a map Δ P ′ : Q S y m k ⟶ Q S y m k ⊗ Q S y m k \Delta'_{P}: \mathrm{QSym}_{k} \longrightarrow\mathrm{QSym}_{k} \otimes\mathrm{QSym}_{k} Δ P ′ : QSym k ⟶ QSym k ⊗ QSym k Definition 4.2 (e). The diagram
H → β H H ⊗ Q S y m k β H ↓ ↓ β H ⊗ id H ⊗ Q S y m k → id ⊗ Δ P ′ H ⊗ Q S y m k ⊗ Q S y m k \begin{CD}
H @>{\beta_{H}}>> H \otimes\mathrm{QSym}_{k} \\
@V{\beta_{H}}VV @VV{\beta_{H}\otimes\operatorname{id}}V \\
H \otimes\mathrm{QSym}_{k} @>{\operatorname{id}\otimes\Delta'_{P}}>> H \otimes\mathrm{QSym}_{k} \otimes\mathrm{QSym}_{k}
\end{CD} H β H ↓ ⏐ H ⊗ QSym k β H id ⊗ Δ P ′ H ⊗ QSym k ↓ ⏐ β H ⊗ id H ⊗ QSym k ⊗ QSym k is commutative.
(d) If the k-coalgebra H H H β H ( H ) \beta_H(H) β H ( H ) H ⊗ Λ k H \otimes\Lambda_k H ⊗ Λ k H ⊗ QSym k H \otimes\operatorname{QSym}_k H ⊗ QSym k Λ k \Lambda_k Λ k k k k
Parts (b) and (c) of Theorem 5.7 can be combined into “β H \beta_H β H H H H QSym 2 \operatorname{QSym}_2 QSym 2 QSym 2 \operatorname{QSym}_2 QSym 2 ( QSym , Δ P , ε P ) (\operatorname{QSym}, \Delta_P,\varepsilon_P) ( QSym , Δ P , ε P ) QSym 2 \operatorname{QSym}_2 QSym 2 Proposition 4.4 ).
What Hazewinkel actually calls the Bernstein homomorphism in [Haz08, §18.24] is the k-algebra homomorphism H → H ⊗ Λ k H \to H \otimes\Lambda_k H → H ⊗ Λ k β H : H → H ⊗ QSym k \beta_H : H \to H \otimes\operatorname{QSym}_k β H : H → H ⊗ QSym k H H H 21 Λ k \Lambda_k Λ k QSym k \operatorname{QSym}_k QSym k Proposition 5.3 .
Proof of Theorem 5.7. Set A = H A = H A = H ζ = id \zeta= \operatorname{id} ζ = id ζ α \zeta_\alpha ζ α Corollary 3.12 (c) is precisely the map ζ α \zeta_\alpha ζ α Definition 5.1 (because ζ ⊗ k = id ⊗ k = id \zeta^{\otimes k} = \operatorname{id}^{\otimes k} = \operatorname{id} ζ ⊗ k = id ⊗ k = id ζ α \zeta_\alpha ζ α
(a) Corollary 3.12 (a) shows that there exists a unique graded ( k , A ‾ ) (k,\underline{A}) ( k , A ) Ξ : H → A ‾ ⊗ QSym k \Xi: H \to\underline{A} \otimes\operatorname{QSym}_k Ξ : H → A ⊗ QSym k A = H A = H A = H ζ = id \zeta= \operatorname{id} ζ = id ( k , H ‾ ) (k,\underline{H}) ( k , H ) Ξ : H → H ⊗ QSym k \Xi: H \to H \otimes\operatorname{QSym}_k Ξ : H → H ⊗ QSym k
H → Ξ H ⊗ QSym k id ↓ ↓ id H ⊗ ε P H = H (28) \begin{CD}
H @>{\Xi}>> H \otimes\operatorname{QSym}_k \\
@V{\operatorname{id}}VV @VV{\operatorname{id}_H \otimes\varepsilon_P}V \\
H @= H \tag*{(28)}
\end{CD} H id ↓ ⏐ H Ξ H ⊗ QSym k ↓ ⏐ id H ⊗ ε P H (28) is commutative. Consider this Ξ \Xi Ξ Corollary 3.12 (c) shows that this homomorphism Ξ \Xi Ξ
Ξ ( h ) = ∑ α ∈ Comp ζ α ( h ) ⊗ M α for every h ∈ H . \Xi(h) = \sum_{\alpha\in\operatorname{Comp}} \zeta_\alpha(h) \otimes M_\alpha\qquad\text{for every } h \in H. Ξ ( h ) = α ∈ Comp ∑ ζ α ( h ) ⊗ M α for every h ∈ H . Comparing this equality with (13), we obtain Ξ ( h ) = β H ( h ) \Xi(h) = \beta_H(h) Ξ ( h ) = β H ( h ) h ∈ H h \in H h ∈ H Ξ = β H \Xi= \beta_H Ξ = β H β H \beta_H β H ( k , H ‾ ) (k,\underline{H}) ( k , H ) Ξ \Xi Ξ ( k , H ‾ ) (k,\underline{H}) ( k , H )
Corollary 3.12 (b) shows that Ξ \Xi Ξ β H \beta_H β H Ξ = β H \Xi= \beta_H Ξ = β H Theorem 5.7 (a).
(b) Consider the map Ξ \Xi Ξ Theorem 5.7 (a). We have shown that Ξ = β H \Xi= \beta_H Ξ = β H ( id ⊗ ε P ) ∘ Ξ = id (\operatorname{id} \otimes\varepsilon_P) \circ\Xi= \operatorname{id} ( id ⊗ ε P ) ∘ Ξ = id ( id ⊗ ε P ) ∘ β H = id (\operatorname{id} \otimes\varepsilon_P) \circ\beta_H = \operatorname{id} ( id ⊗ ε P ) ∘ β H = id Ξ = β H \Xi= \beta_H Ξ = β H Theorem 5.7 (b).
(c) Theorem 5.7 (a) shows that the map β H \beta_H β H k k k H → H ⊗ QSym k H \to H \otimes\operatorname{QSym}_k H → H ⊗ QSym k ( k , H ) (k,H) ( k , H ) Theorem 5.7 (a) (applied to QSym k \operatorname{QSym}_k QSym k H H H β QSym k \beta_{\operatorname{QSym}_k} β QSym k k k k QSym k → QSym k ⊗ QSym k \operatorname{QSym}_k \to\operatorname{QSym}_k \otimes\operatorname{QSym}_k QSym k → QSym k ⊗ QSym k ( k , QSym k ) (k,\operatorname{QSym}_k) ( k , QSym k ) Δ P ′ = β QSym k \Delta'_P = \beta_{\operatorname{QSym}_k} Δ P ′ = β QSym k Δ P ′ \Delta'_P Δ P ′ k k k QSym k → QSym k ⊗ QSym k \operatorname{QSym}_k \to\operatorname{QSym}_k \otimes\operatorname{QSym}_k QSym k → QSym k ⊗ QSym k ( k , QSym k ) (k,\operatorname{QSym}_k) ( k , QSym k )
Applying Corollary 3.12 (a) to H ⊗ QSym k H \otimes\operatorname{QSym}_k H ⊗ QSym k β H \beta_H β H A A A ζ \zeta ζ ( k , H ⊗ QSym k ) (k,H \otimes\operatorname{QSym}_k) ( k , H ⊗ QSym k ) Ξ : H → H ⊗ QSym k ⊗ QSym k \Xi: H \to H \otimes\operatorname{QSym}_k \otimes\operatorname{QSym}_k Ξ : H → H ⊗ QSym k ⊗ QSym k
H → Ξ H ⊗ QSym k ⊗ QSym k β H ↓ ↓ id H ⊗ QSym k ⊗ ε P H ⊗ QSym k = H ⊗ QSym k \begin{CD}
H @>{\Xi}>> H \otimes\operatorname{QSym}_k \otimes\operatorname{QSym}_k \\
@V{\beta_H}VV @VV{\operatorname{id}_{H \otimes\operatorname{QSym}_k} \otimes\varepsilon_P}V \\
H \otimes\operatorname{QSym}_k @= H \otimes\operatorname{QSym}_k
\end{CD} H β H ↓ ⏐ H ⊗ QSym k Ξ H ⊗ QSym k ⊗ QSym k ↓ ⏐ id H ⊗ QSym k ⊗ ε P H ⊗ QSym k (30)
is commutative. Thus, if we have two graded ( k , H ⊗ QSym k ) (k,H \otimes\operatorname{QSym}_k) ( k , H ⊗ QSym k ) Ξ : H → H ⊗ QSym k ⊗ QSym k \Xi: H \to H \otimes\operatorname{QSym}_k \otimes\operatorname{QSym}_k Ξ : H → H ⊗ QSym k ⊗ QSym k ( β H ⊗ id ) ∘ β H (\beta_H \otimes\operatorname{id}) \circ\beta_H ( β H ⊗ id ) ∘ β H ( id ⊗ Δ P ′ ) ∘ β H (\operatorname{id} \otimes\Delta'_P) \circ\beta_H ( id ⊗ Δ P ′ ) ∘ β H ( β H ⊗ id ) ∘ β H = ( id ⊗ Δ P ′ ) ∘ β H (\beta_H \otimes\operatorname{id}) \circ\beta_H = (\operatorname{id} \otimes\Delta'_P) \circ\beta_H ( β H ⊗ id ) ∘ β H = ( id ⊗ Δ P ′ ) ∘ β H Theorem 5.7 (c) will follow.
Recall that β H \beta_H β H Δ P ′ \Delta'_P Δ P ′ ( β H ⊗ id ) ∘ β H (\beta_H \otimes\operatorname{id}) \circ\beta_H ( β H ⊗ id ) ∘ β H ( id ⊗ Δ P ′ ) ∘ β H (\operatorname{id} \otimes\Delta'_P) \circ\beta_H ( id ⊗ Δ P ′ ) ∘ β H β H \beta_H β H ( k , H ) (k,H) ( k , H ) Δ P ′ \Delta'_P Δ P ′ ( k , QSym k ) (k,\operatorname{QSym}_k) ( k , QSym k ) ( β H ⊗ id ) ∘ β H (\beta_H \otimes\operatorname{id}) \circ\beta_H ( β H ⊗ id ) ∘ β H ( id ⊗ Δ P ′ ) ∘ β H (\operatorname{id} \otimes\Delta'_P) \circ\beta_H ( id ⊗ Δ P ′ ) ∘ β H ( k , H ⊗ QSym k ) (k,H \otimes\operatorname{QSym}_k) ( k , H ⊗ QSym k ) 22
H → ( β H ⊗ i d ) ∘ β H H ‾ ⊗ Q S y m k ‾ ⊗ Q S y m k β H ↓ ↓ i d H ⊗ Q S y m k ⊗ ε P H ‾ ⊗ Q S y m k ‾ = H ‾ ⊗ Q S y m k ‾ (29) \begin{CD}
H @>{(\beta_H\otimes\mathrm{id})\circ\beta_H}>> \underline{H}\otimes\underline{\mathrm{QSym}_k}\otimes\mathrm{QSym}_k\\
@V{\beta_H}VV @VV{\mathrm{id}_{H\otimes\mathrm{QSym}_k}\otimes\varepsilon_P}V\\
\underline{H}\otimes\underline{\mathrm{QSym}_k} @= \underline{H}\otimes\underline{\mathrm{QSym}_k} \tag*{(29)}
\end{CD} H β H ↓ ⏐ H ⊗ QSym k ( β H ⊗ id ) ∘ β H H ⊗ QSym k ⊗ QSym k ↓ ⏐ id H ⊗ QSym k ⊗ ε P H ⊗ QSym k (29) and
H → ( i d ⊗ Δ P ′ ) ∘ β H H ‾ ⊗ Q S y m k ‾ ⊗ Q S y m k β H ↓ ↓ i d H ⊗ Q S y m k ⊗ ε P H ‾ ⊗ Q S y m k ‾ = H ‾ ⊗ Q S y m k ‾ (30) \begin{CD}
H @>{(\mathrm{id}\otimes\Delta'_P)\circ\beta_H}>> \underline{H}\otimes\underline{\mathrm{QSym}_k}\otimes\mathrm{QSym}_k\\
@V{\beta_H}VV @VV{\mathrm{id}_{H\otimes\mathrm{QSym}_k}\otimes\varepsilon_P}V\\
\underline{H}\otimes\underline{\mathrm{QSym}_k} @= \underline{H}\otimes\underline{\mathrm{QSym}_k} \tag*{(30)}
\end{CD} H β H ↓ ⏐ H ⊗ QSym k ( id ⊗ Δ P ′ ) ∘ β H H ⊗ QSym k ⊗ QSym k ↓ ⏐ id H ⊗ QSym k ⊗ ε P H ⊗ QSym k (30) are commutative. This follows from the computations
( i d H ⊗ Q S y m k ⊗ ε P ) ∘ ( β H ⊗ i d ) ∘ β H ⏟ = β H ⊗ ε P = β H ∘ ( i d ⊗ ε P ) = β H ∘ ( i d ⊗ ε P ) ∘ β H ⏟ = i d (by Theorem 5.7 (b)) = β H \underbrace{\left(\mathrm{id}_{H\otimes\mathrm{QSym}_k}\otimes\varepsilon_P\right)\circ(\beta_H\otimes\mathrm{id})\circ\beta_H}_{=\beta_H\otimes\varepsilon_P=\beta_H\circ(\mathrm{id}\otimes\varepsilon_P)}
=\beta_H\circ\underbrace{(\mathrm{id}\otimes\varepsilon_P)\circ\beta_H}_{=\mathrm{id}\ \text{(by Theorem 5.7 (b))}}
=\beta_H = β H ⊗ ε P = β H ∘ ( id ⊗ ε P ) ( id H ⊗ QSym k ⊗ ε P ) ∘ ( β H ⊗ id ) ∘ β H = β H ∘ = id (by Theorem 5.7 (b)) ( id ⊗ ε P ) ∘ β H = β H and
( i d H ⊗ Q S y m k ⏟ = i d H ⊗ i d Q S y m k ⊗ ε P ) ∘ ( i d H ⊗ Δ P ′ ⏟ = β Q S y m k ) ∘ β H = ( i d H ⊗ i d Q S y m k ⊗ ε P ) ∘ ( i d H ⊗ β Q S y m k ) ⏟ = i d H ⊗ ( ( i d Q S y m k ⊗ ε P ) ∘ β Q S y m k ) ∘ β H = ( i d H ⊗ ( ( i d Q S y m k ⊗ ε P ) ∘ β Q S y m k ⏟ = i d (by Theorem 5.7 (b), applied to QSym k instead of H ) ) ∘ β H = ( i d H ⊗ i d ) ∘ β H = β H . \begin{aligned}
&\left(\underbrace{\mathrm{id}_{H\otimes\mathrm{QSym}_k}}_{=\mathrm{id}_H\otimes\mathrm{id}_{\mathrm{QSym}_k}}\otimes\varepsilon_P\right)\circ\left(\underbrace{\mathrm{id}_H\otimes\Delta'_P}_{=\beta_{\mathrm{QSym}_k}}\right)\circ\beta_H\\
&=\underbrace{\left(\mathrm{id}_H\otimes\mathrm{id}_{\mathrm{QSym}_k}\otimes\varepsilon_P\right)\circ\left(\mathrm{id}_H\otimes\beta_{\mathrm{QSym}_k}\right)}_{=\mathrm{id}_H\otimes\left((\mathrm{id}_{\mathrm{QSym}_k}\otimes\varepsilon_P)\circ\beta_{\mathrm{QSym}_k}\right)}\circ\beta_H\\
&=\left(\mathrm{id}_H\otimes\left(\underbrace{(\mathrm{id}_{\mathrm{QSym}_k}\otimes\varepsilon_P)\circ\beta_{\mathrm{QSym}_k}}_{=\mathrm{id}\ \text{(by Theorem 5.7 (b), applied to QSym}_k\text{ instead of }H}\right)\right)\circ\beta_H\\
&=(\mathrm{id}_H\otimes\mathrm{id})\circ\beta_H=\beta_H.
\end{aligned} = id H ⊗ id QSym k id H ⊗ QSym k ⊗ ε P ∘ = β QSym k id H ⊗ Δ P ′ ∘ β H = = id H ⊗ ( ( id QSym k ⊗ ε P ) ∘ β QSym k ) ( id H ⊗ id QSym k ⊗ ε P ) ∘ ( id H ⊗ β QSym k ) ∘ β H = id H ⊗ = id (by Theorem 5.7 (b), applied to QSym k instead of H ( id QSym k ⊗ ε P ) ∘ β QSym k ∘ β H = ( id H ⊗ id ) ∘ β H = β H . Thus, we know that ( β H ⊗ i d ) ∘ β H (\beta_H\otimes\mathrm{id})\circ\beta_H ( β H ⊗ id ) ∘ β H ( i d ⊗ Δ P ′ ) ∘ β H (\mathrm{id}\otimes\Delta'_P)\circ\beta_H ( id ⊗ Δ P ′ ) ∘ β H ( k , H ‾ ⊗ Q S y m k ‾ ) (\mathbf{k},\underline{H}\otimes\underline{\mathrm{QSym}_k}) ( k , H ⊗ QSym k ) Ξ : H ⟶ H ‾ ⊗ Q S y m k ‾ ⊗ Q S y m k \Xi:H\longrightarrow\underline{H}\otimes\underline{\mathrm{QSym}_k}\otimes\mathrm{QSym}_k Ξ : H ⟶ H ⊗ QSym k ⊗ QSym k
But we have shown before that any two such homomorphisms must be identical. Thus, we conclude that ( β H ⊗ id ) ∘ β H = ( id ⊗ Δ P ′ ) ∘ β H (\beta_H \otimes\operatorname{id}) \circ\beta_H = (\operatorname{id} \otimes\Delta'_P) \circ\beta_H ( β H ⊗ id ) ∘ β H = ( id ⊗ Δ P ′ ) ∘ β H
(d) Consider the map Ξ \Xi Ξ Ξ = β H \Xi= \beta_H Ξ = β H
Assume that H H H Ξ ( H ) \Xi(H) Ξ ( H ) A ⊗ Λ k A \otimes\Lambda_k A ⊗ Λ k A ⊗ QSym k A \otimes\operatorname{QSym}_k A ⊗ QSym k β H ( H ) \beta_H(H) β H ( H ) H ⊗ Λ k H \otimes\Lambda_k H ⊗ Λ k H ⊗ QSym k H \otimes\operatorname{QSym}_k H ⊗ QSym k Ξ = β H \Xi= \beta_H Ξ = β H A = H A = H A = H
(Alternatively, we could prove (d) by checking that for any element h h h H H H ξ α ( h ) \xi_\alpha(h) ξ α ( h ) H H H α \alpha α α \alpha α
Proof of Proposition 4.4. Let τ \tau τ τ QSym k , QSym k : QSym k ⊗ QSym k → QSym k ⊗ QSym k \tau_{\operatorname{QSym}_k,\operatorname{QSym}_k} : \operatorname{QSym}_k \otimes\operatorname{QSym}_k \to\operatorname{QSym}_k \otimes\operatorname{QSym}_k τ QSym k , QSym k : QSym k ⊗ QSym k → QSym k ⊗ QSym k τ ∘ τ = id \tau\circ\tau= \operatorname{id} τ ∘ τ = id τ ∘ Δ P ′ = τ ∘ Δ P ′ ⏟ = τ ∘ Δ P \tau\circ\Delta'_P = \underbrace{\tau\circ\Delta'_P}_{=\tau\circ\Delta_P} τ ∘ Δ P ′ = = τ ∘ Δ P τ ∘ Δ P ′
τ ∘ τ ⏟ = id ∘ Δ P = Δ P . \underbrace{\tau\circ\tau}_{=\operatorname{id}}\circ\Delta_P=\Delta_P. = id τ ∘ τ ∘ Δ P = Δ P . Theorem 5.7 (c) (applied to H = QSym k H = \operatorname{QSym}_k H = QSym k
QSym k → β QSym k QSym k ⊗ QSym k β QSym k ↓ ↓ β QSym k ⊗ id QSym k ⊗ QSym k → id ⊗ Δ P ′ QSym k ⊗ QSym k ⊗ QSym k \begin{CD}
\operatorname{QSym}_k @>{\beta_{\operatorname{QSym}_k}}>> \operatorname{QSym}_k \otimes\operatorname{QSym}_k \\
@V{\beta_{\operatorname{QSym}_k}}VV @VV{\beta_{\operatorname{QSym}_k}\otimes\operatorname{id}}V \\
\operatorname{QSym}_k \otimes\operatorname{QSym}_k @>>{\operatorname{id}\otimes\Delta'_P}> \operatorname{QSym}_k \otimes\operatorname{QSym}_k \otimes\operatorname{QSym}_k
\end{CD} QSym k β QSym k ↓ ⏐ QSym k ⊗ QSym k β QSym k id ⊗ Δ P ′ QSym k ⊗ QSym k ↓ ⏐ β QSym k ⊗ id QSym k ⊗ QSym k ⊗ QSym k is commutative. In other words, ( id ⊗ Δ P ′ ) ∘ β QSym k = ( β QSym k ⊗ id ) ∘ β QSym k (\operatorname{id} \otimes\Delta'_P) \circ\beta_{\operatorname{QSym}_k} = (\beta_{\operatorname{QSym}_k} \otimes\operatorname{id}) \circ\beta_{\operatorname{QSym}_k} ( id ⊗ Δ P ′ ) ∘ β QSym k = ( β QSym k ⊗ id ) ∘ β QSym k β QSym k = Δ P ′ \beta_{\operatorname{QSym}_k} = \Delta'_P β QSym k = Δ P ′ ( id ⊗ Δ P ′ ) ∘ Δ P ′ = ( Δ P ′ ⊗ id ) ∘ Δ P ′ (\operatorname{id} \otimes\Delta'_P) \circ\Delta'_P = (\Delta'_P \otimes\operatorname{id}) \circ\Delta'_P ( id ⊗ Δ P ′ ) ∘ Δ P ′ = ( Δ P ′ ⊗ id ) ∘ Δ P ′ Δ P ′ \Delta'_P Δ P ′ Δ P = τ ∘ Δ P ′ \Delta_P = \tau\circ\Delta'_P Δ P = τ ∘ Δ P ′ H → H ⊗ H H \to H \otimes H H → H ⊗ H τ H , H : H ⊗ H → H ⊗ H \tau_{H,H} : H \otimes H \to H \otimes H τ H , H : H ⊗ H → H ⊗ H ε P \varepsilon_P ε P Δ P \Delta_P Δ P k k k QSym k \operatorname{QSym}_k QSym k Δ P \Delta_P Δ P ε P \varepsilon_P ε P k k k k k k Δ P \Delta_P Δ P ε P \varepsilon_P ε P k k k ε P \varepsilon_P ε P ε P \varepsilon_P ε P f ∈ QSym k f \in\operatorname{QSym}_k f ∈ QSym k f ( 1 , 0 , 0 , 0 , … ) f(1,0,0,0,\ldots) f ( 1 , 0 , 0 , 0 , … ) Δ P \Delta_P Δ P k k k
The map β QSym k \beta_{\operatorname{QSym}_k} β QSym k k k k QSym k → QSym k ⊗ QSym k \operatorname{QSym}_k \to\operatorname{QSym}_k \otimes\operatorname{QSym}_k QSym k → QSym k ⊗ QSym k H = QSym k H = \operatorname{QSym}_k H = QSym k Δ P ′ \Delta'_P Δ P ′ k k k QSym k → QSym k ⊗ QSym k \operatorname{QSym}_k \to\operatorname{QSym}_k \otimes\operatorname{QSym}_k QSym k → QSym k ⊗ QSym k β QSym k = Δ P ′ \beta_{\operatorname{QSym}_k} = \Delta'_P β QSym k = Δ P ′ QSym k = QSym k \operatorname{QSym}_k = \operatorname{QSym}_k QSym k = QSym k k k k Δ P = τ ∘ Δ P ′ \Delta_P = \tau\circ\Delta'_P Δ P = τ ∘ Δ P ′ k k k τ \tau τ Δ P ′ \Delta'_P Δ P ′ k k k
This completes the proof of Proposition 4.4.
Remark on antipodes
We have hitherto not really used the antipode of a Hopf algebra; thus, we could just as well have replaced the words “Hopf algebra” by “bialgebra” throughout the entire preceding text24
The antipode of any Hopf algebra H H H S H S_H S H
Proposition 6.1. Let k \mathbf{k} k A A A k \mathbf{k} k H H H k \mathbf{k} k G G G A A A k \mathbf{k} k f : H → G f:H\to G f : H → G ( k , A ) (\mathbf{k},A) ( k , A ) f ∘ S H = S G ∘ f f\circ S_H=S_G\circ f f ∘ S H = S G ∘ f
Proof of Proposition 6.1. We know that H H H k \mathbf{k} k A ⊗ H A\otimes H A ⊗ H A A A S A ⊗ H = id A ⊗ S H S_{A\otimes H}=\operatorname{id}_A\otimes S_H S A ⊗ H = id A ⊗ S H
Let f : H → G f:H\to G f : H → G k \mathbf{k} k ( k , A ) (\mathbf{k},A) ( k , A ) f ♯ : A ⊗ H → G f^\sharp:A\otimes H\to G f ♯ : A ⊗ H → G A A A f f f ( k , A ) (\mathbf{k},A) ( k , A ) A A A f ♯ f^\sharp f ♯ A A A A A A A A A A A A A A A f ♯ f^\sharp f ♯ f ♯ ∘ S A ⊗ H = S G ∘ f ♯ f^\sharp\circ S_{A\otimes H}=S_G\circ f^\sharp f ♯ ∘ S A ⊗ H = S G ∘ f ♯
Now, let ι \iota ι k \mathbf{k} k H → A ⊗ H H\to A\otimes H H → A ⊗ H h ↦ 1 ⊗ h h\mapsto1\otimes h h ↦ 1 ⊗ h ( id A ⊗ S H ) ∘ ι = ι ∘ S H (\operatorname{id}_A\otimes S_H)\circ\iota=\iota\circ S_H ( id A ⊗ S H ) ∘ ι = ι ∘ S H f ♯ ∘ ι = f f^\sharp\circ\iota=f f ♯ ∘ ι = f
f ∘ S H = f ♯ ∘ ι ∘ S H = f ♯ ∘ ( id A ⊗ S H ) ∘ ι = f ♯ ∘ S A ⊗ H ∘ ι = S G ∘ f ♯ ∘ ι = S G ∘ f . \begin{aligned}
f\circ S_H
&=f^\sharp\circ\iota\circ S_H\\
&=f^\sharp\circ(\operatorname{id}_A\otimes S_H)\circ\iota\\
&=f^\sharp\circ S_{A\otimes H}\circ\iota\\
&=S_G\circ f^\sharp\circ\iota\\
&=S_G\circ f.
\end{aligned} f ∘ S H = f ♯ ∘ ι ∘ S H = f ♯ ∘ ( id A ⊗ S H ) ∘ ι = f ♯ ∘ S A ⊗ H ∘ ι = S G ∘ f ♯ ∘ ι = S G ∘ f . This proves Proposition 6.1.
Corollary 6.2. Let k \mathbf{k} k H H H k \mathbf{k} k β H : H → H ⊗ QSym k \beta_H:H\to H\otimes\operatorname{QSym}_{\mathbf{k}} β H : H → H ⊗ QSym k
β H ∘ S H = ( id H ⊗ S QSym k ) ∘ β H . \beta_H\circ S_H=(\operatorname{id}_H\otimes S_{\operatorname{QSym}_{\mathbf{k}}})\circ\beta_H. β H ∘ S H = ( id H ⊗ S QSym k ) ∘ β H . Proof of Corollary 6.2. Theorem 5.7 (a) shows that the map β H \beta_H β H k \mathbf{k} k H → H ⊗ QSym k H \to H \otimes\operatorname{QSym}_{\mathbf{k}} H → H ⊗ QSym k ( k , H ) (\mathbf{k},H) ( k , H ) A = H A=H A = H G = H ⊗ QSym k G=H \otimes\operatorname{QSym}_{\mathbf{k}} G = H ⊗ QSym k f = β H f=\beta_H f = β H β H ∘ S H = S H ⊗ QSym k ∘ β H \beta_H \circ S_H=S_{H\otimes\operatorname{QSym}_{\mathbf{k}}}\circ\beta_H β H ∘ S H = S H ⊗ QSym k ∘ β H
But the H H H H ⊗ QSym k H \otimes\operatorname{QSym}_{\mathbf{k}} H ⊗ QSym k S H ⊗ QSym k = id H ⊗ S QSym k S_{H\otimes\operatorname{QSym}_{\mathbf{k}}}=\operatorname{id}_H\otimes S_{\operatorname{QSym}_{\mathbf{k}}} S H ⊗ QSym k = id H ⊗ S QSym k
β H ∘ S H = S H ⊗ QSym k ⏟ = id H ⊗ S QSym k ∘ β H = ( id H ⊗ S QSym k ) ∘ β H . \beta_H\circ S_H
=
\underbrace{S_{H\otimes\operatorname{QSym}_{\mathbf{k}}}}_{\mathrel{=\operatorname{id}_H\otimes S_{\operatorname{QSym}_{\mathbf{k}}}}}\circ\beta_H
=
(\operatorname{id}_H\otimes S_{\operatorname{QSym}_{\mathbf{k}}})\circ\beta_H. β H ∘ S H = = id H ⊗ S QSym k S H ⊗ QSym k ∘ β H = ( id H ⊗ S QSym k ) ∘ β H . This proves Corollary 6.2.
Corollary 6.3. Let k \mathbf{k} k H H H k \mathbf{k} k β H : H → H ⊗ QSym k \beta_H:H\to H\otimes\operatorname{QSym}_{\mathbf{k}} β H : H → H ⊗ QSym k
S H = ( id H ⊗ ( ε P ∘ S QSym k ) ) ∘ β H . S_H=\left(\operatorname{id}_H\otimes(\varepsilon_P\circ S_{\operatorname{QSym}_{\mathbf{k}}})\right)\circ\beta_H. S H = ( id H ⊗ ( ε P ∘ S QSym k ) ) ∘ β H . Proof of Corollary 6.3. We have
( id H ⊗ ( ε P ∘ S QSym k ) ) ∘ β H ⏟ = ( id H ⊗ ε P ) ∘ ( id H ⊗ S QSym k ) ∘ β H = ( id H ⊗ ε P ) ∘ ( id H ⊗ S QSym k ) ∘ β H ⏟ = β H ∘ S H (by Corollary 6.2) = ( id H ⊗ ε P ) ∘ β H ⏟ = id (by Theorem 5.7 (b)) ∘ S H = S H , \begin{aligned}
\underbrace{\left(\operatorname{id}_H\otimes(\varepsilon_P\circ S_{\operatorname{QSym}_{\mathbf{k}}})\right)\circ\beta_H}_{=(\operatorname{id}_H\otimes\varepsilon_P)\circ(\operatorname{id}_H\otimes S_{\operatorname{QSym}_{\mathbf{k}}})\circ\beta_H}
=
(\operatorname{id}_H\otimes\varepsilon_P)\circ
\underbrace{(\operatorname{id}_H\otimes S_{\operatorname{QSym}_{\mathbf{k}}})\circ\beta_H}_{=\beta_H\circ S_H\ \text{(by Corollary 6.2)}} \\
&=
\underbrace{(\operatorname{id}_H\otimes\varepsilon_P)\circ\beta_H}_{=\operatorname{id}\ \text{(by Theorem 5.7 (b))}}\circ S_H
=
S_H,
\end{aligned} = ( id H ⊗ ε P ) ∘ ( id H ⊗ S QSym k ) ∘ β H ( id H ⊗ ( ε P ∘ S QSym k ) ) ∘ β H = ( id H ⊗ ε P ) ∘ = β H ∘ S H (by Corollary 6.2) ( id H ⊗ S QSym k ) ∘ β H = = id (by Theorem 5.7 (b)) ( id H ⊗ ε P ) ∘ β H ∘ S H = S H , and thus Corollary 6.3 is proven.
Remark 6.4. What I find remarkable about Corollary 6.3 is that it provides a formula for the antipode S H S_H S H H H H β H \beta_H β H QSym k \operatorname{QSym}_{\mathbf{k}} QSym k H H H β H \beta_H β H QSym k \operatorname{QSym}_{\mathbf{k}} QSym k
Similar claims can be made about other endomorphisms of H H H k \mathbf{k} k Q \mathbb{Q} Q β H : H → H ⊗ QSym k \beta_H:H\to H\otimes\operatorname{QSym}_{\mathbf{k}} β H : H → H ⊗ QSym k k \mathbf{k} k H H H H H H H ⊗ QSym k ≅ QSym H H\otimes\operatorname{QSym}_{\mathbf{k}}\cong\operatorname{QSym}_H H ⊗ QSym k ≅ QSym H β H \beta_H β H ( β H ) ♯ : H ⊗ H → H ⊗ QSym k (\beta_H)^\sharp:H\otimes H\to H\otimes\operatorname{QSym}_{\mathbf{k}} ( β H ) ♯ : H ⊗ H → H ⊗ QSym k a ⊗ h a\otimes h a ⊗ h a β H ( h ) a\beta_H(h) a β H ( h ) H H H H H H H H H
this shows that β H \beta_H β H
Note that Corollary 6.3 is not completely new. Indeed, it can also be obtained from Takeuchi’s formula ([GriRei14, Proposition 1.4.24]) by breaking up the map f = id − u ϵ f=\operatorname{id}-u\epsilon f = id − u ϵ π 1 + π 2 + π 3 + ⋯ \pi_1+\pi_2+\pi_3+\cdots π 1 + π 2 + π 3 + ⋯
Questions and final remarks
I shall finish with some remarks and open questions which may or may not be worth further study.
Dualizing *QSym2
It is well-known (see, e.g., [GriRei14, §5.4]) that the graded Hopf-algebraic dual of the graded Hopf algebra QSym is a graded Hopf algebra NSym. The second comultiplication Δ P \Delta_P Δ P ε P \varepsilon_P ε P m P m_P m P u P u_P u P u P u_P u P k \mathbb{k} k k \mathbb{k} k k \mathbb{k} k ( N S y m , m P , u P , Δ , ε ) (\mathrm{NSym},m_P,u_P,\Delta,\varepsilon) ( NSym , m P , u P , Δ , ε ) u P u_P u P N S y m ( 2 ) \mathrm{NSym}^{(2)} NSym ( 2 )
m P ( H β ⊗ H γ ) = ∑ A ∈ N r e d ∙ , ∙ ; row A = β ; column A = γ H ( read A ) r e d m_P\left(H_\beta\otimes H_\gamma\right)=\sum_{\substack{A\in\mathbb{N}^{\bullet,\bullet}_{\mathrm{red}};\\ \operatorname{row} A=\beta;\\ \operatorname{column} A=\gamma}} H_{(\operatorname{read} A)^{\mathrm{red}}} m P ( H β ⊗ H γ ) = A ∈ N red ∙ , ∙ ; row A = β ; column A = γ ∑ H ( read A ) red for all β , γ ∈ Comp \beta,\gamma\in\operatorname{Comp} β , γ ∈ Comp
where ( H α ) α ∈ Comp (H_\alpha)_{\alpha\in\operatorname{Comp}} ( H α ) α ∈ Comp ( M α ) α ∈ Comp (M_\alpha)_{\alpha\in\operatorname{Comp}} ( M α ) α ∈ Comp internal product ∗ \ast ∗ N S y m → Λ \mathrm{NSym}\to\Lambda NSym → Λ H α H_\alpha H α h α h_\alpha h α m P m_P m P Λ \Lambda Λ
u P = H ( ) + H ( 1 ) + H ( 2 ) + H ( 3 ) + ⋯ u_P=H_{()}+H_{(1)}+H_{(2)}+H_{(3)}+\cdots u P = H ( ) + H ( 1 ) + H ( 2 ) + H ( 3 ) + ⋯ of the completion N S y m ^ \widehat{\mathrm{NSym}} NSym
Forgetting Δ \Delta Δ ε \varepsilon ε NSym ( 2 ) = ( NSym , m P , u P ) \operatorname{NSym}^{(2)} = (\operatorname{NSym},m_P,u_P) NSym ( 2 ) = ( NSym , m P , u P ) descent algebras of the symmetric groups S 0 , S 1 , S 2 , … S_0,S_1,S_2,\ldots S 0 , S 1 , S 2 , …
We can (more or less) dualize Theorem 5.7. As a result, instead of a QSym 2 \operatorname{QSym}_2 QSym 2 H H H NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) H H H H α ∈ NSym ( 2 ) H_\alpha\in\operatorname{NSym}^{(2)} H α ∈ NSym ( 2 )
π a 1 ⋆ π a 2 ⋆ ⋯ ⋆ π a k ∈ End H \pi_{a_1}\star\pi_{a_2}\star\cdots\star\pi_{a_k}\in\operatorname{End}H π a 1 ⋆ π a 2 ⋆ ⋯ ⋆ π a k ∈ End H for every composition α = ( a 1 , a 2 , … , a k ) \alpha=(a_1,a_2,\ldots,a_k) α = ( a 1 , a 2 , … , a k ) ⋆ \star ⋆ End H \operatorname{End}H End H NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) W \mathfrak{W} W NSym 2 \operatorname{NSym}_2 NSym 2 ( End g r a d e d H , ∘ ) (\operatorname{End}_{\mathrm{graded}} H,\circ) ( End graded H , ∘ ) H H H
Question 7.1. Is it possible to prove that this works using universal properties like I have done above for Theorem 5.7? (Just saying “dualize Theorem 5.7” is not enough, because dualization over arbitrary commutative rings is a heuristic, not a proof strategy; there does not seem to be a general theorem stating that “the dual of a correct result is correct”, at least when the result has assumptions about gradedness and similar things.)
If the answer is positive, can we use this to give a slick proof of Solomon’s Mackey formula? (I am not saying that there is a need for slick proofs of this formula – not after those by Gessel and Bidigare –, but it would be interesting to have a new one. I am thinking of letting both NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) T ( V ) T(V) T ( V ) k k k V V V
Note that if u u u v v v NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) NSym \operatorname{NSym} NSym u v uv uv NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) H H H u u u v v v NSym ( 2 ) → End H \operatorname{NSym}^{(2)}\to\operatorname{End}H NSym ( 2 ) → End H NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) NSym \operatorname{NSym} NSym
Natural transformations
Question 7.2. In Question 7.1, we found a k k k NSym ( 2 ) → ( End H , ∘ ) \operatorname{NSym}^{(2)}\to(\operatorname{End}H,\circ) NSym ( 2 ) → ( End H , ∘ ) H H H H H H NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 )
{ cocommutative connected graded Hopf algebras } ⟶ { k -modules } , H ↦ End H . \{\text{cocommutative connected graded Hopf algebras}\}\longrightarrow\{k\text{-modules}\}, \qquad H\mapsto\operatorname{End}H. { cocommutative connected graded Hopf algebras } ⟶ { k -modules } , H ↦ End H . Does the image of this action span (up to topology) the whole functor? I guess I am badly abusing categorical language here, so let me restate the question in simpler terms: If a natural endomorphism of the k k k H H H H H H H m H_m H m m m m v v v NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 ) v v v
If the answer is “No”, then does it change if we require the endomorphism of H H H k k k
What if we restrict ourselves to commutative cocommutative connected graded Hopf algebras? At least then, if k k k F q \mathbb{F}_q F q H H H x ↦ x q x \mapsto x^q x ↦ x q H H H k k k NSym ( 2 ) \operatorname{NSym}^{(2)} NSym ( 2 )
Dropping commutativity
Question 7.3. What are the natural endomorphisms of connected graded Hopf algebras, without any cocommutativity or commutativity assumption? I suspect that they will form a connected graded Hopf algebra, with two multiplications (one for composition and the other for convolution), but now with a basis indexed by “mopiscotions” (i.e., pairs ( α , σ ) (\alpha,\sigma) ( α , σ ) α \alpha α σ ∈ S ℓ ( α ) \sigma\in S_{\ell(\alpha)} σ ∈ S ℓ ( α )
Other combinatorial Hopf algebras?
Question 7.4. Can we extend the map β H : H → H ⊗ QSym k \beta_H : H \to H \otimes\operatorname{QSym}_k β H : H → H ⊗ QSym k H → H ⊗ U H \to H \otimes U H → H ⊗ U U U U QSym k \operatorname{QSym}_k QSym k H H H U = NCQSym k U = \operatorname{NCQSym}_k U = NCQSym k U = DoublePosets k U = \operatorname{DoublePosets}_k U = DoublePosets k k = Z k = \mathbb{Z} k = Z Z D \mathbb{Z}\mathrm{D} Z D k k k NCQSym k \operatorname{NCQSym}_k NCQSym k H → H ⊗ NCQSym k H \to H \otimes\operatorname{NCQSym}_k H → H ⊗ NCQSym k [1 Proposition 88] .
Further consequences?
Question 7.5. Do we gain anything from applying Corollary 6.2 to H = QSym k H = \operatorname{QSym}_k H = QSym k Δ p ′ \Delta'_p Δ p ′ Δ p ′ \Delta'_p Δ p ′ S ( a ∗ b ) = a ∗ S ( b ) S(a * b) = a * S(b) S ( a ∗ b ) = a ∗ S ( b ) a , b ∈ NSym k a,b \in\operatorname{NSym}_k a , b ∈ NSym k ∗ * ∗
Question 7.6. From Theorem 5.7 (a) and Proposition 5.3 , we can conclude that Δ p ′ \Delta'_p Δ p ′ ( k , QSym k ) (k,\operatorname{QSym}_k) ( k , QSym k )
( A B ) ∗ G = ∑ ( G ) ( A ∗ G ( 1 ) ) ( B ∗ G ( 2 ) ) (AB) * G = \sum_{(G)} \left(A * G_{(1)}\right)\left(B * G_{(2)}\right) ( A B ) ∗ G = ( G ) ∑ ( A ∗ G ( 1 ) ) ( B ∗ G ( 2 ) ) (using Sweedler’s notation) for any three elements A A A B B B G G G NSym \operatorname{NSym} NSym [3 §7] that the same splitting formula holds when A A A B B B FQSym \operatorname{FQSym} FQSym NSym \operatorname{NSym} NSym G G G NSym \operatorname{NSym} NSym NSym \operatorname{NSym} NSym Theorem 2.1 ?
References 