Manifest. This work was written by GPT-6 based on a proof found by GPT-5.5. It has then been extensively edited and fully proofread by myself. This work is in the public domain. –DG
Definitions and statements
Integral and rational symmetric functions
Let S y m m Z \mathbf{Symm}_{\mathbb{Z}} Symm Z Z \mathbb{Z} Z S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q Q \mathbb{Q} Q
S y m m Q = Q ⊗ Z S y m m Z . \mathbf{Symm}_{\mathbb{Q}}=\mathbb{Q}\otimes_{\mathbb{Z}}\mathbf{Symm}_{\mathbb{Z}}. Symm Q = Q ⊗ Z Symm Z . We write S y m m Z , n \mathbf{Symm}_{\mathbb{Z},n} Symm Z , n S y m m Q , n \mathbf{Symm}_{\mathbb{Q},n} Symm Q , n n n n S y m m Z \mathbf{Symm}_{\mathbb{Z}} Symm Z S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q
For r ≥ 1 r\geq1 r ≥ 1 p r p_r p r r r r λ = ( λ 1 , … , λ ℓ ) \lambda=(\lambda_1,\ldots,\lambda_\ell) λ = ( λ 1 , … , λ ℓ )
p λ = p λ 1 ⋯ p λ ℓ . p_\lambda=p_{\lambda_1}\cdots p_{\lambda_\ell}. p λ = p λ 1 ⋯ p λ ℓ . Each p λ p_\lambda p λ S y m m Z \mathbf{Symm}_{\mathbb{Z}} Symm Z ( p λ ) λ (p_\lambda)_\lambda ( p λ ) λ Q \mathbb{Q} Q S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q Z \mathbb{Z} Z S y m m Z \mathbf{Symm}_{\mathbb{Z}} Symm Z
h 2 = p 1 2 + p 2 2 . h_2=\frac{p_1^2+p_2}{2}. h 2 = 2 p 1 2 + p 2 . Equivalently,
S y m m Q = Q [ p 1 , p 2 , p 3 , … ] , \mathbf{Symm}_{\mathbb{Q}}=\mathbb{Q}[p_1,p_2,p_3,\ldots], Symm Q = Q [ p 1 , p 2 , p 3 , … ] , whereas the integral ring is
S y m m Z = Z [ h 1 , h 2 , h 3 , … ] = Z [ e 1 , e 2 , e 3 , … ] . \mathbf{Symm}_{\mathbb{Z}}=\mathbb{Z}[h_1,h_2,h_3,\ldots]=\mathbb{Z}[e_1,e_2,e_3,\ldots]. Symm Z = Z [ h 1 , h 2 , h 3 , … ] = Z [ e 1 , e 2 , e 3 , … ] . Consequently, it is very easy to define multilinear operations on S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q S y m m Z \mathbf{Symm}_{\mathbb{Z}} Symm Z 1 [21 Exercise 2.9.4] ; see also the symmetric-function chapters of Stanley [41, Chapter 7] and Macdonald [28, Chapter I]. The two operations considered below provide another illustration of precisely this integrality issue.
The arithmetic and anti-arithmetic products
Since the p λ p_\lambda p λ Q \mathbb{Q} Q S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q Q \mathbb{Q} Q
Definition 1.1. The arithmetic product ⊡ \boxdot ⊡ Q \mathbb{Q} Q S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q
p λ ⊡ p μ = ∏ i = 1 r ∏ j = 1 s p lcm ( λ i , μ j ) gcd ( λ i , μ j ) (1) p_\lambda\boxdot p_\mu=\prod_{i=1}^{r}\prod_{j=1}^{s}p_{\operatorname{lcm}(\lambda_i,\mu_j)}^{\operatorname{gcd}(\lambda_i,\mu_j)} \tag*{(1)} p λ ⊡ p μ = i = 1 ∏ r j = 1 ∏ s p lcm ( λ i , μ j ) gcd ( λ i , μ j ) (1) for all pairs of partitions λ = ( λ 1 , … , λ r ) \lambda=(\lambda_1,\ldots,\lambda_r) λ = ( λ 1 , … , λ r ) μ = ( μ 1 , … , μ s ) \mu=(\mu_1,\ldots,\mu_s) μ = ( μ 1 , … , μ s )
The anti-arithmetic product ⋄ \diamond ⋄ Q \mathbb{Q} Q S y m m Q \mathbf{Symm}_{\mathbb{Q}} Symm Q
p λ ⋄ p μ = ∏ i = 1 r ∏ j = 1 s p gcd ( λ i , μ j ) lcm ( λ i , μ j ) (2) p_{\lambda}\diamond p_{\mu}=\prod_{i=1}^{r}\prod_{j=1}^{s}p_{\gcd(\lambda_i,\mu_j)}^{\operatorname{lcm}(\lambda_i,\mu_j)} \tag*{(2)} p λ ⋄ p μ = i = 1 ∏ r j = 1 ∏ s p g c d ( λ i , μ j ) lcm ( λ i , μ j ) (2) for all pairs of partitions λ = ( λ 1 , … , λ r ) \lambda=(\lambda_1,\ldots,\lambda_r) λ = ( λ 1 , … , λ r ) μ = ( μ 1 , … , μ s ) \mu=(\mu_1,\ldots,\mu_s) μ = ( μ 1 , … , μ s )
If ∣ λ ∣ = m |\lambda|=m ∣ λ ∣ = m ∣ μ ∣ = n |\mu|=n ∣ μ ∣ = n m n mn mn ( λ i , μ j ) (\lambda_i,\mu_j) ( λ i , μ j ) λ i μ j \lambda_i\mu_j λ i μ j m , n ≥ 0 m,n\geq0 m , n ≥ 0
⊡ , ⋄ : S y m m Q , m × S y m m Q , n ⟶ S y m m Q , m n . \boxdot,\diamond:\mathbf{Symm}_{\mathbb{Q},m}\times\mathbf{Symm}_{\mathbb{Q},n}\longrightarrow\mathbf{Symm}_{\mathbb{Q},mn}. ⊡ , ⋄ : Symm Q , m × Symm Q , n ⟶ Symm Q , mn . The arithmetic product is the symmetric-function shadow of the arithmetic product of combinatorial species of Maia and Méndez [29 ] ; its integrality is discussed in [21 Exercise 4.4.9] and in the MathOverflow question [17 ] . For a further development of the species-theoretic side, Li [27 ] introduced an exponential composition based on the arithmetic product and used it to study prime graphs under Cartesian product. The anti-arithmetic product was introduced in the MathOverflow question [16 ] , where its integrality was asked for. Unlike the arithmetic product, it is not Schur-positive on pairs of Schur functions; for example, s ( 2 , 1 , 1 ) ⋄ s ( 2 , 1 , 1 ) s_{(2,1,1)}\diamond s_{(2,1,1)} s ( 2 , 1 , 1 ) ⋄ s ( 2 , 1 , 1 ) [16 ] .2
Our main result is the following.
Theorem 1.2 (Integrality). For all m , n ≥ 0 m,n\geq0 m , n ≥ 0
(a) We have
S y m m Z , m ⊡ S y m m Z , n ⊆ S y m m Z , m n . \mathbf{Symm}_{\mathbb{Z},m}\boxdot\mathbf{Symm}_{\mathbb{Z},n}\subseteq\mathbf{Symm}_{\mathbb{Z},mn}. Symm Z , m ⊡ Symm Z , n ⊆ Symm Z , mn . (b) If f ∈ S y m m Z , m f\in\mathbf{Symm}_{\mathbb{Z},m} f ∈ Symm Z , m g ∈ S y m m Z , n g\in\mathbf{Symm}_{\mathbb{Z},n} g ∈ Symm Z , n f ⊡ g f\boxdot g f ⊡ g
(c) We have
S y m m Z , m ⋄ S y m m Z , n ⊆ S y m m Z , m n . \mathbf{Symm}_{\mathbb{Z},m}\diamond\mathbf{Symm}_{\mathbb{Z},n}\subseteq\mathbf{Symm}_{\mathbb{Z},mn}. Symm Z , m ⋄ Symm Z , n ⊆ Symm Z , mn . Parts (a) and (b) are known. We include them because the proof of (a) is parallel to our proof of part (c), except at exactly one point. Both proofs proceed using the classical Frobenius correspondence between symmetric functions and representations/class functions of symmetric groups. For the arithmetic product ⊡ \boxdot ⊡ 3
S m × S n ⟶ S m n S_m\times S_n\longrightarrow S_{mn} S m × S n ⟶ S mn provides the required pullback on representations. For the anti-arithmetic product no such homomorphism is available; instead, we construct a certain virtual representation by Adams operations and Möbius inversion.
Before we present these proofs, we shall develop the machinery needed for them. Very little of it is new, but not all of it is found in textbooks.
Exterior powers, Adams operations, and class functions
We will use the basic language of λ \lambda λ λ \lambda λ [25 Chapter I, especially §§1 and 4] , Yau [45 Chapters 1 and 3] , or Hazewinkel [24 Section 16] . None of the general structure theorems from these references will be used. For the symmetric-group representation rings specifically, Thibon [42 ] and Scharf–Thibon [38 ] develop Adams operations and inner plethysm directly in symmetric-function language. A modern structural perspective, emphasizing Adams operations as natural transformations of the representation-ring functor, is given by Meir–Szymik [31 ] .
Representation rings and class functions
For every finite group G G G Q \mathbb{Q} Q Q \mathbb{Q} Q Q \mathbb{Q} Q § 3 .
Let Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G ) Q \mathbb{Q} Q Q \mathbb{Q} Q G G G χ V \chi_V χ V V V V G G G Q \mathbb{Q} Q Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G ) R ( G ) R(G) R ( G ) Q G \mathbb{Q}G Q G [40 Chapter 9] ). Thus R ( G ) R(G) R ( G ) Z \mathbb{Z} Z [ V ] [V] [ V ] V V V V V V
χ : R ( G ) ↪ Cl Q ( G ) , [ V ] ↦ χ V , (3) \chi\colon R(G)\hookrightarrow\operatorname{Cl}_{\mathbb{Q}}(G),\qquad[V]\mapsto\chi_V, \tag*{(3)} χ : R ( G ) ↪ Cl Q ( G ) , [ V ] ↦ χ V , (3) which we shall call the character embedding . We shall usually identify R ( G ) R(G) R ( G ) [9 §4.2] , Serre [40 Chapter 2] , or the lecture notes of Lassueur [26 §9] .
A weak notion of lambda-ring
Our notion of a λ \lambda λ λ 0 ( x ) \lambda^0(x) λ 0 ( x ) λ 1 ( x ) \lambda^1(x) λ 1 ( x ) λ n ( x + y ) \lambda^n(x+y) λ n ( x + y ) λ \lambda λ pre-λ \lambda λ , and only deem them worthy of the name “λ \lambda λ λ n ( x y ) \lambda^n(xy) λ n ( x y ) λ n ( λ m ( y ) ) \lambda^n(\lambda^m(y)) λ n ( λ m ( y ))
Definition 2.1. A λ \lambda λ in this note (often called a pre-λ \lambda λ commutative unital ring A A A non-linear ) maps
λ r : A ⟶ A for all r ≥ 0 \lambda^r : A \longrightarrow A \qquad\text{for all } r \ge0 λ r : A ⟶ A for all r ≥ 0 such that all x , y ∈ A x,y \in A x , y ∈ A
λ 0 ( x ) = 1 , (4) \lambda^0(x)=1, \tag*{(4)} λ 0 ( x ) = 1 , (4) λ 1 ( x ) = x , (5) \lambda^1(x)=x, \tag*{(5)} λ 1 ( x ) = x , (5) λ n ( x + y ) = ∑ i = 0 n λ i ( x ) λ n − i ( y ) . (6) \lambda^n(x+y)=\sum_{i=0}^{n}\lambda^i(x)\lambda^{n-i}(y). \tag*{(6)} λ n ( x + y ) = i = 0 ∑ n λ i ( x ) λ n − i ( y ) . (6) We can encode these maps λ r \lambda^r λ r
λ t ( x ) = ∑ r ≥ 0 λ r ( x ) t r ∈ A [ [ t ] ] (7) \lambda_t(x)=\sum_{r\geq0}\lambda^r(x)t^r\in A[[t]] \tag*{(7)} λ t ( x ) = r ≥ 0 ∑ λ r ( x ) t r ∈ A [[ t ]] (7) defined for all x ∈ A x\in A x ∈ A λ t : A → A [ [ t ] ] \lambda_t:A\to A[[t]] λ t : A → A [[ t ]]
λ t ( x ) = 1 + x t + ( higher powers of t ) , \lambda_t(x)=1+xt+(\text{higher powers of }t), λ t ( x ) = 1 + x t + ( higher powers of t ) , whereas the axiom (6) can be equivalently rewritten as
λ t ( x + y ) = λ t ( x ) λ t ( y ) . (8) \lambda_t(x+y)=\lambda_t(x)\lambda_t(y). \tag*{(8)} λ t ( x + y ) = λ t ( x ) λ t ( y ) . (8) In other words, λ t \lambda_t λ t A A A 1 1 1 A A A
d d t λ t ( x ) ∣ t = 0 = x \left.\frac{d}{dt}\lambda_t(x)\right|_{t=0}=x d t d λ t ( x ) t = 0 = x for each x ∈ A x\in A x ∈ A
A λ \lambda λ f : A → B f:A\to B f : A → B λ \lambda λ λ r \lambda^r λ r f ∘ λ r = λ r ∘ f f\circ\lambda^r=\lambda^r\circ f f ∘ λ r = λ r ∘ f r ≥ 0 ) . A r\geq0). A r ≥ 0 ) . A λ \lambda λ A A A λ r \lambda^r λ r λ \lambda λ A A A X ⊆ A X\subseteq A X ⊆ A λ \lambda λ A A A X X X
Note that the operations λ r \lambda^r λ r λ \lambda λ A A A λ t \lambda_t λ t λ \lambda λ λ t \lambda_t λ t λ r \lambda^r λ r
Example 2.2 (The binomial λ \lambda λ The ring Z \mathbb{Z} Z λ \lambda λ
λ r ( a ) = ( a r ) , λ t ( a ) = ( 1 + t ) a . \lambda^r(a)=\binom{a}{r},\qquad\lambda_t(a)=(1+t)^a. λ r ( a ) = ( r a ) , λ t ( a ) = ( 1 + t ) a . The binomial theorem shows that these two equalities fit together with (7); the equality (6) is the Chu–Vandermonde convolution.
Example 2.3 (Representation rings). Let G G G R ( G ) R(G) R ( G ) G G G V V V
λ t ( [ V ] ) = ∑ r ≥ 0 [ ⋀ r V ] t r ∈ R ( G ) [ [ t ] ] . (9) \lambda_t([V])=\sum_{r\geq0}[\mathop{\bigwedge}\nolimits^r V]t^r\in R(G)[[t]]. \tag*{(9)} λ t ([ V ]) = r ≥ 0 ∑ [ ⋀ r V ] t r ∈ R ( G ) [[ t ]] . (9) The canonical decomposition
⋀ r ( V ⊕ W ) ≅ ⨁ i + j = r ⋀ i V ⊗ ⋀ j W \mathop{\bigwedge}\nolimits^r(V\oplus W)\cong\bigoplus_{i+j=r}\mathop{\bigwedge}\nolimits^i V\otimes\mathop{\bigwedge}\nolimits^j W ⋀ r ( V ⊕ W ) ≅ i + j = r ⨁ ⋀ i V ⊗ ⋀ j W gives
λ t ( [ V ⊕ W ] ) = λ t ( [ V ] ) λ t ( [ W ] ) . \lambda_t([V\oplus W])=\lambda_t([V])\lambda_t([W]). λ t ([ V ⊕ W ]) = λ t ([ V ]) λ t ([ W ]) . Thus, (9) defines a monoid homomorphism x ↦ λ t ( x ) x\mapsto\lambda_t(x) x ↦ λ t ( x ) G G G 1 + t R ( G ) [ [ t ] ] 1+tR(G)[[t]] 1 + tR ( G ) [[ t ]] R ( G ) R(G) R ( G )
λ t : R ( G ) ⟶ 1 + t R ( G ) [ [ t ] ] \lambda_t:R(G)\longrightarrow1+tR(G)[[t]] λ t : R ( G ) ⟶ 1 + tR ( G ) [[ t ]] by the rule
λ t ( x − y ) = λ t ( x ) λ t ( y ) . \lambda_t(x-y)=\frac{\lambda_t(x)}{\lambda_t(y)}. λ t ( x − y ) = λ t ( y ) λ t ( x ) . Thus R ( G ) R(G) R ( G ) λ \lambda λ
ψ \psi ψ Adams operations are even easier to axiomatize.
Definition 2.4. A ψ \psi ψ A A A
ψ r : A ⟶ A for all r ≥ 1 \psi^r:A\longrightarrow A\qquad\text{for all }r\geq1 ψ r : A ⟶ A for all r ≥ 1 such that
ψ 1 = id and ψ r s = ψ r ∘ ψ s for all r , s ≥ 1. \psi^1=\operatorname{id}\qquad\text{and}\qquad\psi^{rs}=\psi^r\circ\psi^s\quad\text{for all }r,s\geq1. ψ 1 = id and ψ r s = ψ r ∘ ψ s for all r , s ≥ 1. The endomorphisms ψ r \psi^r ψ r A A A ψ \psi ψ ψ r \psi^r ψ r
Proposition 2.5. For every finite group G G G Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G ) ψ \psi ψ ψ \psi ψ ψ r \psi^r ψ r
( ψ r f ) ( g ) = f ( g r ) for all r ≥ 1 and f ∈ Cl Q ( G ) and g ∈ G . (10) (\psi^r f)(g)=f(g^r)\qquad\text{for all }r\ge1\text{ and }f\in\operatorname{Cl}_{\mathbb{Q}}(G)\text{ and }g\in G. \tag*{(10)} ( ψ r f ) ( g ) = f ( g r ) for all r ≥ 1 and f ∈ Cl Q ( G ) and g ∈ G . (10) Proof. The function ψ r f \psi^r f ψ r f (10) is a class function, because if g g g h h h G G G r r r g r g^r g r h r h^r h r ψ r \psi^r ψ r 1 1 1 r , s ≥ 1 r,s\ge1 r , s ≥ 1 f ∈ Cl Q ( G ) f\in\operatorname{Cl}_{\mathbb{Q}}(G) f ∈ Cl Q ( G ) g ∈ G g\in G g ∈ G
ψ r ( ψ s f ) ( g ) = ( ψ s f ) ( g r ) = f ( ( g r ) s ) = f ( g r s ) = ( ψ r s f ) ( g ) . \psi^r(\psi^s f)(g)=(\psi^s f)(g^r)=f((g^r)^s)=f(g^{rs})=(\psi^{rs}f)(g). ψ r ( ψ s f ) ( g ) = ( ψ s f ) ( g r ) = f (( g r ) s ) = f ( g r s ) = ( ψ r s f ) ( g ) . Thus, ψ r s = ψ r ∘ ψ s \psi^{rs}=\psi^r\circ\psi^s ψ r s = ψ r ∘ ψ s
From ψ \psi ψ λ \lambda λ Q \mathbb{Q} Q
Here is the only general construction we need from the theory of λ \lambda λ
Proposition 2.6. Let A A A ψ \psi ψ Q \mathbb{Q} Q Q \mathbb{Q} Q ψ \psi ψ x ∈ A x\in A x ∈ A λ t ( x ) ∈ A [ [ t ] ] \lambda_t(x)\in A[[t]] λ t ( x ) ∈ A [[ t ]]
λ t ( x ) = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ψ k ( x ) t k k ) . (11) \lambda_t(x)=\exp\left(\sum_{k\ge1}(-1)^{k-1}\psi^k(x)\frac{t^k}{k}\right). \tag*{(11)} λ t ( x ) = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ψ k ( x ) k t k ) . (11) Then the following statements hold.
(a) The operations λ r \lambda^r λ r λ t \lambda_t λ t A A A λ \lambda λ
(b) Every morphism of ψ \psi ψ Q \mathbb{Q} Q λ \lambda λ
The familiar symmetric-function identity
∑ r ≥ 0 e r t r = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 p k t k k ) (12) \sum_{r\ge0}e_rt^r=\exp\left(\sum_{k\ge1}(-1)^{k-1}p_k\frac{t^k}{k}\right) \tag*{(12)} r ≥ 0 ∑ e r t r = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 p k k t k ) (12) is one way to remember the formula (11) ; comparing the two identities shows that the Q \mathbb{Q} Q Symm Q → A \operatorname{Symm}_{\mathbb{Q}}\to A Symm Q → A p k p_k p k ψ k ( x ) \psi^k(x) ψ k ( x ) x ∈ A x\in A x ∈ A e r e_r e r λ r ( x ) \lambda^r(x) λ r ( x )
Proof of Proposition 2.6. (a) Since ψ 1 = id \psi^1=\operatorname{id} ψ 1 = id t t t (11) is x x x 1 1 1 ψ k \psi^k ψ k
λ t ( x + y ) = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ( ψ k ( x ) + ψ k ( y ) ) t k k ) = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ψ k ( x ) t k k ) ⋅ exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ψ k ( y ) t k k ) = λ t ( x ) λ t ( y ) . \begin{aligned}
\lambda_t(x+y)=\exp\left(\sum_{k\ge1}(-1)^{k-1}(\psi^k(x)+\psi^k(y))\frac{t^k}{k}\right) \\
&=\exp\left(\sum_{k\ge1}(-1)^{k-1}\psi^k(x)\frac{t^k}{k}\right)\cdot\exp\left(\sum_{k\ge1}(-1)^{k-1}\psi^k(y)\frac{t^k}{k}\right) \\
&=\lambda_t(x)\lambda_t(y).
\end{aligned} λ t ( x + y ) = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ( ψ k ( x ) + ψ k ( y )) k t k ) = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ψ k ( x ) k t k ) ⋅ exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ψ k ( y ) k t k ) = λ t ( x ) λ t ( y ) . This proves the λ \lambda λ A A A λ \lambda λ φ : A → B \varphi: A \to B φ : A → B ψ k \psi^k ψ k φ \varphi φ λ r \lambda^r λ r
Remark 2.7. The proof of Proposition 2.6 uses much less than a ψ \psi ψ λ \lambda λ A A A Q \mathbb{Q} Q
ψ r : A ⟶ A for all r ≥ 1 \psi^r : A \longrightarrow A \qquad\text{for all } r \ge1 ψ r : A ⟶ A for all r ≥ 1 with ψ 1 = id \psi^1 = \operatorname{id} ψ 1 = id ψ k \psi^k ψ k ψ r s = ψ r ∘ ψ s \psi^{rs} = \psi^r \circ\psi^s ψ r s = ψ r ∘ ψ s Q \mathbb{Q} Q ψ \psi ψ λ \lambda λ λ \lambda λ
Differentiating the logarithm of (11) gives
( d d t λ t ( x ) ) / λ t ( x ) = ∑ k ≥ 1 ( − 1 ) k − 1 ψ k ( x ) t k − 1 , (13) \left(\frac{d}{dt}\lambda_t(x)\right) / \lambda_t(x) = \sum_{k \ge1} (-1)^{k-1}\psi^k(x)t^{k-1}, \tag*{(13)} ( d t d λ t ( x ) ) / λ t ( x ) = k ≥ 1 ∑ ( − 1 ) k − 1 ψ k ( x ) t k − 1 , (13) that is,
d d t λ t ( x ) = λ t ( x ) ⋅ ∑ k ≥ 1 ( − 1 ) k − 1 ψ k ( x ) t k − 1 . (14) \frac{d}{dt}\lambda_t(x) = \lambda_t(x) \cdot\sum_{k \ge1} (-1)^{k-1}\psi^k(x)t^{k-1}. \tag*{(14)} d t d λ t ( x ) = λ t ( x ) ⋅ k ≥ 1 ∑ ( − 1 ) k − 1 ψ k ( x ) t k − 1 . (14) Comparing t r − 1 t^{r-1} t r − 1
r λ r ( x ) = ∑ k = 1 r ( − 1 ) k − 1 λ r − k ( x ) ψ k ( x ) (15) r\lambda^r(x) = \sum_{k=1}^{r}(-1)^{k-1}\lambda^{r-k}(x)\psi^k(x) \tag*{(15)} r λ r ( x ) = k = 1 ∑ r ( − 1 ) k − 1 λ r − k ( x ) ψ k ( x ) (15) for each r ≥ 1 r \ge1 r ≥ 1 x ∈ A x \in A x ∈ A ψ r ( x ) \psi^r(x) ψ r ( x )
ψ r ( x ) − λ 1 ( x ) ψ r − 1 ( x ) + λ 2 ( x ) ψ r − 2 ( x ) − ⋯ + ( − 1 ) r r λ r ( x ) = 0. (16) \psi^r(x) - \lambda^1(x)\psi^{r-1}(x) + \lambda^2(x)\psi^{r-2}(x) - \cdots+ (-1)^r r\lambda^r(x) = 0. \tag*{(16)} ψ r ( x ) − λ 1 ( x ) ψ r − 1 ( x ) + λ 2 ( x ) ψ r − 2 ( x ) − ⋯ + ( − 1 ) r r λ r ( x ) = 0. (16) Thus, ψ r ( x ) \psi^r(x) ψ r ( x ) ψ 1 ( x ) , ψ 2 ( x ) , … , ψ r − 1 ( x ) \psi^1(x),\psi^2(x),\ldots,\psi^{r-1}(x) ψ 1 ( x ) , ψ 2 ( x ) , … , ψ r − 1 ( x ) λ 1 ( x ) , λ 2 ( x ) , … , λ r ( x ) \lambda^1(x),\lambda^2(x),\ldots,\lambda^r(x) λ 1 ( x ) , λ 2 ( x ) , … , λ r ( x ) ψ r ( x ) \psi^r(x) ψ r ( x ) integral polynomial in the elements
λ 1 ( x ) , λ 2 ( x ) , … , λ r ( x ) \lambda^1(x),\lambda^2(x),\ldots,\lambda^r(x) λ 1 ( x ) , λ 2 ( x ) , … , λ r ( x ) (a version of the usual Newton identities for power-sum symmetric functions). For example,
ψ 1 ( x ) = λ 1 ( x ) , ψ 2 ( x ) = λ 1 ( x ) 2 − 2 λ 2 ( x ) . \psi^1(x) = \lambda^1(x), \qquad\psi^2(x) = \lambda^1(x)^2 - 2\lambda^2(x). ψ 1 ( x ) = λ 1 ( x ) , ψ 2 ( x ) = λ 1 ( x ) 2 − 2 λ 2 ( x ) . The absence of denominators in this direction will be useful below, as it shows that the λ \lambda λ A A A ψ \psi ψ
Characters respect exterior powers
Consider again a finite group G G G Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G ) G G G ψ \psi ψ Q \mathbb{Q} Q Proposition 2.6 ) becomes a λ \lambda λ R ( G ) R(G) R ( G ) λ \lambda λ λ \lambda λ Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G ) λ \lambda λ R ( G ) R(G) R ( G ) ψ \psi ψ Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G )
Proposition 2.8. (a) The character embedding
R ( G ) ↪ Cl Q ( G ) R(G) \hookrightarrow\operatorname{Cl}_{\mathbb{Q}}(G) R ( G ) ↪ Cl Q ( G ) is a morphism of λ \lambda λ
(b) The operations ψ r \psi^{r} ψ r (10) preserve R ( G ) R(G) R ( G )
(c) For every virtual representation x ∈ R ( G ) x \in R(G) x ∈ R ( G ) g ∈ G g \in G g ∈ G
χ ψ r ( x ) ( g ) = χ x ( g r ) , (17) \chi_{\psi^{r}(x)}(g)=\chi_x(g^r), \tag*{(17)} χ ψ r ( x ) ( g ) = χ x ( g r ) , (17) where χ y \chi_y χ y y ∈ R ( G ) y \in R(G) y ∈ R ( G )
Proof. (a) Let V V V G G G A A A g ∈ G g \in G g ∈ G V V V 4
det ( I + t A ) = ∑ j ≥ 0 tr ( ∧ j A ) t j . (18) \det(I+tA)=\sum_{j\geq0}\operatorname{tr}(\wedge^j A)t^j. \tag*{(18)} det ( I + t A ) = j ≥ 0 ∑ tr ( ∧ j A ) t j . (18) On the other hand, if the eigenvalues of A A A α 1 , … , α d \alpha_1,\ldots,\alpha_d α 1 , … , α d
det ( I + t A ) = ∏ i = 1 d ( 1 + α i t ) = ∑ r ≥ 0 e r ( α 1 , α 2 , … , α d ) t r = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ( ∑ i α i k ) t k k ) (by (12)) = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ( ψ k ( χ V ) ) ( g ) t k k ) , (19) \begin{aligned}
\det(I+tA)&=\prod_{i=1}^{d}(1+\alpha_i t)=\sum_{r\geq0}e_r(\alpha_1,\alpha_2,\ldots,\alpha_d)t^r\\
&=\exp\left(\sum_{k\geq1}(-1)^{k-1}\left(\sum_i\alpha_i^k\right)\frac{t^k}{k}\right)\quad\text{(by (12))}\\
&=\exp\left(\sum_{k\geq1}(-1)^{k-1}(\psi^k(\chi_V))(g)\frac{t^k}{k}\right), \tag*{(19)}
\end{aligned} det ( I + t A ) = i = 1 ∏ d ( 1 + α i t ) = r ≥ 0 ∑ e r ( α 1 , α 2 , … , α d ) t r = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ( i ∑ α i k ) k t k ) (by (12)) = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ( ψ k ( χ V )) ( g ) k t k ) , (19) since each k ≥ 1 k\geq1 k ≥ 1 ∑ i α i k = tr ( A k ) = χ V ( g k ) = ( ψ k ( χ V ) ) ( g ) \sum_i\alpha_i^k=\operatorname{tr}(A^k)=\chi_V(g^k)=(\psi^k(\chi_V))(g) ∑ i α i k = tr ( A k ) = χ V ( g k ) = ( ψ k ( χ V )) ( g ) (18) , we find
∑ j ≥ 0 tr ( ∧ j A ) t j = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ( ψ k ( χ V ) ) ( g ) t k k ) . \sum_{j\geq0}\operatorname{tr}(\wedge^j A)t^j=\exp\left(\sum_{k\geq1}(-1)^{k-1}(\psi^k(\chi_V))(g)\frac{t^k}{k}\right). j ≥ 0 ∑ tr ( ∧ j A ) t j = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ( ψ k ( χ V )) ( g ) k t k ) . On the other hand, applying (11) to x = χ V x=\chi_V x = χ V g g g
∑ j ≥ 0 ( λ j ( χ V ) ) ( g ) t j = exp ( ∑ k ≥ 1 ( − 1 ) k − 1 ( ψ k ( χ V ) ) ( g ) t k k ) . \sum_{j\geq0}(\lambda^j(\chi_V))(g)t^j=\exp\left(\sum_{k\geq1}(-1)^{k-1}(\psi^k(\chi_V))(g)\frac{t^k}{k}\right). j ≥ 0 ∑ ( λ j ( χ V )) ( g ) t j = exp ( k ≥ 1 ∑ ( − 1 ) k − 1 ( ψ k ( χ V )) ( g ) k t k ) . (since the algebra structure on Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G )
∑ j ≥ 0 ( λ j ( χ V ) ) ( g ) t j = ∑ j ≥ 0 tr ( Λ j A ) t j . \sum_{j\ge0}(\lambda^{j}(\chi_V))(g)t^j=\sum_{j\ge0}\operatorname{tr}(\Lambda^j A)t^j. j ≥ 0 ∑ ( λ j ( χ V )) ( g ) t j = j ≥ 0 ∑ tr ( Λ j A ) t j . Thus, for each j ≥ 0 j\ge0 j ≥ 0 λ j ( χ V ) ( g ) = tr ( Λ j A ) = χ Λ j V ( g ) = χ λ j ( [ V ] ) ( g ) \lambda^j(\chi_V)(g)=\operatorname{tr}(\Lambda^j A)=\chi_{\Lambda^j V}(g)=\chi_{\lambda^j([V])}(g) λ j ( χ V ) ( g ) = tr ( Λ j A ) = χ Λ j V ( g ) = χ λ j ([ V ]) ( g ) g g g λ j ( χ V ) \lambda^j(\chi_V) λ j ( χ V ) χ λ j ( [ V ] ) \chi_{\lambda^j([V])} χ λ j ([ V ])
This shows that the character embedding R ( G ) → Cl Q ( G ) R(G)\to\operatorname{Cl}_{\mathbb{Q}}(G) R ( G ) → Cl Q ( G ) λ j \lambda^j λ j λ t \lambda_t λ t λ t \lambda_t λ t R ( G ) R(G) R ( G ) λ \lambda λ
(b) Let x ∈ R ( G ) x\in R(G) x ∈ R ( G ) ψ r ( x ) \psi^r(x) ψ r ( x ) x x x ψ r ( x ) ∈ R ( G ) \psi^r(x)\in R(G) ψ r ( x ) ∈ R ( G )
(c) The character embedding is a λ \lambda λ ψ r \psi^r ψ r x ∈ R ( G ) x\in R(G) x ∈ R ( G ) r ≥ 1 r\ge1 r ≥ 1 χ ψ r ( x ) = ψ r ( χ x ) \chi_{\psi^r(x)}=\psi^r(\chi_x) χ ψ r ( x ) = ψ r ( χ x ) g ∈ G g\in G g ∈ G
Thus we may regard R ( G ) R(G) R ( G ) λ \lambda λ ψ \psi ψ Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G )
A word about special λ \lambda λ
In the usual terminology, one often reserves “λ \lambda λ λ r ( x y ) \lambda^r(xy) λ r ( x y ) λ r ( λ s ( x ) ) \lambda^r(\lambda^s(x)) λ r ( λ s ( x )) special λ \lambda λ . Representation rings are special, and a ψ \psi ψ Q \mathbb{Q} Q λ \lambda λ [25 Chapter I] , Yau [45 Chapters 1–3] , and Hazewinkel [24 Section 16] for the full theory.
The representation ring of the symmetric group
Frobenius characteristic
For any partition λ = ( 1 m 1 2 m 2 ⋯ ) ⊢ n \lambda=(1^{m_1}2^{m_2}\cdots)\vdash n λ = ( 1 m 1 2 m 2 ⋯ ) ⊢ n
z λ = ∏ i ≥ 1 i m i m i ! . z_\lambda=\prod_{i\ge1}i^{m_i}m_i!. z λ = i ≥ 1 ∏ i m i m i ! . For a class function f ∈ Cl Q ( S n ) f\in\operatorname{Cl}_{\mathbb{Q}}(S_n) f ∈ Cl Q ( S n ) f ( λ ) f(\lambda) f ( λ ) λ \lambda λ Frobenius characteristic (also known as the characteristic map in [36 §4.7] , or as the characteristic isomorphism in [43 (5.5)] ) is the Q \mathbb{Q} Q
ch n : Cl Q ( S n ) ⟶ Symm Q , n , ch n ( f ) = ∑ λ ⊢ n f ( λ ) p λ z λ . (20) \operatorname{ch}_n:\operatorname{Cl}_{\mathbb{Q}}(S_n)\longrightarrow\operatorname{Symm}_{\mathbb{Q},n},\qquad\operatorname{ch}_n(f)=\sum_{\lambda\vdash n}f(\lambda)\frac{p_\lambda}{z_\lambda}. \tag*{(20)} ch n : Cl Q ( S n ) ⟶ Symm Q , n , ch n ( f ) = λ ⊢ n ∑ f ( λ ) z λ p λ . (20) It is an isomorphism of Q \mathbb{Q} Q p λ p_\lambda p λ Symm Q , n \operatorname{Symm}_{\mathbb{Q},n} Symm Q , n [36 just before Proposition 4.7.2] ) says that
ch n ( χ λ ) = s λ . (21) \operatorname{ch}_n(\chi^\lambda)=s_\lambda. \tag*{(21)} ch n ( χ λ ) = s λ . (21) where χ λ \chi^\lambda χ λ S λ S^\lambda S λ Q \mathbb{Q} Q Q S n \mathbb{Q}S_n Q S n (20) gives an isomorphism of abelian groups
R ( S n ) ⟶ ∼ Symm Z , n . (22) R(S_n) \overset{\sim}{\longrightarrow} \operatorname{Symm}_{\mathbb{Z},n}. \tag*{(22)} R ( S n ) ⟶ ∼ Symm Z , n . (22) Thus we have a commutative square
R ( S n ) → Cl Q ( S n ) ch n ↓ ↓ ch n Symm Z , n → Symm Q , n . \begin{CD}
R(S_n) @>>> \operatorname{Cl}_{\mathbb{Q}}(S_n) \\
@V{\operatorname{ch}_n}VV @VV{\operatorname{ch}_n}V \\
\operatorname{Symm}_{\mathbb{Z},n} @>>> \operatorname{Symm}_{\mathbb{Q},n}.
\end{CD} R ( S n ) ch n ↓ ⏐ Symm Z , n Cl Q ( S n ) ↓ ⏐ ch n Symm Q , n . This is the bridge between the integrality problem for symmetric functions in Symm Q , n \operatorname{Symm}_{\mathbb{Q},n} Symm Q , n Cl Q ( S n ) \operatorname{Cl}_{\mathbb{Q}}(S_n) Cl Q ( S n ) R ( S n ) R(S_n) R ( S n ) Cl Q ( S n ) \operatorname{Cl}_{\mathbb{Q}}(S_n) Cl Q ( S n ) Kronecker product (also known as the internal product ) on the symmetric functions; but we will not gain anything from this fact.
The natural representation generates as a λ \lambda λ
Let
M n = Q n M_n = \mathbb{Q}^n M n = Q n be the natural permutation representation of S n S_n S n e 1 , … , e n e_1,\ldots,e_n e 1 , … , e n σ ( e i ) = e σ ( i ) \sigma(e_i)=e_{\sigma(i)} σ ( e i ) = e σ ( i ) σ ∈ S n \sigma\in S_n σ ∈ S n i ∈ [ n ] i\in[n] i ∈ [ n ] V n V_n V n S n S_n S n M n M_n M n n ≥ 2 n\ge2 n ≥ 2 S ( n − 1 , 1 ) S^{(n-1,1)} S ( n − 1 , 1 ) n = 1 n=1 n = 1
M n ≅ 1 ⊕ V n . (23) M_n \cong\mathbf{1}\oplus V_n. \tag*{(23)} M n ≅ 1 ⊕ V n . (23) We will need a 1975 result of Evelyn Boorman [3 ] : the representation M n M_n M n R ( S n ) R(S_n) R ( S n ) λ \lambda λ V n V_n V n R ( S n ) R(S_n) R ( S n ) [30 ] ; his proof uses a formula of Dvir and points out earlier equivalent symmetric-function results of Butler [7 ] and Boorman. We record the theorem in the form needed later. Three self-contained proofs are given in the appendices.
Theorem 3.1 (Boorman; Marin). For every n ≥ 1 n\ge1 n ≥ 1 λ \lambda λ R ( S n ) R(S_n) R ( S n ) [ M n ] [M_n] [ M n ] R ( S n ) R(S_n) R ( S n )
Boorman’s proof is a more general categorical argument, based on a backward induction on stabilizers of tuples and applicable also to Burnside rings. Marin’s proof filters by the number n − λ 1 n-\lambda_1 n − λ 1 [23 ] . Appendix B gives a second and a third proof, obtained from stronger integral generation theorems for Solomon’s descent algebra (or, equivalently, for noncommutative symmetric functions). For a readable introduction to Solomon’s descent algebra, including its realization through the face semigroup algebra of the braid arrangement, see Saliola [37 ] , §2, especially §2.1.
The scalar product and integral self-duality
For any finite group G G G ⟨ ⋅ , ⋅ ⟩ G \langle\cdot,\cdot\rangle_G ⟨ ⋅ , ⋅ ⟩ G Cl Q ( G ) \operatorname{Cl}_{\mathbb{Q}}(G) Cl Q ( G )
⟨ f , g ⟩ G = 1 ∣ G ∣ ∑ x ∈ G f ( x ) g ( x − 1 ) for all f , g ∈ Cl Q ( G ) . (24) \langle f,g\rangle_G=\frac{1}{\lvert G\rvert}\sum_{x\in G}f(x)g(x^{-1}) \qquad\text{for all } f,g\in\operatorname{Cl}_{\mathbb{Q}}(G). \tag*{(24)} ⟨ f , g ⟩ G = ∣ G ∣ 1 x ∈ G ∑ f ( x ) g ( x − 1 ) for all f , g ∈ Cl Q ( G ) . (24) For symmetric groups, every conjugacy class is invariant under inversion, so this is simply the usual character scalar product without any need for complex conjugation. For G = S n G=S_n G = S n χ λ \chi^\lambda χ λ λ ⊢ n \lambda\vdash n λ ⊢ n Cl Q ( S n ) \operatorname{Cl}_{\mathbb{Q}}(S_n) Cl Q ( S n ) G G G G G G U U U V V V
⟨ χ U , χ V ⟩ G = dim Hom G ( U , V ) , (25) \langle\chi_U,\chi_V\rangle_G=\dim\operatorname{Hom}_G(U,V), \tag*{(25)} ⟨ χ U , χ V ⟩ G = dim Hom G ( U , V ) , (25) which (by Schur’s lemma) is 0 when U U U V V V U U U V V V G = S n G=S_n G = S n S λ S^\lambda S λ Hom S n ( S λ , S λ ) ≅ Q \operatorname{Hom}_{S_n}(S^\lambda,S^\lambda)\cong\mathbb{Q} Hom S n ( S λ , S λ ) ≅ Q [9 last paragraph of §5.13] ) and therefore ⟨ χ λ , χ λ ⟩ S n = 1 \langle\chi^\lambda,\chi^\lambda\rangle_{S_n}=1 ⟨ χ λ , χ λ ⟩ S n = 1
The Hall scalar product on S y m m Q , n \mathbf{Symm}_{\mathbb{Q},n} Symm Q , n
⟨ p λ , p μ ⟩ = δ λ μ z λ . (26) \langle p_\lambda,p_\mu\rangle=\delta_{\lambda\mu}z_\lambda. \tag*{(26)} ⟨ p λ , p μ ⟩ = δ λ μ z λ . (26) From (20) one checks immediately that Frobenius characteristic is an isometry:
⟨ ch n ( f ) , ch n ( g ) ⟩ = ⟨ f , g ⟩ S n for all f , g ∈ Cl Q ( S n ) . (27) \langle\operatorname{ch}_n(f),\operatorname{ch}_n(g)\rangle=\langle f,g\rangle_{S_n}\qquad\text{for all }f,g\in\operatorname{Cl}_{\mathbb{Q}}(S_n). \tag*{(27)} ⟨ ch n ( f ) , ch n ( g )⟩ = ⟨ f , g ⟩ S n for all f , g ∈ Cl Q ( S n ) . (27) Equivalently, the Schur functions form an orthonormal basis:
⟨ s λ , s μ ⟩ = δ λ μ . \langle s_\lambda,s_\mu\rangle=\delta_{\lambda\mu}. ⟨ s λ , s μ ⟩ = δ λ μ . In particular, the lattice S y m m Z , n \mathbf{Symm}_{\mathbb{Z},n} Symm Z , n
Lemma 3.2. If F ∈ S y m m Q , n F\in\mathbf{Symm}_{\mathbb{Q},n} F ∈ Symm Q , n
⟨ F , H ⟩ ∈ Z for every H ∈ S y m m Z , n , \langle F,H\rangle\in\mathbb{Z}\qquad\text{for every }H\in\mathbf{Symm}_{\mathbb{Z},n}, ⟨ F , H ⟩ ∈ Z for every H ∈ Symm Z , n , then F ∈ S y m m Z , n F\in\mathbf{Symm}_{\mathbb{Z},n} F ∈ Symm Z , n
Proof. Write F = ∑ λ ⊢ n c λ s λ F=\sum_{\lambda\vdash n}c_\lambda s_\lambda F = ∑ λ ⊢ n c λ s λ ⟨ F , H ⟩ ∈ Z \langle F,H\rangle\in\mathbb{Z} ⟨ F , H ⟩ ∈ Z H ∈ S y m m Z , n H\in\mathbf{Symm}_{\mathbb{Z},n} H ∈ Symm Z , n ⟨ F , s λ ⟩ ∈ Z \langle F,s_\lambda\rangle\in\mathbb{Z} ⟨ F , s λ ⟩ ∈ Z λ ⊢ n \lambda\vdash n λ ⊢ n ⟨ F , s λ ⟩ = c λ \langle F,s_\lambda\rangle=c_\lambda ⟨ F , s λ ⟩ = c λ c λ ∈ Z c_\lambda\in\mathbb{Z} c λ ∈ Z λ \lambda λ F ∈ S y m m Z , n F\in\mathbf{Symm}_{\mathbb{Z},n} F ∈ Symm Z , n
We note one simple fact connecting the scalar product with the Frobenius characteristic ch n \operatorname{ch}_n ch n h ∈ Cl Q ( S n ) h\in\operatorname{Cl}_{\mathbb{Q}}(S_n) h ∈ Cl Q ( S n )
⟨ p ν , ch n ( h ) ⟩ = h ( ν ) (28) \langle p_\nu,\operatorname{ch}_n(h)\rangle=h(\nu) \tag*{(28)} ⟨ p ν , ch n ( h )⟩ = h ( ν ) (28) for every partition ν \nu ν n n n (20) and (26) .
Products of symmetric groups
For any finite groups G G G H H H
Cl Q ( G ) ⊗ Q Cl Q ( H ) ⟶ Cl Q ( G × H ) , f ⊗ h ⟼ ( ( g , k ) ⟼ f ( g ) h ( k ) ) (29) \operatorname{Cl}_{\mathbb{Q}}(G) \otimes_{\mathbb{Q}} \operatorname{Cl}_{\mathbb{Q}}(H) \longrightarrow\operatorname{Cl}_{\mathbb{Q}}(G \times H), \qquad f \otimes h \longmapsto\bigl((g,k) \longmapsto f(g)h(k)\bigr) \tag*{(29)} Cl Q ( G ) ⊗ Q Cl Q ( H ) ⟶ Cl Q ( G × H ) , f ⊗ h ⟼ ( ( g , k ) ⟼ f ( g ) h ( k ) ) (29) is an isomorphism of Q \mathbb{Q} Q G × H G \times H G × H Q X ⊗ Q Q Y ≅ Q X × Y \mathbb{Q}^{X} \otimes_{\mathbb{Q}} \mathbb{Q}^{Y} \cong\mathbb{Q}^{X \times Y} Q X ⊗ Q Q Y ≅ Q X × Y X X X Y Y Y
Cl Q ( S m ) ⊗ Q Cl Q ( S n ) → ≅ Cl Q ( S m × S n ) (30) \operatorname{Cl}_{\mathbb{Q}}(S_m) \otimes_{\mathbb{Q}} \operatorname{Cl}_{\mathbb{Q}}(S_n) \xrightarrow{\ \cong\ } \operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) \tag*{(30)} Cl Q ( S m ) ⊗ Q Cl Q ( S n ) ≅ Cl Q ( S m × S n ) (30) which sends every f ⊗ g f \otimes g f ⊗ g S m × S n S_m \times S_n S m × S n ( σ , τ ) ⟼ f ( σ ) g ( τ ) (\sigma,\tau) \longmapsto f(\sigma)g(\tau) ( σ , τ ) ⟼ f ( σ ) g ( τ ) f ⊠ g f \boxtimes g f ⊠ g f 1 , f 2 ∈ Cl Q ( S m ) f_1,f_2 \in\operatorname{Cl}_{\mathbb{Q}}(S_m) f 1 , f 2 ∈ Cl Q ( S m ) g 1 , g 2 ∈ Cl Q ( S n ) g_1,g_2 \in\operatorname{Cl}_{\mathbb{Q}}(S_n) g 1 , g 2 ∈ Cl Q ( S n )
⟨ f 1 ⊗ g 1 , f 2 ⊗ g 2 ⟩ S m × S n = ⟨ f 1 , f 2 ⟩ S m ⟨ g 1 , g 2 ⟩ S n . (31) \left\langle f_1 \otimes g_1,f_2 \otimes g_2 \right\rangle_{S_m \times S_n} = \left\langle f_1,f_2 \right\rangle_{S_m}\left\langle g_1,g_2 \right\rangle_{S_n}. \tag*{(31)} ⟨ f 1 ⊗ g 1 , f 2 ⊗ g 2 ⟩ S m × S n = ⟨ f 1 , f 2 ⟩ S m ⟨ g 1 , g 2 ⟩ S n . (31) This follows from the definition of the scalar product using a simple double-sum computation.
On the other hand, if U U U Q [ S m ] \mathbb{Q}[S_m] Q [ S m ] V V V Q [ S n ] \mathbb{Q}[S_n] Q [ S n ] Q [ S m × S n ] \mathbb{Q}[S_m \times S_n] Q [ S m × S n ] U ⊠ V U \boxtimes V U ⊠ V external tensor product of U U U V V V U ⊗ V U \otimes V U ⊗ V S m S_m S m S n S_n S n
R ( S m ) ⊗ Z R ( S n ) ⟶ R ( S m × S n ) (32) R(S_m) \otimes_{\mathbb{Z}} R(S_n) \longrightarrow R(S_m \times S_n) \tag*{(32)} R ( S m ) ⊗ Z R ( S n ) ⟶ R ( S m × S n ) (32) that sends each [ U ] ⊗ [ V ] [U] \otimes[V] [ U ] ⊗ [ V ] [ U ⊠ V ] [U \boxtimes V] [ U ⊠ V ]
There is furthermore a canonical ring homomorphism
R ( S m ) ⊗ Z R ( S n ) ⟶ Cl Q ( S m ) ⊗ Z Cl Q ( S n ) = Cl Q ( S m ) ⊗ Q Cl Q ( S n ) (33) R(S_m) \otimes_{\mathbb{Z}} R(S_n) \longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m) \otimes_{\mathbb{Z}} \operatorname{Cl}_{\mathbb{Q}}(S_n) = \operatorname{Cl}_{\mathbb{Q}}(S_m) \otimes_{\mathbb{Q}} \operatorname{Cl}_{\mathbb{Q}}(S_n) \tag*{(33)} R ( S m ) ⊗ Z R ( S n ) ⟶ Cl Q ( S m ) ⊗ Z Cl Q ( S n ) = Cl Q ( S m ) ⊗ Q Cl Q ( S n ) (33) obtained by tensoring the character embeddings of S m S_m S m S n S_n S n Q \mathbb{Q} Q Z \mathbb{Z} Z Q \mathbb{Q} Q
Proposition 3.3. The ring homomorphism (32) is an isomorphism, and the ring homomorphism (33) is injective. These two homomorphisms as well as the isomorphism (30) and the character embedding R ( S m × S n ) ⟶ Cl Q ( S m × S n ) R(S_m \times S_n) \longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) R ( S m × S n ) ⟶ Cl Q ( S m × S n )
R ( S m ) ⊗ Z R ( S n ) → ∼ R ( S m × S n ) ↓ ↓ Cl Q ( S m ) ⊗ Q Cl Q ( S n ) → ∼ Cl Q ( S m × S n ) (34) \begin{array}{ccc}
R(S_m) \otimes_{\mathbb{Z}} R(S_n) & \xrightarrow{\ \sim\ } & R(S_m \times S_n) \\
\downarrow& & \downarrow\\
\operatorname{Cl}_{\mathbb{Q}}(S_m) \otimes_{\mathbb{Q}} \operatorname{Cl}_{\mathbb{Q}}(S_n) & \xrightarrow{\ \sim\ } & \operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) \tag*{(34)}
\end{array} R ( S m ) ⊗ Z R ( S n ) ↓ Cl Q ( S m ) ⊗ Q Cl Q ( S n ) ∼ ∼ R ( S m × S n ) ↓ Cl Q ( S m × S n ) (34) Proof. Any Q [ S m ] \mathbb{Q}[S_m] Q [ S m ] U U U Q [ S n ] \mathbb{Q}[S_n] Q [ S n ] V V V
χ U ⊠ V = χ U ⊠ χ V . \chi_{U \boxtimes V} = \chi_U \boxtimes\chi_V. χ U ⊠ V = χ U ⊠ χ V . Thus, the diagram (34) is commutative. Moreover, this shows that any partitions λ , ν \lambda,\nu λ , ν m m m μ , ρ \mu,\rho μ , ρ n n n
⟨ χ S λ ⊠ S μ , χ S ν ⊠ S ρ ⟩ S m × S n = ⟨ χ λ , χ ν ⟩ S m ⟨ χ μ , χ ρ ⟩ S n (35) \left\langle\chi^{S^\lambda\boxtimes S^\mu},\chi^{S^\nu\boxtimes S^\rho}\right\rangle_{S_m\times S_n}
=\left\langle\chi^\lambda,\chi^\nu\right\rangle_{S_m}\left\langle\chi^\mu,\chi^\rho\right\rangle_{S_n} \tag*{(35)} ⟨ χ S λ ⊠ S μ , χ S ν ⊠ S ρ ⟩ S m × S n = ⟨ χ λ , χ ν ⟩ S m ⟨ χ μ , χ ρ ⟩ S n (35) (by (31))
= δ λ ν δ μ ρ =\delta_{\lambda\nu}\delta_{\mu\rho} = δ λ ν δ μ ρ (since the χ λ \chi^\lambda χ λ
In particular, each S λ ⊠ S μ S^\lambda\boxtimes S^\mu S λ ⊠ S μ 5 6 Q [ S m × S n ] \mathbb{Q}[S_m\times S_n] Q [ S m × S n ] S λ ⊠ S μ S^\lambda\boxtimes S^\mu S λ ⊠ S μ ( λ , μ ) (\lambda,\mu) ( λ , μ ) p ( m ) p ( n ) p(m)p(n) p ( m ) p ( n ) χ S λ ⊠ S μ \chi^{S^\lambda\boxtimes S^\mu} χ S λ ⊠ S μ Q \mathbb{Q} Q Cl Q ( S m × S n ) \operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n) Cl Q ( S m × S n ) p ( k ) p(k) p ( k ) k k k p ( m ) p ( n ) p(m)p(n) p ( m ) p ( n ) Q \mathbb{Q} Q S m × S n S_m\times S_n S m × S n p ( m ) p ( n ) p(m)p(n) p ( m ) p ( n ) p ( m ) p ( n ) p(m)p(n) p ( m ) p ( n ) Cl Q ( S m × S n ) \operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n) Cl Q ( S m × S n ) Q [ S m × S n ] \mathbb{Q}[S_m\times S_n] Q [ S m × S n ] (25) shows that any such module would have a character orthogonal to all the χ S λ ⊠ S μ \chi^{S^\lambda\boxtimes S^\mu} χ S λ ⊠ S μ Cl Q ( S m × S n ) \operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n) Cl Q ( S m × S n ) p ( m ) p ( n ) p(m)p(n) p ( m ) p ( n ) S λ ⊠ S μ S^\lambda\boxtimes S^\mu S λ ⊠ S μ S m × S n S_m\times S_n S m × S n (32) is an isomorphism.
So the top row of the diagram (34) is an isomorphism, while the right column is injective (being the character embedding). Hence, the left column must be injective as well. In other words, (33) is injective. This completes the proof.
Note that we could have saved ourselves some trouble in the above proof if we recalled the general result that if G G G H H H K K K G = S m G=S_m G = S m H = S n H=S_n H = S n K K K G × H G\times H G × H K K K U ⊠ V U\boxtimes V U ⊠ V U U U G G G V V V H H H
Proposition 3.3 allows us to move freely between representations/class functions of S m × S n S_m\times S_n S m × S n
Arithmetic and anti-arithmetic products
We now prove Theorem 1.2. The argument is most transparent after passing to the adjoints of the two products.
Two pullback maps on conjugacy classes
First, we define two maps from S m × S n S_m\times S_n S m × S n S m n S_{mn} S mn arithmetic rule B \mathcal{B} B anti-arithmetic rule A \mathcal{A} A
To define them, we let σ ∈ S m \sigma\in S_m σ ∈ S m τ ∈ S n \tau\in S_n τ ∈ S n
For the arithmetic rule, let B ( σ , τ ) \mathcal{B}(\sigma,\tau) B ( σ , τ ) [ m ] × [ n ] [m]\times[n] [ m ] × [ n ]
B ( σ , τ ) ( i , j ) = ( σ ( i ) , τ ( j ) ) . (36) \mathcal{B}(\sigma,\tau)(i,j)=(\sigma(i),\tau(j)). \tag*{(36)} B ( σ , τ ) ( i , j ) = ( σ ( i ) , τ ( j )) . (36) This permutation B ( σ , τ ) \mathcal{B}(\sigma,\tau) B ( σ , τ ) σ × τ \sigma\times\tau σ × τ a a a ( x 1 , x 2 , … , x a ) (x_1,x_2,\ldots,x_a) ( x 1 , x 2 , … , x a ) σ \sigma σ b b b ( y 1 , y 2 , … , y b ) (y_1,y_2,\ldots,y_b) ( y 1 , y 2 , … , y b ) τ \tau τ
gcd ( a , b ) cycles of length lcm ( a , b ) (37) \gcd(a,b)\text{ cycles of length }\operatorname{lcm}(a,b) \tag*{(37)} g cd( a , b ) cycles of length lcm ( a , b ) (37) in the cycle decomposition of B ( σ , τ ) \mathcal{B}(\sigma,\tau) B ( σ , τ ) 7
B : S m × S n ⟶ S m n (38) \mathcal{B}:S_m\times S_n\longrightarrow S_{mn} \tag*{(38)} B : S m × S n ⟶ S mn (38) is a genuine group homomorphism (after choosing an identification [ m ] × [ n ] ≅ [ m n ] [m]\times[n]\cong[mn] [ m ] × [ n ] ≅ [ mn ] (1) .
For the anti-arithmetic rule, we define A ( σ , τ ) ∈ S m n \mathcal{A}(\sigma,\tau)\in S_{mn} A ( σ , τ ) ∈ S mn A ( σ , τ ) \mathcal{A}(\sigma,\tau) A ( σ , τ ) a a a σ \sigma σ b b b τ \tau τ A ( σ , τ ) \mathcal{A}(\sigma,\tau) A ( σ , τ )
lcm ( a , b ) cycles of length gcd ( a , b ) . (39) \operatorname{lcm}(a,b)\text{ cycles of length }\gcd(a,b). \tag*{(39)} lcm ( a , b ) cycles of length g cd( a , b ) . (39) The total number of letters contributed by this pair is again a b ab ab m n mn mn S m n S_{mn} S mn A ( σ , τ ) \mathcal{A}(\sigma,\tau) A ( σ , τ ) A \mathcal{A} A
The two class maps define pullbacks
Δ m , n ⊡ : Cl Q ( S m n ) ⟶ Cl Q ( S m × S n ) , (40) \Delta^{\boxdot}_{m,n}:\operatorname{Cl}_{\mathbb{Q}}(S_{mn})\longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n), \tag*{(40)} Δ m , n ⊡ : Cl Q ( S mn ) ⟶ Cl Q ( S m × S n ) , (40) Δ m , n ⋄ : Cl Q ( S m n ) ⟶ Cl Q ( S m × S n ) (41) \Delta^{\diamond}_{m,n}:\operatorname{Cl}_{\mathbb{Q}}(S_{mn})\longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n) \tag*{(41)} Δ m , n ⋄ : Cl Q ( S mn ) ⟶ Cl Q ( S m × S n ) (41) by
( Δ m , n ⊡ f ) ( σ , τ ) = f ( B ( σ , τ ) ) , ( Δ m , n ⋄ f ) ( σ , τ ) = f ( A ( σ , τ ) ) . \begin{aligned}
(\Delta^{\boxdot}_{m,n}f)(\sigma,\tau)&=f(\mathcal{B}(\sigma,\tau)),\\
(\Delta^{\diamond}_{m,n}f)(\sigma,\tau)&=f(\mathcal{A}(\sigma,\tau)).
\end{aligned} ( Δ m , n ⊡ f ) ( σ , τ ) ( Δ m , n ⋄ f ) ( σ , τ ) = f ( B ( σ , τ )) , = f ( A ( σ , τ )) . Both Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ Δ m , n ⋄ \Delta^{\diamond}_{m,n} Δ m , n ⋄
These pullbacks are the adjoints of the two products
Recall that the Frobenius characteristic is an isomorphism ch k : Cl Q ( S k ) ⟶ Symm Q , k \operatorname{ch}_k : \operatorname{Cl}_{\mathbb{Q}}(S_k) \longrightarrow\operatorname{Symm}_{\mathbb{Q},k} ch k : Cl Q ( S k ) ⟶ Symm Q , k k ≥ 0 k \ge0 k ≥ 0 Q \mathbb{Q} Q
Δ m , n ⊡ : Symm Q , m n ⟶ Symm Q , m ⊗ Q Symm Q , n (42) \Delta^{\boxdot}_{m,n} : \operatorname{Symm}_{\mathbb{Q},mn} \longrightarrow\operatorname{Symm}_{\mathbb{Q},m} \otimes_{\mathbb{Q}} \operatorname{Symm}_{\mathbb{Q},n} \tag*{(42)} Δ m , n ⊡ : Symm Q , mn ⟶ Symm Q , m ⊗ Q Symm Q , n (42) Δ m , n ⋄ : Symm Q , m n ⟶ Symm Q , m ⊗ Q Symm Q , n (43) \Delta^{\diamond}_{m,n} : \operatorname{Symm}_{\mathbb{Q},mn} \longrightarrow\operatorname{Symm}_{\mathbb{Q},m} \otimes_{\mathbb{Q}} \operatorname{Symm}_{\mathbb{Q},n} \tag*{(43)} Δ m , n ⋄ : Symm Q , mn ⟶ Symm Q , m ⊗ Q Symm Q , n (43) (we use the same symbols as for the original two maps (39) and (40)) so that the following diagram commutes for each ∘ ∈ { ⊡ , ⋄ } \circ\in\{\boxdot, \diamond\} ∘ ∈ { ⊡ , ⋄ }
\begin{CD}
\operatorname{Cl}_{\mathbb{Q}}(S_{mn}) @>{\Delta^{\circ}_{m,n}}>> \operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) \\
@V{\operatorname{ch}_{mn}}V{\cong}V @V{\cong}V \\
\operatorname{Symm}_{\mathbb{Q},mn} @>{\Delta^{\circ}_{m,n}}>> \operatorname{Symm}_{\mathbb{Q},m} \otimes_{\mathbb{Q}} \operatorname{Symm}_{\mathbb{Q},n} \tag*{(44)}
\end{CD}
where the right vertical arrow is the composition of the inverse of (30) with ch m ⊗ ch n \operatorname{ch}_m \otimes\operatorname{ch}_n ch m ⊗ ch n Cl Q ( S m ) ⊗ Q Cl Q ( S n ) ⟶ Symm Q , m ⊗ Q Symm Q , n \operatorname{Cl}_{\mathbb{Q}}(S_m) \otimes_{\mathbb{Q}} \operatorname{Cl}_{\mathbb{Q}}(S_n) \longrightarrow\operatorname{Symm}_{\mathbb{Q},m} \otimes_{\mathbb{Q}} \operatorname{Symm}_{\mathbb{Q},n} Cl Q ( S m ) ⊗ Q Cl Q ( S n ) ⟶ Symm Q , m ⊗ Q Symm Q , n Q \mathbb{Q} Q
⟨ F 1 ⊗ G 1 , F 2 ⊗ G 2 ⟩ = ⟨ F 1 , F 2 ⟩ ⟨ G 1 , G 2 ⟩ . (45) \langle F_1 \otimes G_1, F_2 \otimes G_2 \rangle= \langle F_1, F_2 \rangle\langle G_1, G_2 \rangle. \tag*{(45)} ⟨ F 1 ⊗ G 1 , F 2 ⊗ G 2 ⟩ = ⟨ F 1 , F 2 ⟩ ⟨ G 1 , G 2 ⟩ . (45) Proposition 4.1. For F ∈ Symm Q , m F \in\operatorname{Symm}_{\mathbb{Q},m} F ∈ Symm Q , m G ∈ Symm Q , n G \in\operatorname{Symm}_{\mathbb{Q},n} G ∈ Symm Q , n H ∈ Symm Q , m n H \in\operatorname{Symm}_{\mathbb{Q},mn} H ∈ Symm Q , mn
(a) We have
⟨ F ⊡ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ⊡ H ⟩ . (46) \langle F \boxdot G, H \rangle= \langle F \otimes G, \Delta^{\boxdot}_{m,n} H \rangle. \tag*{(46)} ⟨ F ⊡ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ⊡ H ⟩ . (46) (b) We have
⟨ F ⋄ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ⋄ H ⟩ . (47) \langle F \diamond G, H \rangle= \langle F \otimes G, \Delta^{\diamond}_{m,n} H \rangle. \tag*{(47)} ⟨ F ⋄ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ⋄ H ⟩ . (47) Proof. (a) By bilinearity, it suffices to take F = p λ F = p_\lambda F = p λ G = p μ G = p_\mu G = p μ λ = ( λ 1 , λ 2 , … , λ r ) \lambda= (\lambda_1, \lambda_2, \ldots, \lambda_r) λ = ( λ 1 , λ 2 , … , λ r ) μ = ( μ 1 , μ 2 , … , μ s ) \mu= (\mu_1, \mu_2, \ldots, \mu_s) μ = ( μ 1 , μ 2 , … , μ s )
F ⊡ G = p λ ⊡ p μ = ∏ i = 1 r ∏ j = 1 s p lcm ( λ i , μ j ) gcd ( λ i , μ j ) = p ω , F \boxdot G = p_\lambda\boxdot p_\mu= \prod_{i=1}^{r} \prod_{j=1}^{s} p_{\operatorname{lcm}(\lambda_i,\mu_j)}^{\operatorname{gcd}(\lambda_i,\mu_j)} = p_\omega, F ⊡ G = p λ ⊡ p μ = i = 1 ∏ r j = 1 ∏ s p lcm ( λ i , μ j ) gcd ( λ i , μ j ) = p ω , where ω \omega ω gcd ( λ i , μ j ) \operatorname{gcd}(\lambda_i,\mu_j) gcd ( λ i , μ j ) lcm ( λ i , μ j ) \operatorname{lcm}(\lambda_i,\mu_j) lcm ( λ i , μ j ) ( i , j ) (i,j) ( i , j ) ω \omega ω B ( σ , τ ) B(\sigma,\tau) B ( σ , τ ) σ \sigma σ λ \lambda λ τ \tau τ μ \mu μ
( Δ m , n ⊡ f ) ( λ , μ ) = f ( ω ) for any f ∈ Cl Q ( S m n ) (48) (\Delta^{\boxdot}_{m,n}f)(\lambda,\mu) = f(\omega) \qquad\text{for any } f \in\operatorname{Cl}_{\mathbb{Q}}(S_{mn}) \tag*{(48)} ( Δ m , n ⊡ f ) ( λ , μ ) = f ( ω ) for any f ∈ Cl Q ( S mn ) (48) (by the definition of Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡
Now, let h ∈ Cl Q ( S m n ) h \in\operatorname{Cl}_{\mathbb{Q}}(S_{mn}) h ∈ Cl Q ( S mn ) ch m n ( h ) = H \operatorname{ch}_{mn}(h) = H ch mn ( h ) = H
⟨ p ν , H ⟩ = h ( ν ) (49) \langle p_\nu, H \rangle= h(\nu) \tag*{(49)} ⟨ p ν , H ⟩ = h ( ν ) (49) for every partition ν \nu ν m n mn mn ν = ω \nu=\omega ν = ω
⟨ F ⊡ G , H ⟩ = h ( ω ) (50) \langle F \boxdot G,H\rangle=h(\omega) \tag*{(50)} ⟨ F ⊡ G , H ⟩ = h ( ω ) (50) (since F ⊡ G = p ω F\boxdot G=p_{\omega} F ⊡ G = p ω (46) as h ( ω ) h(\omega) h ( ω )
To compute the right-hand side, write k = Δ m , n ⊡ h ∈ Cl Q ( S m × S n ) = Cl Q ( S m ) ⊗ Q Cl Q ( S n ) k=\Delta^{\boxdot}_{m,n}h\in\operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n)=\operatorname{Cl}_{\mathbb{Q}}(S_m)\otimes_{\mathbb{Q}}\operatorname{Cl}_{\mathbb{Q}}(S_n) k = Δ m , n ⊡ h ∈ Cl Q ( S m × S n ) = Cl Q ( S m ) ⊗ Q Cl Q ( S n ) (30) used as an identification. Thus, its Frobenius characteristic (i.e., the image of k k k ch m ⊗ ch n \operatorname{ch}_m\otimes\operatorname{ch}_n ch m ⊗ ch n
Δ m , n ⊡ H = ∑ α ⊢ m ∑ β ⊢ n k ( α , β ) p α ⊗ p β z α z β . \Delta^{\boxdot}_{m,n}H=\sum_{\alpha\vdash m}\sum_{\beta\vdash n}k(\alpha,\beta)\frac{p_{\alpha}\otimes p_{\beta}}{z_{\alpha}z_{\beta}}. Δ m , n ⊡ H = α ⊢ m ∑ β ⊢ n ∑ k ( α , β ) z α z β p α ⊗ p β . (by the definitions of ch m \operatorname{ch}_m ch m ch n \operatorname{ch}_n ch n F ⊗ G = p λ ⊗ p μ F\otimes G=p_{\lambda}\otimes p_{\mu} F ⊗ G = p λ ⊗ p μ
⟨ F ⊗ G , Δ m , n ⊡ H ⟩ = ⟨ p λ ⊗ p μ , ∑ α ⊢ m ∑ β ⊢ n k ( α , β ) p α ⊗ p β z α z β ⟩ = ∑ α ⊢ m ∑ β ⊢ n k ( α , β ) z α z β ⟨ p λ , p α ⟩ ⋅ ⟨ p μ , p β ⟩ (by (44)) = ∑ α ⊢ m ∑ β ⊢ n k ( α , β ) z α z β δ λ α z λ ⋅ δ μ β z μ (by (26)) = k ( λ , μ ) z λ z μ z λ z μ = k ( λ , μ ) = ( Δ m , n ⊡ h ) ( λ , μ ) = h ( ω ) (by (47)). \begin{aligned}
\langle F\otimes G,\Delta^{\boxdot}_{m,n}H\rangle=\left\langle p_{\lambda}\otimes p_{\mu},\sum_{\alpha\vdash m}\sum_{\beta\vdash n}k(\alpha,\beta)\frac{p_{\alpha}\otimes p_{\beta}}{z_{\alpha}z_{\beta}}\right\rangle\\
&=\sum_{\alpha\vdash m}\sum_{\beta\vdash n}\frac{k(\alpha,\beta)}{z_{\alpha}z_{\beta}}\langle p_{\lambda},p_{\alpha}\rangle\cdot\langle p_{\mu},p_{\beta}\rangle\qquad\text{(by (44))} \\
&=\sum_{\alpha\vdash m}\sum_{\beta\vdash n}\frac{k(\alpha,\beta)}{z_{\alpha}z_{\beta}}\delta_{\lambda\alpha}z_{\lambda}\cdot\delta_{\mu\beta}z_{\mu}\qquad\text{(by (26))} \\
&=\frac{k(\lambda,\mu)}{z_{\lambda}z_{\mu}}z_{\lambda}z_{\mu}=k(\lambda,\mu)=(\Delta^{\boxdot}_{m,n}h)(\lambda,\mu)=h(\omega)\qquad\text{(by (47)).}
\end{aligned} ⟨ F ⊗ G , Δ m , n ⊡ H ⟩ = ⟨ p λ ⊗ p μ , α ⊢ m ∑ β ⊢ n ∑ k ( α , β ) z α z β p α ⊗ p β ⟩ = α ⊢ m ∑ β ⊢ n ∑ z α z β k ( α , β ) ⟨ p λ , p α ⟩ ⋅ ⟨ p μ , p β ⟩ (by (44)) = α ⊢ m ∑ β ⊢ n ∑ z α z β k ( α , β ) δ λ α z λ ⋅ δ μ β z μ (by (26)) = z λ z μ k ( λ , μ ) z λ z μ = k ( λ , μ ) = ( Δ m , n ⊡ h ) ( λ , μ ) = h ( ω ) (by (47)). Comparing this with (50) , we obtain ⟨ F ⊡ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ⊡ H ⟩ \langle F\boxdot G,H\rangle=\langle F\otimes G,\Delta^{\boxdot}_{m,n}H\rangle ⟨ F ⊡ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ⊡ H ⟩
(b) The anti-arithmetic case is identical, with the anti-arithmetic cycle type in place of the arithmetic one.
Thus integrality of the products follows from integrality of their adjoints.
Proposition 4.2. Let ∘ \circ ∘ ⊡ \boxdot ⊡ ⋄ \diamond ⋄
Δ m , n ∘ ( Symm Z , m n ) ⊆ Symm Z , m ⊗ Z Symm Z , n (51) \Delta^{\circ}_{m,n}\left(\operatorname{Symm}_{\mathbb{Z},mn}\right)\subseteq\operatorname{Symm}_{\mathbb{Z},m}\otimes_{\mathbb{Z}}\operatorname{Symm}_{\mathbb{Z},n} \tag*{(51)} Δ m , n ∘ ( Symm Z , mn ) ⊆ Symm Z , m ⊗ Z Symm Z , n (51) then
Symm Z , m ∘ Symm Z , n ⊆ Symm Z , m n . \operatorname{Symm}_{\mathbb{Z},m}\circ\operatorname{Symm}_{\mathbb{Z},n}\subseteq\operatorname{Symm}_{\mathbb{Z},mn}. Symm Z , m ∘ Symm Z , n ⊆ Symm Z , mn . Proof. Take F ∈ Symm Z , m F\in\operatorname{Symm}_{\mathbb{Z},m} F ∈ Symm Z , m G ∈ Symm Z , n G\in\operatorname{Symm}_{\mathbb{Z},n} G ∈ Symm Z , n H ∈ Symm Z , m n H\in\operatorname{Symm}_{\mathbb{Z},mn} H ∈ Symm Z , mn
⟨ F ∘ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ∘ H ⟩ \langle F\circ G,H\rangle=\langle F\otimes G,\Delta^{\circ}_{m,n}H\rangle ⟨ F ∘ G , H ⟩ = ⟨ F ⊗ G , Δ m , n ∘ H ⟩ ∈ ⟨ Symm Z , m ⊗ Z Symm Z , n , Symm Z , m ⊗ Z Symm Z , n ⟩ (by (50)) \in\langle\operatorname{Symm}_{\mathbb{Z},m}\otimes_{\mathbb{Z}}\operatorname{Symm}_{\mathbb{Z},n},\operatorname{Symm}_{\mathbb{Z},m}\otimes_{\mathbb{Z}}\operatorname{Symm}_{\mathbb{Z},n}\rangle\qquad\text{(by (50))} ∈ ⟨ Symm Z , m ⊗ Z Symm Z , n , Symm Z , m ⊗ Z Symm Z , n ⟩ (by (50)) because the Schur bases on the two tensor factors are orthonormal. Lemma 3.2 now implies F ∘ G ∈ Symm Z , m n F\circ G\in\operatorname{Symm}_{\mathbb{Z},mn} F ∘ G ∈ Symm Z , mn Symm Z , m ∘ Symm Z , n ⊆ Symm Z , m n \operatorname{Symm}_{\mathbb{Z},m}\circ\operatorname{Symm}_{\mathbb{Z},n}\subseteq\operatorname{Symm}_{\mathbb{Z},mn} Symm Z , m ∘ Symm Z , n ⊆ Symm Z , mn
In view of the commutative diagram (43), condition (50) can be rewritten as
Δ m , n ∘ ( R ( S m n ) ) ⊆ R ( S m × S n ) . (52) \Delta^\circ_{m,n}\bigl(R(S_{mn})\bigr) \subseteq R(S_m \times S_n). \tag*{(52)} Δ m , n ∘ ( R ( S mn ) ) ⊆ R ( S m × S n ) . (52) Indeed, the left vertical isomorphism in (43) sends R ( S m n ) R(S_{mn}) R ( S mn ) Symm Z , m n \operatorname{Symm}_{\mathbb{Z},mn} Symm Z , mn Δ m , n ∘ ( R ( S m n ) ) \Delta^\circ_{m,n}\bigl(R(S_{mn})\bigr) Δ m , n ∘ ( R ( S mn ) ) Δ m , n ∘ ( Symm Z , m n ) \Delta^\circ_{m,n}\bigl(\operatorname{Symm}_{\mathbb{Z},mn}\bigr) Δ m , n ∘ ( Symm Z , mn ) R ( S m × S n ) R(S_m \times S_n) R ( S m × S n ) Symm Z , m ⊗ Z Symm Z , n \operatorname{Symm}_{\mathbb{Z},m} \otimes_{\mathbb{Z}} \operatorname{Symm}_{\mathbb{Z},n} Symm Z , m ⊗ Z Symm Z , n S λ ⊠ S μ S^\lambda\boxtimes S^\mu S λ ⊠ S μ Z \mathbb{Z} Z R ( S m × S n ) R(S_m \times S_n) R ( S m × S n ) s λ ⊗ s μ s_\lambda\otimes s_\mu s λ ⊗ s μ Z \mathbb{Z} Z
Power compatibility
The first decisive observation is that both class maps B \mathcal{B} B A \mathcal{A} A
For the arithmetic map B \mathcal{B} B
B ( σ r , τ r ) = B ( ( σ , τ ) r ) = B ( σ , τ ) r . (53) \mathcal{B}(\sigma^r,\tau^r)=\mathcal{B}\bigl((\sigma,\tau)^r\bigr)=\mathcal{B}(\sigma,\tau)^r. \tag*{(53)} B ( σ r , τ r ) = B ( ( σ , τ ) r ) = B ( σ , τ ) r . (53) For the anti-arithmetic map A \mathcal{A} A
Lemma 4.3 (A gcd identity). For all positive integers a , b , r a,b,r a , b , r
gcd ( a gcd ( a , r ) , b gcd ( b , r ) ) = gcd ( a , b ) gcd ( gcd ( a , b ) , r ) . (54) \gcd\left(\frac{a}{\gcd(a,r)},\frac{b}{\gcd(b,r)}\right)=\frac{\gcd(a,b)}{\gcd(\gcd(a,b),r)}. \tag*{(54)} g cd( g cd( a , r ) a , g cd( b , r ) b ) = g cd( g cd( a , b ) , r ) g cd( a , b ) . (54) Proof. A straightforward proof can be done using p p p p p p
α = v p ( a ) , β = v p ( b ) , ρ = v p ( r ) . \alpha=v_p(a),\qquad\beta=v_p(b),\qquad\rho=v_p(r). α = v p ( a ) , β = v p ( b ) , ρ = v p ( r ) . It is classical that any two positive integers s s s t t t v p ( gcd ( s , t ) ) = min ( v p ( s ) , v p ( t ) ) v_p(\gcd(s,t))=\min(v_p(s),v_p(t)) v p ( g cd( s , t )) = min ( v p ( s ) , v p ( t )) m m m
v p ( m / gcd ( m , r ) ) = ( v p ( m ) − ρ ) + , v_p\bigl(m/\gcd(m,r)\bigr)=\bigl(v_p(m)-\rho\bigr)_+, v p ( m / g cd( m , r ) ) = ( v p ( m ) − ρ ) + , where x + = max ( x , 0 ) x_+=\max(x,0) x + = max ( x , 0 ) p p p 8
min ( ( α − ρ ) + , ( β − ρ ) + ) = ( min ( α − ρ , β − ρ ) ) + = ( min ( α , β ) − ρ ) + . \min\bigl((\alpha-\rho)_+,(\beta-\rho)_+\bigr)=\bigl(\min(\alpha-\rho,\beta-\rho)\bigr)_+=\bigl(\min(\alpha,\beta)-\rho\bigr)_+. min ( ( α − ρ ) + , ( β − ρ ) + ) = ( min ( α − ρ , β − ρ ) ) + = ( min ( α , β ) − ρ ) + . The valuation of the right-hand side is
( v p ( gcd ( a , b ) ) − ρ ) + = ( min ( α , β ) − ρ ) + . \bigl(v_p(\gcd(a,b))-\rho\bigr)_+=\bigl(\min(\alpha,\beta)-\rho\bigr)_+. ( v p ( g cd( a , b )) − ρ ) + = ( min ( α , β ) − ρ ) + . Thus the two sides have the same p p p p p p □ \square □
Lemma 4.4. For all r ≥ 1 r \ge1 r ≥ 1
A ( σ r , τ r ) ∼ A ( σ , τ ) r , (55) \mathcal{A}(\sigma^r,\tau^r) \sim\mathcal{A}(\sigma,\tau)^r, \tag*{(55)} A ( σ r , τ r ) ∼ A ( σ , τ ) r , (55) where ∼ \sim ∼ S m n S_{mn} S mn
Proof. Consider one a a a σ \sigma σ b b b τ \tau τ A ( σ r , τ r ) \mathcal{A}(\sigma^r,\tau^r) A ( σ r , τ r ) A ( σ , τ ) r \mathcal{A}(\sigma,\tau)^r A ( σ , τ ) r
Set
g = gcd ( a , b ) , c = gcd ( g , r ) . g = \gcd(a,b), \qquad c = \gcd(g,r). g = g cd( a , b ) , c = g cd( g , r ) . The definition of A ( σ , τ ) \mathcal{A}(\sigma,\tau) A ( σ , τ ) σ \sigma σ τ \tau τ lcm ( a , b ) \operatorname{lcm}(a,b) lcm ( a , b ) g g g A ( σ , τ ) \mathcal{A}(\sigma,\tau) A ( σ , τ ) r r r g g g c c c g / c g/c g / c r r r g g g ( g / c , g / c , … , g / c ) (g/c,g/c,\ldots,g/c) ( g / c , g / c , … , g / c ) A ( σ , τ ) r \mathcal{A}(\sigma,\tau)^r A ( σ , τ ) r
lcm ( a , b ) c cycles of length g / c (56) \operatorname{lcm}(a,b)c \text{ cycles of length } g/c \tag*{(56)} lcm ( a , b ) c cycles of length g / c (56) from our two cycles.
Now take the r r r σ \sigma σ τ \tau τ a a a σ \sigma σ gcd ( a , r ) \gcd(a,r) g cd( a , r ) σ r \sigma^r σ r a / gcd ( a , r ) a/\gcd(a,r) a / g cd( a , r ) b b b τ \tau τ gcd ( b , r ) \gcd(b,r) g cd( b , r ) τ r \tau^r τ r b / gcd ( b , r ) b/\gcd(b,r) b / g cd( b , r ) a / gcd ( a , r ) a/\gcd(a,r) a / g cd( a , r ) σ r \sigma^r σ r b / gcd ( b , r ) b/\gcd(b,r) b / g cd( b , r ) τ r \tau^r τ r A ( σ r , τ r ) \mathcal{A}(\sigma^r,\tau^r) A ( σ r , τ r )
gcd ( a gcd ( a , r ) , b gcd ( b , r ) ) = gcd ( a , b ) gcd ( gcd ( a , b ) , r ) \gcd\left(\frac{a}{\gcd(a,r)},\frac{b}{\gcd(b,r)}\right)
= \frac{\gcd(a,b)}{\gcd(\gcd(a,b),r)} g cd( g cd( a , r ) a , g cd( b , r ) b ) = g cd( g cd( a , b ) , r ) g cd( a , b ) (by Lemma 4.3)
= g gcd ( g , r ) = g c . = \frac{g}{\gcd(g,r)} = \frac{g}{c}. = g cd( g , r ) g = c g . Since the block coming from our initial two cycles of σ \sigma σ τ \tau τ a b ab ab
a b g / c = a b g c = a b gcd ( a , b ) c = lcm ( a , b ) c . \frac{ab}{g/c} = \frac{ab}{g}c = \frac{ab}{\gcd(a,b)}c = \operatorname{lcm}(a,b)c. g / c ab = g ab c = g cd( a , b ) ab c = lcm ( a , b ) c . So A ( σ r , τ r ) \mathcal{A}(\sigma^r,\tau^r) A ( σ r , τ r )
lcm ( a , b ) c cycles of length g / c \operatorname{lcm}(a,b)c \text{ cycles of length } g/c lcm ( a , b ) c cycles of length g / c from our two cycles. This agrees with (55). So the permutations A ( σ r , τ r ) \mathcal{A}(\sigma^r,\tau^r) A ( σ r , τ r ) A ( σ , τ ) r \mathcal{A}(\sigma,\tau)^r A ( σ , τ ) r
Corollary 4.5. On rational class functions , the following statements hold.
(a) The map Δ m , n □ : Cl Q ( S m n ) ⟶ Cl Q ( S m × S n ) \Delta^{\square}_{m,n} : \operatorname{Cl}_{\mathbb{Q}}(S_{mn}) \longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) Δ m , n □ : Cl Q ( S mn ) ⟶ Cl Q ( S m × S n ) ψ \psi ψ λ \lambda λ
(b) The map Δ m , n ⋄ : Cl Q ( S m n ) ⟶ Cl Q ( S m × S n ) \Delta^{\diamond}_{m,n} : \operatorname{Cl}_{\mathbb{Q}}(S_{mn}) \longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) Δ m , n ⋄ : Cl Q ( S mn ) ⟶ Cl Q ( S m × S n ) ψ \psi ψ λ \lambda λ
Proof. For either part, let C C C B \mathcal{B} B A \mathcal{A} A σ ∈ S m \sigma\in S_m σ ∈ S m τ ∈ S n \tau\in S_n τ ∈ S n
C ( σ , τ ) r ∼ C ( σ r , τ r ) (57) C(\sigma,\tau)^r \sim C(\sigma^r,\tau^r) \tag*{(57)} C ( σ , τ ) r ∼ C ( σ r , τ r ) (57) (where ∼ \sim ∼ S m n S_{mn} S mn (53) ; for part (b), it follows from Lemma 4.4 .
Let Δ \Delta Δ Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ Δ m , n ⋄ \Delta^{\diamond}_{m,n} Δ m , n ⋄ Δ \Delta Δ ψ \psi ψ λ \lambda λ
For any class function f ∈ Cl Q ( S m n ) f \in\operatorname{Cl}_{\mathbb{Q}}(S_{mn}) f ∈ Cl Q ( S mn ) r ≥ 1 r \ge1 r ≥ 1 σ ∈ S m \sigma\in S_m σ ∈ S m τ ∈ S n \tau\in S_n τ ∈ S n
( Δ ψ r f ) ( σ , τ ) = ( ψ r f ) ( C ( σ , τ ) ) (by the definition of Δ ) = f ( C ( σ , τ ) r ) (by the definition of ψ r ) = f ( C ( σ r , τ r ) ) (by (56)) = ( Δ f ) ( σ r , τ r ) (by the definition of Δ ) = ( ψ r Δ f ) ( σ , τ ) (by the definition of ψ r ). \begin{aligned}
(\Delta\psi^r f)(\sigma,\tau) &= (\psi^r f)(C(\sigma,\tau)) && \text{(by the definition of }\Delta\text{)}\\
&= f(C(\sigma,\tau)^r) && \text{(by the definition of }\psi^r\text{)}\\
&= f(C(\sigma^r,\tau^r)) && \text{(by (56))}\\
&= (\Delta f)(\sigma^r,\tau^r) && \text{(by the definition of }\Delta\text{)}\\
&= (\psi^r\Delta f)(\sigma,\tau) && \text{(by the definition of }\psi^r\text{).}
\end{aligned} ( Δ ψ r f ) ( σ , τ ) = ( ψ r f ) ( C ( σ , τ )) = f ( C ( σ , τ ) r ) = f ( C ( σ r , τ r )) = ( Δ f ) ( σ r , τ r ) = ( ψ r Δ f ) ( σ , τ ) (by the definition of Δ ) (by the definition of ψ r ) (by (56)) (by the definition of Δ ) (by the definition of ψ r ). Thus, Δ ψ r f = ψ r Δ f \Delta\psi^r f = \psi^r\Delta f Δ ψ r f = ψ r Δ f f f f r r r Δ \Delta Δ ψ \psi ψ Δ \Delta Δ Proposition 2.6 (b) makes Δ \Delta Δ λ \lambda λ
At this point, Theorem 3.1 reduces the entire integrality problem to one calculation: we need only check that the pullback Δ ( [ M m n ] ) \Delta([M_{mn}]) Δ ([ M mn ]) Δ \Delta Δ Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ Δ m , n ⋄ \Delta^{\diamond}_{m,n} Δ m , n ⋄ M m n M_{mn} M mn S m × S n S_m \times S_n S m × S n Corollary 4.5 that Δ \Delta Δ λ \lambda λ R ( S m × S n ) R(S_m \times S_n) R ( S m × S n ) λ \lambda λ λ \lambda λ R ( S m n ) R(S_{mn}) R ( S mn ) [ M m n ] [M_{mn}] [ M mn ] R ( S m × S n ) R(S_m \times S_n) R ( S m × S n ) λ \lambda λ R ( S m n ) R(S_{mn}) R ( S mn ) Δ ( R ( S m n ) ) ⊆ R ( S m × S n ) \Delta(R(S_{mn})) \subseteq R(S_m \times S_n) Δ ( R ( S mn )) ⊆ R ( S m × S n ) (52) will follow. This, in turn, will yield (51) (since (52) is just a restatement of (51) ), and therefore (by Proposition 4.2 ) we will obtain Symm Z , m ∘ Symm Z , n ⊆ Symm Z , m n \operatorname{Symm}_{\mathbb{Z},m} \circ\operatorname{Symm}_{\mathbb{Z},n} \subseteq\operatorname{Symm}_{\mathbb{Z},mn} Symm Z , m ∘ Symm Z , n ⊆ Symm Z , mn
The arithmetic product: an actual pullback
For the arithmetic map there is nothing to construct: Since the map B \mathcal{B} B Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ M m n M_{mn} M mn S m n S_{mn} S mn (38) (and then identifying [ m n ] [mn] [ mn ] [ m ] × [ n ] [m] \times[n] [ m ] × [ n ] S m × S n S_m \times S_n S m × S n [ m ] × [ n ] [m] \times[n] [ m ] × [ n ] e i ⊗ e j e_i \otimes e_j e i ⊗ e j M m ⊠ M n M_m \boxtimes M_n M m ⊠ M n ( σ , τ ) ∈ S m × S n (\sigma,\tau) \in S_m \times S_n ( σ , τ ) ∈ S m × S n
e i ⊗ e j ⟼ e σ ( i ) ⊗ e τ ( j ) , e_i \otimes e_j \longmapsto e_{\sigma(i)} \otimes e_{\tau(j)}, e i ⊗ e j ⟼ e σ ( i ) ⊗ e τ ( j ) , which is exactly the product action on [ m ] × [ n ] [m] \times[n] [ m ] × [ n ]
Δ m , n ⊡ ( M m n ) = M m ⊠ M n ∈ R ( S m × S n ) . (58) \Delta^{\boxdot}_{m,n}(M_{mn}) = M_m \boxtimes M_n \in R(S_m \times S_n). \tag*{(58)} Δ m , n ⊡ ( M mn ) = M m ⊠ M n ∈ R ( S m × S n ) . (58) Corollary 4.5 says that Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ λ \lambda λ R ( S m × S n ) R(S_m \times S_n) R ( S m × S n ) λ \lambda λ Cl Q ( S m × S n ) \operatorname{Cl}_{\mathbb{Q}}(S_m \times S_n) Cl Q ( S m × S n ) (58) , together with the fact that M m n M_{mn} M mn R ( S m n ) R(S_{mn}) R ( S mn ) λ \lambda λ
Δ m , n ⊡ ( R ( S m n ) ) ⊆ R ( S m × S n ) . \Delta^{\boxdot}_{m,n}\left(R(S_{mn})\right) \subseteq R(S_m \times S_n). Δ m , n ⊡ ( R ( S mn ) ) ⊆ R ( S m × S n ) . That is, (52) holds for ∘ = ⊡ \circ= \boxdot ∘ = ⊡ (51) holds for ∘ = ⊡ \circ= \boxdot ∘ = ⊡ (52) is just a restatement of (51) ). Proposition 4.2 thus proves the arithmetic integrality assertion
Symm Z , m ⊡ Symm Z , n ⊆ Symm Z , m n . \operatorname{Symm}_{\mathbb{Z},m} \boxdot\operatorname{Symm}_{\mathbb{Z},n} \subseteq\operatorname{Symm}_{\mathbb{Z},mn}. Symm Z , m ⊡ Symm Z , n ⊆ Symm Z , mn . In other words, Theorem 1.2 (a) is proved.
There is also a stronger conclusion. Since B : S m × S n ↪ S m n \mathcal{B}:S_m \times S_n \hookrightarrow S_{mn} B : S m × S n ↪ S mn Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ ⊡ \boxdot ⊡ (46) shows that this map is the adjoint of Δ m , n ⊡ \Delta^{\boxdot}_{m,n} Δ m , n ⊡ λ ⊢ m \lambda\vdash m λ ⊢ m μ ⊢ n \mu\vdash n μ ⊢ n
s λ ⊡ s μ = ch m n ( Ind S m × S n S m n ( S λ ⊠ S μ ) ) , (59) s_\lambda\boxdot s_\mu= \operatorname{ch}_{mn}\left(\operatorname{Ind}_{S_m \times S_n}^{S_{mn}}\left(S^\lambda\boxtimes S^\mu\right)\right), \tag*{(59)} s λ ⊡ s μ = ch mn ( Ind S m × S n S mn ( S λ ⊠ S μ ) ) , (59) where S m × S n S_m \times S_n S m × S n s λ ⊡ s μ s_\lambda\boxdot s_\mu s λ ⊡ s μ
The same product-action restriction S m n ↓ S m × S n S_{mn} \downarrow S_m \times S_n S mn ↓ S m × S n [35 ] from the viewpoint of stable symmetric-group characters; his Kronecker-comultiplication formulas and stability results give a close representation-theoretic companion to the arithmetic side of the present note.
Detecting exact cycle lengths by Adams operations
For the anti-arithmetic map, the pullback of M m n M_{mn} M mn
For any 1 ≤ a ≤ m 1 \leq a \leq m 1 ≤ a ≤ m
C m , a = ∑ d ∣ a μ ( a / d ) ψ d ( M m ) ∈ R ( S m ) , (60) C_{m,a} = \sum_{d \mid a} \mu(a/d)\,\psi^d(M_m) \in R(S_m), \tag*{(60)} C m , a = d ∣ a ∑ μ ( a / d ) ψ d ( M m ) ∈ R ( S m ) , (60) where μ \mu μ R ( S m ) R(S_m) R ( S m )
If σ ∈ S m \sigma\in S_m σ ∈ S m m a ( σ ) m_a(\sigma) m a ( σ ) a a a
Lemma 4.6. Let σ ∈ S m \sigma\in S_m σ ∈ S m 1 ≤ a ≤ m 1 \leq a \leq m 1 ≤ a ≤ m
χ C m , a ( σ ) = a m a ( σ ) . (61) \chi_{C_{m,a}}(\sigma) = a\,m_a(\sigma). \tag*{(61)} χ C m , a ( σ ) = a m a ( σ ) . (61) Proof. The character χ M m \chi_{M_m} χ M m M m M_m M m S m S_m S m d ≥ 1 d \geq1 d ≥ 1 χ M m ( σ d ) = ∣ Fix ( σ d ) ∣ \chi_{M_m}(\sigma^d) = \lvert\operatorname{Fix}(\sigma^d)\rvert χ M m ( σ d ) = ∣ Fix ( σ d )∣ (17) yields χ ψ d ( M m ) ( σ ) = χ M m ( σ d ) \chi_{\psi^d(M_m)}(\sigma) = \chi_{M_m}(\sigma^d) χ ψ d ( M m ) ( σ ) = χ M m ( σ d )
χ ψ d ( M m ) ( σ ) = ∣ Fix ( σ d ) ∣ = ∑ b ∣ d b m b ( σ ) . (62) \chi_{\psi^d(M_m)}(\sigma) = \lvert\operatorname{Fix}(\sigma^d)\rvert= \sum_{b \mid d} b\,m_b(\sigma). \tag*{(62)} χ ψ d ( M m ) ( σ ) = ∣ Fix ( σ d )∣ = b ∣ d ∑ b m b ( σ ) . (62) Here, the last equality sign is because the fixed points of σ d \sigma^d σ d b b b σ \sigma σ b ∣ d b \mid d b ∣ d b b b b b b σ d \sigma^d σ d
Taking characters in (60) gives
χ C m , a ( σ ) = ∑ d ∣ a μ ( a / d ) χ ψ d ( M m ) ( σ ) = ∑ d ∣ a μ ( a / d ) ∑ b ∣ d b m b ( σ ) (by (61)) = ∑ b ∣ a b m b ( σ ) ∑ d : b ∣ d ∣ a μ ( a / d ) . (63) \begin{aligned}
\chi_{C_{m,a}}(\sigma)=\sum_{d\mid a}\mu(a/d)\chi_{\psi^d(M_m)}(\sigma) \\
&=\sum_{d\mid a}\mu(a/d)\sum_{b\mid d}b\,m_b(\sigma)\qquad\text{(by (61))} \\
&=\sum_{b\mid a}b\,m_b(\sigma)\sum_{\substack{d:\ b\mid d\mid a}}\mu(a/d). \tag*{(63)}
\end{aligned} χ C m , a ( σ ) = d ∣ a ∑ μ ( a / d ) χ ψ d ( M m ) ( σ ) = d ∣ a ∑ μ ( a / d ) b ∣ d ∑ b m b ( σ ) (by (61)) = b ∣ a ∑ b m b ( σ ) d : b ∣ d ∣ a ∑ μ ( a / d ) . (63) The inner sum is 1 for b = a b=a b = a (63) simplifies to χ C m , a ( σ ) = a m a ( σ ) \chi_{C_{m,a}}(\sigma)=a\,m_a(\sigma) χ C m , a ( σ ) = a m a ( σ ) (61) .
Thus C m , a C_{m,a} C m , a exactly length a a a
Example 4.7. For the first few values of a a a (60) gives (whenever the displayed indices are at most m m m
C m , 1 = M m , C m , 2 = ψ 2 ( M m ) − M m , C_{m,1}=M_m,\qquad C_{m,2}=\psi^2(M_m)-M_m, C m , 1 = M m , C m , 2 = ψ 2 ( M m ) − M m , and
C m , 6 = ψ 6 ( M m ) − ψ 3 ( M m ) − ψ 2 ( M m ) + M m . C_{m,6}=\psi^6(M_m)-\psi^3(M_m)-\psi^2(M_m)+M_m. C m , 6 = ψ 6 ( M m ) − ψ 3 ( M m ) − ψ 2 ( M m ) + M m . Accordingly, their characters are respectively m 1 ( σ ) m_1(\sigma) m 1 ( σ ) 2 m 2 ( σ ) 2m_2(\sigma) 2 m 2 ( σ ) 6 m 6 ( σ ) 6m_6(\sigma) 6 m 6 ( σ )
Remark 4.8 (Prior work on C m , a C_{m,a} C m , a ch m \operatorname{ch}_m ch m C m , a C_{m,a} C m , a h m − a p a h_{m-a}p_a h m − a p a [15 Definition 3.7 and Theorem 3.9] . For a partition λ \lambda λ e e e V λ [ e ] V^\lambda[e] V λ [ e ] e e e λ \lambda λ ( − 1 ) ℓ (-1)^\ell ( − 1 ) ℓ ℓ \ell ℓ
s λ p e = ∑ α ( − 1 ) ℓ ( α / λ ) s α . s_\lambda p_e=\sum_{\alpha}(-1)^{\ell(\alpha/\lambda)}s_\alpha. s λ p e = α ∑ ( − 1 ) ℓ ( α / λ ) s α . Their Theorem 3.9 says that, if a permutation σ \sigma σ k k k e e e
V λ [ e ] ( σ ) = k e χ λ ( τ ) , V^\lambda[e](\sigma)=ke\,\chi^\lambda(\tau), V λ [ e ] ( σ ) = k e χ λ ( τ ) , where τ \tau τ σ \sigma σ e e e k = 0 k=0 k = 0 λ = ( m − a ) \lambda=(m-a) λ = ( m − a ) e = a e=a e = a λ \lambda λ a = m a=m a = m
V ( m − a ) [ a ] ( σ ) = a m a ( σ ) . V^{(m-a)}[a](\sigma)=a\,m_a(\sigma). V ( m − a ) [ a ] ( σ ) = a m a ( σ ) . Hence Lemma 4.6 shows that
C m , a = V ( m − a ) [ a ] in R ( S m ) . C_{m,a}=V^{(m-a)}[a]\qquad\text{in }R(S_m). C m , a = V ( m − a ) [ a ] in R ( S m ) . Thus the virtual character C m , a C_{m,a} C m , a (60) gives a different Adams–Möbius expression for this special case.
The anti-arithmetic substitute for the natural representation
Define
W m , n = ∑ 1 ≤ a ≤ m , 1 ≤ b ≤ n ; gcd ( a , b ) = 1 C m , a ⊠ C n , b ∈ R ( S m × S n ) (64) W_{m,n}=\sum_{\substack{1\leq a\leq m,\ 1\leq b\leq n;\\ \gcd(a,b)=1}} C_{m,a}\boxtimes C_{n,b}\in R(S_m\times S_n) \tag*{(64)} W m , n = 1 ≤ a ≤ m , 1 ≤ b ≤ n ; g c d ( a , b ) = 1 ∑ C m , a ⊠ C n , b ∈ R ( S m × S n ) (64) (this holds because C m , a ∈ R ( S m ) C_{m,a}\in R(S_m) C m , a ∈ R ( S m ) C n , b ∈ R ( S n ) C_{n,b}\in R(S_n) C n , b ∈ R ( S n )
Proposition 4.10. Using the character embedding to identify each representation ring R ( G ) R(G) R ( G ) C l Q ( G ) \mathrm{Cl}_{\mathbb{Q}}(G) Cl Q ( G )
(a) In C l Q ( S m × S n ) \mathrm{Cl}_{\mathbb{Q}}(S_m\times S_n) Cl Q ( S m × S n )
Δ m , n ⋄ ( M m n ) = W m , n . (65) \Delta^\diamond_{m,n}(M_{mn})=W_{m,n}. \tag*{(65)} Δ m , n ⋄ ( M mn ) = W m , n . (65) (b) We have Δ m , n ⋄ ( M m n ) ∈ R ( S m × S n ) \Delta^\diamond_{m,n}(M_{mn})\in R(S_m\times S_n) Δ m , n ⋄ ( M mn ) ∈ R ( S m × S n )
Proof. For ( σ , τ ) ∈ S m × S n (\sigma,\tau)\in S_m\times S_n ( σ , τ ) ∈ S m × S n
χ W m , n ( σ , τ ) = ∑ 1 ≤ a ≤ m , 1 ≤ b ≤ n gcd ( a , b ) = 1 a m a ( σ ) b m b ( τ ) . (66) \chi_{W_{m,n}}(\sigma,\tau)=\sum_{\substack{1\leq a\leq m,\ 1\leq b\leq n\\ \gcd(a,b)=1}}a\,m_a(\sigma)\,b\,m_b(\tau). \tag*{(66)} χ W m , n ( σ , τ ) = 1 ≤ a ≤ m , 1 ≤ b ≤ n g c d ( a , b ) = 1 ∑ a m a ( σ ) b m b ( τ ) . (66) Indeed, characters multiply under external tensor products:
χ C m , a ⊠ C n , b ( σ , τ ) = χ C m , a ( σ ) χ C n , b ( τ ) , \chi_{C_{m,a}\boxtimes C_{n,b}}(\sigma,\tau)=\chi_{C_{m,a}}(\sigma)\chi_{C_{n,b}}(\tau), χ C m , a ⊠ C n , b ( σ , τ ) = χ C m , a ( σ ) χ C n , b ( τ ) , and Lemma 4.6 evaluates these two factors as a m a ( σ ) a\,m_a(\sigma) a m a ( σ ) b m b ( τ ) b\,m_b(\tau) b m b ( τ )
On the other hand, the character of M m n M_{mn} M mn χ M m n ( A ( σ , τ ) ) \chi_{M_{mn}}(A(\sigma,\tau)) χ M mn ( A ( σ , τ )) A ( σ , τ ) A(\sigma,\tau) A ( σ , τ )
A pair consisting of an a a a σ \sigma σ b b b τ \tau τ lcm ( a , b ) \operatorname{lcm}(a,b) lcm ( a , b ) gcd ( a , b ) \gcd(a,b) g cd( a , b ) A ( σ , τ ) A(\sigma,\tau) A ( σ , τ ) gcd ( a , b ) = 1 \gcd(a,b)=1 g cd( a , b ) = 1
lcm ( a , b ) = a b \operatorname{lcm}(a,b)=ab lcm ( a , b ) = ab of them. Thus, the given pair of cycles contributes exactly a b ab ab A ( σ , τ ) A(\sigma,\tau) A ( σ , τ ) gcd ( a , b ) = 1 \gcd(a,b)=1 g cd( a , b ) = 1
χ M m n ( A ( σ , τ ) ) = ∑ 1 ≤ a ≤ m , 1 ≤ b ≤ n gcd ( a , b ) = 1 m a ( σ ) m b ( τ ) ⋅ a b = ∑ 1 ≤ a ≤ m , 1 ≤ b ≤ n gcd ( a , b ) = 1 a m a ( σ ) b m b ( τ ) . \chi_{M_{mn}}(A(\sigma,\tau))=
\sum_{\substack{1\leq a\leq m,\ 1\leq b\leq n\\ \gcd(a,b)=1}}m_a(\sigma)m_b(\tau)\cdot ab
=
\sum_{\substack{1\leq a\leq m,\ 1\leq b\leq n\\ \gcd(a,b)=1}}a\,m_a(\sigma)b\,m_b(\tau). χ M mn ( A ( σ , τ )) = 1 ≤ a ≤ m , 1 ≤ b ≤ n g c d ( a , b ) = 1 ∑ m a ( σ ) m b ( τ ) ⋅ ab = 1 ≤ a ≤ m , 1 ≤ b ≤ n g c d ( a , b ) = 1 ∑ a m a ( σ ) b m b ( τ ) . Comparing this with (65), we find
χ W m , n ( σ , τ ) = χ M m n ( A ( σ , τ ) ) = ( Δ m , n ⋄ ( χ M m n ) ) ( σ , τ ) . \chi_{W_{m,n}}(\sigma,\tau)=\chi_{M_{mn}}(A(\sigma,\tau))=\left(\Delta^\diamond_{m,n}(\chi_{M_{mn}})\right)(\sigma,\tau). χ W m , n ( σ , τ ) = χ M mn ( A ( σ , τ )) = ( Δ m , n ⋄ ( χ M mn ) ) ( σ , τ ) . Since σ \sigma σ τ \tau τ χ W m , n = Δ m , n ⋄ ( χ M m n ) \chi_{W_{m,n}}=\Delta^\diamond_{m,n}(\chi_{M_{mn}}) χ W m , n = Δ m , n ⋄ ( χ M mn ) W m , n ∈ R ( S m × S n ) W_{m,n}\in R(S_m\times S_n) W m , n ∈ R ( S m × S n )
We can now finish the proof of Theorem 1.2 in a few lines of λ \lambda λ
Proposition 4.11. For all m , n ≥ 1 m,n\geq1 m , n ≥ 1
Δ m , n ⋄ ( R ( S m n ) ) ⊆ R ( S m × S n ) . (67) \Delta^\diamond_{m,n}(R(S_{mn}))\subseteq R(S_m\times S_n). \tag*{(67)} Δ m , n ⋄ ( R ( S mn )) ⊆ R ( S m × S n ) . (67) Proof. By Corollary 4.5 (b), the map Δ m , n ⋄ \Delta^\diamond_{m,n} Δ m , n ⋄ λ \lambda λ
Cl Q ( S m n ) ⟶ Cl Q ( S m × S n ) . \operatorname{Cl}_{\mathbb{Q}}(S_{mn})\longrightarrow\operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n). Cl Q ( S mn ) ⟶ Cl Q ( S m × S n ) . By Proposition 4.10 (b), it sends the natural representation M m n M_{mn} M mn λ \lambda λ R ( S m × S n ) ⊆ Cl Q ( S m × S n ) R(S_m\times S_n)\subseteq\operatorname{Cl}_{\mathbb{Q}}(S_m\times S_n) R ( S m × S n ) ⊆ Cl Q ( S m × S n ) m n mn mn n n n M m n M_{mn} M mn R ( S m n ) R(S_{mn}) R ( S mn ) λ \lambda λ R ( S m n ) R(S_{mn}) R ( S mn ) R ( S m × S n ) R(S_m\times S_n) R ( S m × S n )
Proof of Theorem 1.2. We have already proved parts (a) and (b) in Subsection 4.4.
(c) Proposition 4.11 shows that (51) holds for ∘ = ⋄ \circ=\diamond ∘ = ⋄ ∘ = ⋄ \circ=\diamond ∘ = ⋄ Sym Z , m ⋄ Sym Z , n ⊆ Sym Z , m n \operatorname{Sym}_{\mathbb{Z},m}\diamond\operatorname{Sym}_{\mathbb{Z},n}\subseteq\operatorname{Sym}_{\mathbb{Z},mn} Sym Z , m ⋄ Sym Z , n ⊆ Sym Z , mn
What the proof is really using
It may be useful to isolate the short core of the above argument for the anti-arithmetic product. There are four ingredients.
(i) The anti-arithmetic map on conjugacy classes is compatible with powers:
A ( σ r , τ r ) ∼ A ( σ , τ ) r . \mathcal{A}(\sigma^r,\tau^r) \sim\mathcal{A}(\sigma,\tau)^r. A ( σ r , τ r ) ∼ A ( σ , τ ) r . Therefore its pullback is a ψ \psi ψ λ \lambda λ
(ii) The Adams operations of the natural S m S_m S m
χ ψ d ( M m ) ( σ ) = ∣ Fix ( σ d ) ∣ . \chi_{\psi^d(M_m)}(\sigma)=|\operatorname{Fix}(\sigma^d)|. χ ψ d ( M m ) ( σ ) = ∣ Fix ( σ d ) ∣. Möbius inversion therefore produces the exact-cycle virtual representations C m , a C_{m,a} C m , a
(iii) Coprime pairs of cycle lengths are exactly the pairs that produce fixed points under the anti-arithmetic rule. This gives the virtual representation W m , n W_{m,n} W m , n (64) , which is the anti-arithmetic pullback of M m n M_{mn} M mn
(iv) The single representation M m n M_{mn} M mn R ( S m n ) R(S_{mn}) R ( S mn ) λ \lambda λ
Some of these ingredients have their natural habitat in the Burnside ring of G G G R ( G ) R(G) R ( G ) G G G R ( S n ) R(S_n) R ( S n )
A general class-map criterion
The proof also isolates a general mechanism that may be useful elsewhere. Let G G G H H H
a : { conjugacy classes of G } ⟶ { conjugacy classes of H } a:\{\text{conjugacy classes of }G\}\longrightarrow\{\text{conjugacy classes of }H\} a : { conjugacy classes of G } ⟶ { conjugacy classes of H } a class map . Pullback gives an algebra homomorphism
a ∗ : Cl Q ( H ) ⟶ Cl Q ( G ) . a^*:\operatorname{Cl}_{\mathbb{Q}}(H)\longrightarrow\operatorname{Cl}_{\mathbb{Q}}(G). a ∗ : Cl Q ( H ) ⟶ Cl Q ( G ) . For a conjugacy class C C C H H H r ≥ 1 r\geq1 r ≥ 1 C [ r ] C^{[r]} C [ r ] h r h^r h r h h h C C C
a ( [ g r ] ) = a ( [ g ] ) [ r ] for all g ∈ G and r ≥ 1 , (68) a([g^r])=a([g])^{[r]} \qquad\text{for all }g\in G\text{ and }r\geq1, \tag*{(68)} a ([ g r ]) = a ([ g ] ) [ r ] for all g ∈ G and r ≥ 1 , (68) then a ∗ a^* a ∗ λ \lambda λ
a ∗ ( R ( H ) ) ⊆ R ( G ) . a^*(R(H))\subseteq R(G). a ∗ ( R ( H )) ⊆ R ( G ) . The anti-arithmetic proof establishes this condition by checking the image of one λ \lambda λ (68) ; integrality has to be manufactured separately, and Proposition 4.10 is exactly the missing step.
Example 4.12. A basic non-homomorphic example is the power class map
[ g ] ⟼ [ g q ] for a given q ≥ 1 [g] \longmapsto[g^q] \qquad\text{for a given } q \ge1 [ g ] ⟼ [ g q ] for a given q ≥ 1 from the conjugacy classes of a finite group G G G ψ q \psi^q ψ q R ( G ) R(G) R ( G )
Further precedents and nearby literature
Several parts of the proof have close relatives in the literature, although we do not know a previous occurrence of the anti-arithmetic integrality argument itself.
First, the use of Adams operations on R ( S n ) R(S_n) R ( S n ) r r r
f ⟼ p r { f } f \longmapsto p_r\{f\} f ⟼ p r { f } where { } \{\ \} { } [42 ] and Scharf–Thibon [38 ] use the Hopf algebra of symmetric functions to study precisely these Adams operators and to recover Littlewood’s formulas for inner plethysm. The present proof uses only the easiest part of that theory, namely the power-trace identity χ ψ r V ( g ) = χ V ( g r ) \chi^{\psi^r V}(g)=\chi^V(g^r) χ ψ r V ( g ) = χ V ( g r ) [31 ] give a useful modern account of Adams operations on finite group representation rings and characterize them as natural operations on the representation-ring functor.
Second, the cycle-length arithmetic behind the ordinary arithmetic product belongs to a broader gcd/lcm circle of ideas. The necklace ring of Metropolis–Rota [32 ] has multiplication whose structure constants involve gcd and lcm, and Dress–Siebeneicher [8 ] identify closely related necklace and Burnside-ring constructions with the big Witt vectors and λ \lambda λ [29 ] and Li [27 ] show how the ordinary arithmetic product interacts with Dirichlet series, Cartesian products, and prime decompositions of combinatorial structures. Ryba [35 ] studies the corresponding restriction S m n ↓ S m × S n S_{mn}\downarrow S_m\times S_n S mn ↓ S m × S n
Finally, the generation result for R ( S n ) R(S_n) R ( S n ) [33 ] studied generation of irreducible representations under Kronecker products already in 1955. Butler [7 ] and especially Boorman [3 ] obtained forms of the one-generator result in the language of S S S λ \lambda λ [30 ] later gave a short proof based on Dvir’s formula. Harman [23 ] extends this circle of results to representation rings of certain wreath products.
A triangular proof of Theorem 3.1
As we promised, we shall now give three proofs of Theorem 3.1, both to keep this paper self-contained and to explore the “roads less traveled” around this result.
Depth and tail
For a partition λ ⊢ n \lambda\vdash n λ ⊢ n depth by
depth ( λ ) = n − λ 1 . (69) \operatorname{depth}(\lambda)=n-\lambda_1. \tag*{(69)} depth ( λ ) = n − λ 1 . (69) Thus, if d = depth ( λ ) d=\operatorname{depth}(\lambda) d = depth ( λ ) 10
λ = ( n − d , α ) , where α ⊢ d with α 1 ≤ n − d . \lambda=(n-d,\alpha), \qquad\text{where }\alpha\vdash d\text{ with }\alpha_1\le n-d. λ = ( n − d , α ) , where α ⊢ d with α 1 ≤ n − d . Here ( n − d , α ) (n-d,\alpha) ( n − d , α ) n − d n-d n − d α \alpha α α \alpha α λ \lambda λ
α ′ = ( c 1 , c 2 , … , c r ) (70) \alpha'=(c_1,c_2,\ldots,c_r) \tag*{(70)} α ′ = ( c 1 , c 2 , … , c r ) (70) be the conjugate partition of α \alpha α
T λ = ⨂ j = 1 r ⋀ c j M n . (71) T_\lambda=\bigotimes_{j=1}^{r}\bigwedge^{c_j} M_n. \tag*{(71)} T λ = j = 1 ⨂ r ⋀ c j M n . (71) Clearly [ T λ ] [T_\lambda] [ T λ ] λ \lambda λ R ( S n ) R(S_n) R ( S n ) M n M_n M n T λ T_\lambda T λ S λ S^\lambda S λ S μ S^\mu S μ T λ T_\lambda T λ
Reminder on Schur functors
We will use Schur functors to create new representations of S n S_n S n V V V Q \mathbb{Q} Q β ⊢ d \beta\vdash d β ⊢ d V ⊗ d V^{\otimes d} V ⊗ d S d S_d S d
( v 1 ⊗ ⋯ ⊗ v d ) ⋅ g = v g ( 1 ) ⊗ ⋯ ⊗ v g ( d ) for all g ∈ S d (v_1\otimes\cdots\otimes v_d)\cdot g=v_{g(1)}\otimes\cdots\otimes v_{g(d)}\qquad\text{for all }g\in S_d ( v 1 ⊗ ⋯ ⊗ v d ) ⋅ g = v g ( 1 ) ⊗ ⋯ ⊗ v g ( d ) for all g ∈ S d (that is, the place-permutation action). It also carries the corresponding left action g ⋅ w = w ⋅ g − 1 g\cdot w=w\cdot g^{-1} g ⋅ w = w ⋅ g − 1
S H o m β ( V ) : = Hom S d ( S β , V ⊗ d ) . (72) S^\beta_{\mathrm{Hom}}(V):=\operatorname{Hom}_{S_d}(S^\beta,V^{\otimes d}). \tag*{(72)} S Hom β ( V ) := Hom S d ( S β , V ⊗ d ) . (72) Using the right action, we can instead define it by a balanced tensor product:
S ⊗ β ( V ) : = V ⊗ d ⊗ Q [ S d ] S β . (73) S^\beta_{\otimes}(V):=V^{\otimes d}\otimes_{\mathbb{Q}[S_d]}S^\beta. \tag*{(73)} S ⊗ β ( V ) := V ⊗ d ⊗ Q [ S d ] S β . (73) The latter is the construction used by Fulton [11 §8.3] . Both constructions are functorial in V V V V V V V = M n V=M_n V = M n S n S_n S n
Lemma A.1 (Multiplicity spaces and balanced tensor products).
(a) Let G G G W W W E E E Q [ G ] \mathbb{Q}[G] Q [ G ] E E E W W W G G G w ⋅ g = g − 1 w w\cdot g=g^{-1}w w ⋅ g = g − 1 w
W ⊗ Q [ G ] E ∗ ⟶ Hom G ( E , W ) , w ⊗ φ ⟼ ( e ⟼ 1 ∣ G ∣ ∑ g ∈ G φ ( g − 1 e ) g w ) . (74) W\otimes_{\mathbb{Q}[G]}E^*\longrightarrow\operatorname{Hom}_G(E,W),\qquad w\otimes\varphi\longmapsto\left(e\longmapsto\frac{1}{|G|}\sum_{g\in G}\varphi(g^{-1}e)gw\right). \tag*{(74)} W ⊗ Q [ G ] E ∗ ⟶ Hom G ( E , W ) , w ⊗ φ ⟼ e ⟼ ∣ G ∣ 1 g ∈ G ∑ φ ( g − 1 e ) g w . (74) It is natural in W W W E E E W W W G G G
(b) Consequently, a choice of G G G E ≅ E ∗ E \cong E^* E ≅ E ∗ W ⊗ Q [ G ] E ≅ Hom G ( E , W ) W \otimes_{\mathbb{Q}[G]} E \cong\operatorname{Hom}_G(E,W) W ⊗ Q [ G ] E ≅ Hom G ( E , W ) W W W
(c) In particular, if G = S d G=S_d G = S d S β ≅ ( S β ) ∗ S^\beta\cong(S^\beta)^* S β ≅ ( S β ) ∗
S ⊗ β ( V ) ≅ S Hom β ( V ) S^\beta_{\otimes}(V) \cong S^\beta_{\operatorname{Hom}}(V) S ⊗ β ( V ) ≅ S Hom β ( V ) naturally in V V V
Proof. (a) This is a combination of two standard isomorphisms in group representation theory (over fields of characteristic 0).
For any G G G U U U invariant space
U G : = { u ∈ U ∣ g u = u for all g ∈ G } U^G := \{u \in U \mid gu=u \text{ for all } g \in G\} U G := { u ∈ U ∣ g u = u for all g ∈ G } and its coinvariant space
U G : = U / span Q { g u − u ∣ u ∈ U and g ∈ G } . U_G := U \mathbin{/} \operatorname{span}_{\mathbb{Q}}\{gu-u \mid u \in U \text{ and } g \in G\}. U G := U / span Q { g u − u ∣ u ∈ U and g ∈ G } . Then, the averaging operator P U : U → U P_U:U \to U P U : U → U
P U ( u ) = 1 ∣ G ∣ ∑ g ∈ G g u P_U(u)=\frac{1}{\lvert G\rvert}\sum_{g\in G}gu P U ( u ) = ∣ G ∣ 1 g ∈ G ∑ g u is a projection onto U G U^G U G g u − u gu-u g u − u U G U_G U G
P ‾ U : U G → U G , \overline{P}_U:U_G \to U^G, P U : U G → U G , which is easily seen to be a vector space isomorphism (its inverse simply sends each u ∈ U G u \in U^G u ∈ U G U G U_G U G
The second is even more basic (and holds over any field): The ordinary tensor product W ⊗ Q E ∗ W \otimes_{\mathbb{Q}} E^* W ⊗ Q E ∗ G G G G G G E ∗ E^* E ∗ ( g φ ) ( e ) = φ ( g − 1 e ) (g\varphi)(e)=\varphi(g^{-1}e) ( g φ ) ( e ) = φ ( g − 1 e ) Hom Q ( E , W ) \operatorname{Hom}_{\mathbb{Q}}(E,W) Hom Q ( E , W ) G G G ( g f ) ( e ) = g f ( g − 1 e ) (gf)(e)=gf(g^{-1}e) ( g f ) ( e ) = g f ( g − 1 e ) f ∈ Hom Q ( E , W ) f \in\operatorname{Hom}_{\mathbb{Q}}(E,W) f ∈ Hom Q ( E , W ) g ∈ G g \in G g ∈ G e ∈ E e \in E e ∈ E
Q : W ⊗ Q E ∗ ⟶ Hom Q ( E , W ) , w ⊗ φ ⟼ ( e ↦ φ ( e ) w ) Q:W \otimes_{\mathbb{Q}} E^* \longrightarrow\operatorname{Hom}_{\mathbb{Q}}(E,W), \qquad w \otimes\varphi\longmapsto(e \mapsto\varphi(e)w) Q : W ⊗ Q E ∗ ⟶ Hom Q ( E , W ) , w ⊗ φ ⟼ ( e ↦ φ ( e ) w ) is G G G
Q G : ( W ⊗ Q E ∗ ) G ⟶ ( Hom Q ( E , W ) ) G Q^G:(W \otimes_{\mathbb{Q}} E^*)^G \longrightarrow(\operatorname{Hom}_{\mathbb{Q}}(E,W))^G Q G : ( W ⊗ Q E ∗ ) G ⟶ ( Hom Q ( E , W ) ) G on the invariant spaces.
Now, we combine the two ingredients. Applying the above-constructed isomorphism P ‾ U : U G → U G \overline{P}_U:U_G \to U^G P U : U G → U G U = W ⊗ Q E ∗ U=W \otimes_{\mathbb{Q}} E^* U = W ⊗ Q E ∗ Q G Q^G Q G
( W ⊗ Q E ∗ ) G → ≅ P ‾ W ⊗ Q E ∗ ( W ⊗ Q E ∗ ) G → ≅ Q G ( Hom Q ( E , W ) ) G . (W \otimes_{\mathbb{Q}} E^*)_G \xrightarrow[\cong]{\overline{P}_{W\otimes_{\mathbb{Q}}E^*}} (W \otimes_{\mathbb{Q}} E^*)^G \xrightarrow[\cong]{Q^G} (\operatorname{Hom}_{\mathbb{Q}}(E,W))^G. ( W ⊗ Q E ∗ ) G P W ⊗ Q E ∗ ≅ ( W ⊗ Q E ∗ ) G Q G ≅ ( Hom Q ( E , W ) ) G . But the coinvariant space ( W ⊗ Q E ∗ ) G (W \otimes_{\mathbb{Q}} E^*)_G ( W ⊗ Q E ∗ ) G W ⊗ Q [ G ] E ∗ W \otimes_{\mathbb{Q}[G]} E^* W ⊗ Q [ G ] E ∗ ( g − 1 w ) ⊗ φ = w ⊗ ( g φ ) (g^{-1}w) \otimes\varphi= w \otimes(g\varphi) ( g − 1 w ) ⊗ φ = w ⊗ ( g φ ) W ⊗ Q [ G ] E ∗ W \otimes_{\mathbb{Q}[G]} E^* W ⊗ Q [ G ] E ∗ g u = u gu = u g u = u u = ( g − 1 w ) ⊗ φ u = (g^{-1}w) \otimes\varphi u = ( g − 1 w ) ⊗ φ ( Hom Q ( E , W ) ) G (\operatorname{Hom}_{\mathbb{Q}}(E,W))^G ( Hom Q ( E , W ) ) G Hom G ( E , W ) \operatorname{Hom}_G(E,W) Hom G ( E , W ) G G G f ∈ Hom Q ( E , W ) f \in\operatorname{Hom}_{\mathbb{Q}}(E,W) f ∈ Hom Q ( E , W ) G G G g f ( g − 1 e ) = f ( e ) gf(g^{-1}e) = f(e) g f ( g − 1 e ) = f ( e ) g ∈ G g \in G g ∈ G e ∈ E e \in E e ∈ E f f f G G G W ⊗ Q [ G ] E ∗ ⟶ Hom G ( E , W ) W \otimes_{\mathbb{Q}[G]} E^* \longrightarrow\operatorname{Hom}_G(E,W) W ⊗ Q [ G ] E ∗ ⟶ Hom G ( E , W ) (74) (this follows from the definitions of P U P_U P U Q Q Q
These constructions are natural and commute with any additional action on W W W G G G
(b) This is automatic.
(c) Rational Specht modules are self-dual. One can see this directly by taking a positive definite rational bilinear form on S β S^\beta S β S d S_d S d S d S_d S d S d S_d S d S β S^\beta S β ( S β ) ∗ (S^\beta)^* ( S β ) ∗ 11 G = S d G = S_d G = S d E = S β E = S^\beta E = S β W = V ⊗ d W = V^{\otimes d} W = V ⊗ d V ⊗ d ⊗ Q [ S d ] S β ≅ Hom S d ( S β , V ⊗ d ) V^{\otimes d} \otimes_{\mathbb{Q}[S_d]} S^\beta\cong\operatorname{Hom}_{S_d}(S^\beta,V^{\otimes d}) V ⊗ d ⊗ Q [ S d ] S β ≅ Hom S d ( S β , V ⊗ d ) S ⊗ β ( V ) ≅ S Hom β ( V ) S^\beta_{\otimes}(V) \cong S^\beta_{\operatorname{Hom}}(V) S ⊗ β ( V ) ≅ S Hom β ( V )
We write S β ( V ) S^\beta(V) S β ( V ) S Hom β ( V ) S^\beta_{\operatorname{Hom}}(V) S Hom β ( V ) (72) , and use Lemma A.1 (c) to pass to the tensor definition (73) when convenient. The isomorphism between them is natural in V V V S β S^\beta S β (74) requires no such choice.
A triangular decomposition of S β ( M n ) S^\beta(M_n) S β ( M n )
Lemma A.2. Let β ⊢ d \beta\vdash d β ⊢ d β 1 ≤ n − d \beta_1 \le n-d β 1 ≤ n − d R ( S n ) R(S_n) R ( S n )
[ S β ( M n ) ] = [ S ( n − d , β ) ] + ∑ μ ⊢ n ; depth ( μ ) < d a β μ [ S μ ] (75) [S^\beta(M_n)] = [S^{(n-d,\beta)}] + \sum_{\substack{\mu\vdash n;\\ \operatorname{depth}(\mu)<d}} a_{\beta\mu}[S^\mu] \tag*{(75)} [ S β ( M n )] = [ S ( n − d , β ) ] + μ ⊢ n ; depth ( μ ) < d ∑ a β μ [ S μ ] (75) for some nonnegative integers a β μ a_{\beta\mu} a β μ
Proof. We will use the notation Inj ( X , Y ) \operatorname{Inj}(X,Y) Inj ( X , Y ) X X X Y Y Y S Y S_Y S Y S X S_X S X
Step 1: the lower part of the filtration. Filter M n ⊗ d M_n^{\otimes d} M n ⊗ d F r F_r F r
e i 1 ⊗ ⋯ ⊗ e i d for which ∣ { i 1 , … , i d } ∣ ≤ r . e_{i_1} \otimes\cdots\otimes e_{i_d} \qquad\text{for which } \lvert\{i_1,\ldots,i_d\}\rvert\le r. e i 1 ⊗ ⋯ ⊗ e i d for which ∣{ i 1 , … , i d }∣ ≤ r . This span F r F_r F r S n S_n S n S d S_d S d
0 = F − 1 ⊆ F 0 ⊆ F 1 ⊆ ⋯ ⊆ F d = M n ⊗ d 0 = F_{-1} \subseteq F_0 \subseteq F_1 \subseteq\cdots\subseteq F_d = M_n^{\otimes d} 0 = F − 1 ⊆ F 0 ⊆ F 1 ⊆ ⋯ ⊆ F d = M n ⊗ d of M n ⊗ d M_n^{\otimes d} M n ⊗ d S n S_n S n
For a tuple i = ( i 1 , … , i d ) ∈ [ n ] d \mathbf{i}=(i_1,\ldots,i_d)\in[n]^d i = ( i 1 , … , i d ) ∈ [ n ] d π ( i ) \pi(\mathbf{i}) π ( i ) [ d ] [d] [ d ] j ↦ i j j\mapsto i_j j ↦ i j j j j k k k π ( i ) \pi(\mathbf{i}) π ( i ) i j = i k i_j=i_k i j = i k π ( i ) \pi(\mathbf{i}) π ( i ) kernel partition of i \mathbf{i} i ( 3 , 7 , 3 , 3 , 7 ) (3,7,3,3,7) ( 3 , 7 , 3 , 3 , 7 ) { { 1 , 3 , 4 } , { 2 , 5 } } \{\{1,3,4\},\{2,5\}\} {{ 1 , 3 , 4 } , { 2 , 5 }}
Now fix a set partition π \pi π [ d ] [d] [ d ] r r r e i 1 ⊗ ⋯ ⊗ e i d e_{i_1}\otimes\cdots\otimes e_{i_d} e i 1 ⊗ ⋯ ⊗ e i d π ( i ) = π \pi(\mathbf{i})=\pi π ( i ) = π S n S_n S n r r r π \pi π [ n ] [n] [ n ] i j i_j i j S n S_n S n Inj ( π , [ n ] ) \operatorname{Inj}(\pi,[n]) Inj ( π , [ n ]) S n − r S_{n-r} S n − r r r r n − r n-r n − r [ n ] [n] [ n ] Q [ Inj ( π , [ n ] ) ] \mathbb{Q}[\operatorname{Inj}(\pi,[n])] Q [ Inj ( π , [ n ])]
Ind S n − r S n 1. \operatorname{Ind}_{S_{n-r}}^{S_n} \mathbf{1}. Ind S n − r S n 1 . By the branching rule12 S μ S^\mu S μ
μ 1 ≥ n − r , \mu_1\ge n-r, μ 1 ≥ n − r , and hence depth ( μ ) ≤ r \operatorname{depth}(\mu)\le r depth ( μ ) ≤ r
Thus, as an S n S_n S n
F r = ⨁ π a set partition of [ d ] ; ∣ π ∣ ≤ r Q [ Inj ( π , [ n ] ) ] , F_r=\bigoplus_{\substack{\pi\text{ a set partition of }[d];\\|\pi|\le r}}\mathbb{Q}[\operatorname{Inj}(\pi,[n])], F r = π a set partition of [ d ] ; ∣ π ∣ ≤ r ⨁ Q [ Inj ( π , [ n ])] , where ∣ π ∣ |\pi| ∣ π ∣ π \pi π π \pi π S n S_n S n
Ind S n − ∣ π ∣ S n 1. \operatorname{Ind}_{S_{n-|\pi|}}^{S_n}\mathbf{1}. Ind S n − ∣ π ∣ S n 1 . By the branching rule, every Specht module S μ S^\mu S μ
depth ( μ ) ≤ ∣ π ∣ ≤ r . \operatorname{depth}(\mu)\le|\pi|\le r. depth ( μ ) ≤ ∣ π ∣ ≤ r . Hence every S n S_n S n F r F_r F r r r r S ν S^\nu S ν ν \nu ν S n S_n S n F d − 1 F_{d-1} F d − 1 d − 1 d-1 d − 1 S n S_n S n Hom S d ( S β , F d − 1 ) \operatorname{Hom}_{S_d}(S^\beta,F_{d-1}) Hom S d ( S β , F d − 1 ) d − 1 d-1 d − 1 Hom S d ( S β , F d − 1 ) \operatorname{Hom}_{S_d}(S^\beta,F_{d-1}) Hom S d ( S β , F d − 1 ) S n S_n S n Hom Q ( S β , F d − 1 ) ≅ ⨁ F d − 1 ⊕ dim S β \operatorname{Hom}_{\mathbb{Q}}(S^\beta,F_{d-1})\cong\bigoplus F_{d-1}^{\oplus\dim S^\beta} Hom Q ( S β , F d − 1 ) ≅ ⨁ F d − 1 ⊕ d i m S β
Step 2: the top quotient. The top quotient F d / F d − 1 F_d/F_{d-1} F d / F d − 1
F d / F d − 1 ≅ Q [ Inj ( [ d ] , [ n ] ) ] (76) F_d/F_{d-1}\cong\mathbb{Q}[\operatorname{Inj}([d],[n])] \tag*{(76)} F d / F d − 1 ≅ Q [ Inj ([ d ] , [ n ])] (76) equivariantly for both commuting actions of S n S_n S n S d S_d S d S β S^\beta S β H = S d × S n − d ⊆ S n H=S_d\times S_{n-d}\subseteq S_n H = S d × S n − d ⊆ S n S d S_d S d 1 , … , d 1,\ldots,d 1 , … , d S n − d S_{n-d} S n − d d + 1 , … , n d+1,\ldots,n d + 1 , … , n f 0 : [ d ] ↪ [ n ] f_0:[d]\hookrightarrow[n] f 0 : [ d ] ↪ [ n ] f 0 ( i ) = i f_0(i)=i f 0 ( i ) = i Inj ( [ d ] , [ n ] ) \operatorname{Inj}([d],[n]) Inj ([ d ] , [ n ]) S n S_n S n S n / S n − d S_n/S_{n-d} S n / S n − d g S n − d ↦ g f 0 gS_{n-d}\mapsto gf_0 g S n − d ↦ g f 0 ( S n , S d ) (S_n,S_d) ( S n , S d )
Q [ S n ] ⊗ Q [ H ] Q [ S d ] ≅ Q [ Inj ( [ d ] , [ n ] ) ] , q ⊗ a ⟼ q f 0 a . (77) \mathbb{Q}[S_n]\otimes_{\mathbb{Q}[H]}\mathbb{Q}[S_d]\cong\mathbb{Q}[\operatorname{Inj}([d],[n])],\qquad q\otimes a\longmapsto qf_0a. \tag*{(77)} Q [ S n ] ⊗ Q [ H ] Q [ S d ] ≅ Q [ Inj ([ d ] , [ n ])] , q ⊗ a ⟼ q f 0 a . (77) where ( b , c ) ∈ H (b,c)\in H ( b , c ) ∈ H Q [ S d ] \mathbb{Q}[S_d] Q [ S d ] b b b S d S_d S d ( b , c ) f 0 = f 0 b (b,c)f_0=f_0b ( b , c ) f 0 = f 0 b S n / H S_n/H S n / H a ∈ S d a\in S_d a ∈ S d S n − d S_{n-d} S n − d S n . ) S_n.) S n . ) S d S_d S d Q [ Inj ( [ d ] , [ n ] ) ] \mathbb{Q}[\operatorname{Inj}([d],[n])] Q [ Inj ([ d ] , [ n ])] S d S_d S d g ⋅ f = f ⋅ g − 1 g\cdot f=f\cdot g^{-1} g ⋅ f = f ⋅ g − 1 g ∈ S d g\in S_d g ∈ S d f ∈ Q [ Inj ( [ d ] , [ n ] ) ] f\in\mathbb{Q}[\operatorname{Inj}([d],[n])] f ∈ Q [ Inj ([ d ] , [ n ])]
Now, we have a chain of left S n S_n S n
Hom S d ( S β , F d / F d − 1 ) ≅ Hom S d ( S β , Q [ Inj ( [ d ] , [ n ] ) ] ) (by (75)) ≅ Q [ Inj ( [ d ] , [ n ] ) ] ⊗ Q [ S d ] S β (by Lemma A.1 (b)) ≅ ( Q [ S n ] ⊗ Q [ H ] Q [ S d ] ) ⊗ Q [ S d ] S β (by (76)) ≅ Q [ S n ] ⊗ Q [ H ] ( Q [ S d ] ⊗ Q [ S d ] S β ) (by associativity of ⊗ ) ≅ Q [ S n ] ⊗ Q [ H ] ( S β ⊠ 1 ) \begin{aligned}
\operatorname{Hom}_{S_d}(S^\beta,F_d/F_{d-1})&\cong\operatorname{Hom}_{S_d}(S^\beta,\mathbb{Q}[\operatorname{Inj}([d],[n])]) &&\text{(by (75))}\\
&\cong\mathbb{Q}[\operatorname{Inj}([d],[n])]\otimes_{\mathbb{Q}[S_d]}S^\beta&&\text{(by Lemma A.1 (b))}\\
&\cong\left(\mathbb{Q}[S_n]\otimes_{\mathbb{Q}[H]}\mathbb{Q}[S_d]\right)\otimes_{\mathbb{Q}[S_d]}S^\beta&&\text{(by (76))}\\
&\cong\mathbb{Q}[S_n]\otimes_{\mathbb{Q}[H]}\left(\mathbb{Q}[S_d]\otimes_{\mathbb{Q}[S_d]}S^\beta\right) &&\text{(by associativity of }\otimes\text{)}\\
&\cong\mathbb{Q}[S_n]\otimes_{\mathbb{Q}[H]}(S^\beta\boxtimes1)
\end{aligned} Hom S d ( S β , F d / F d − 1 ) ≅ Hom S d ( S β , Q [ Inj ([ d ] , [ n ])]) ≅ Q [ Inj ([ d ] , [ n ])] ⊗ Q [ S d ] S β ≅ ( Q [ S n ] ⊗ Q [ H ] Q [ S d ] ) ⊗ Q [ S d ] S β ≅ Q [ S n ] ⊗ Q [ H ] ( Q [ S d ] ⊗ Q [ S d ] S β ) ≅ Q [ S n ] ⊗ Q [ H ] ( S β ⊠ 1 ) (by (75)) (by Lemma A.1 (b)) (by (76)) (by associativity of ⊗ ) where 1 1 1 S n − d S_{n-d} S n − d S β ⊠ 1 S^\beta\boxtimes1 S β ⊠ 1 S d × S n − d = H S_d\times S_{n-d}=H S d × S n − d = H
= Ind S d × S n − d S n ( S β ⊠ 1 ) . (78) =\operatorname{Ind}_{S_d\times S_{n-d}}^{S_n}(S^\beta\boxtimes1). \tag*{(78)} = Ind S d × S n − d S n ( S β ⊠ 1 ) . (78) Here, the last ≅ \cong ≅ H H H Q [ S d ] ⊗ Q [ S d ] S β ≅ S β ⊠ 1 \mathbb{Q}[S_d]\otimes_{\mathbb{Q}[S_d]}S^\beta\cong S^\beta\boxtimes1 Q [ S d ] ⊗ Q [ S d ] S β ≅ S β ⊠ 1 a ⊗ s ↦ a s ⊗ 1 a\otimes s\mapsto as\otimes1 a ⊗ s ↦ a s ⊗ 1 S n − d S_{n-d} S n − d S ν S^\nu S ν ν ⊢ n \nu\vdash n ν ⊢ n ν / β \nu/\beta ν / β n − d n-d n − d
The strip has n − d n-d n − d ν \nu ν ν 1 \nu_1 ν 1 ν \nu ν ν 1 ≥ n − d \nu_1\ge n-d ν 1 ≥ n − d depth ( ν ) ≤ d \operatorname{depth}(\nu)\le d depth ( ν ) ≤ d ν 1 = n − d \nu_1=n-d ν 1 = n − d β ⊆ ν \beta\subseteq\nu β ⊆ ν
ν i + 1 ≤ β i for all i ≥ 1. \nu_{i+1}\le\beta_i\qquad\text{for all }i\ge1. ν i + 1 ≤ β i for all i ≥ 1. Since
∑ i ≥ 1 ν i + 1 = n − ν 1 = d = ∣ β ∣ = ∑ i ≥ 1 β i , \sum_{i\ge1}\nu_{i+1}=n-\nu_1=d=|\beta|=\sum_{i\ge1}\beta_i, i ≥ 1 ∑ ν i + 1 = n − ν 1 = d = ∣ β ∣ = i ≥ 1 ∑ β i , all these inequalities must be equalities. Hence
ν = ( n − d , β ) . \nu=(n-d,\beta). ν = ( n − d , β ) . Thus, S ( n − d , β ) S^{(n-d,\beta)} S ( n − d , β ) 1 1 1 S n S_n S n
Hom S d ( S β , F d / F d − 1 ) , \operatorname{Hom}_{S_d}(S^\beta,F_d/F_{d-1}), Hom S d ( S β , F d / F d − 1 ) , whereas all other constituents of this multiplicity space have depth < d <d < d S β S^\beta S β F d / F d − 1 F_d/F_{d-1} F d / F d − 1 S ( n − d , β ) S^{(n-d,\beta)} S ( n − d , β ) dim S β \dim S^\beta dim S β
Step 3: take the S β S^{\beta} S β We have
M n ⊗ d = F d ≅ F d − 1 ⊕ ( F d / F d − 1 ) M_n^{\otimes d}=F_d\cong F_{d-1}\oplus(F_d/F_{d-1}) M n ⊗ d = F d ≅ F d − 1 ⊕ ( F d / F d − 1 ) as S n × S d S_n\times S_d S n × S d Hom S d ( S β , − ) \operatorname{Hom}_{S_d}(S^{\beta},-) Hom S d ( S β , − )
Hom S d ( S β , M n ⊗ d ) ≅ Hom S d ( S β , F d − 1 ) ⊕ Hom S d ( S β , F d / F d − 1 ) . \operatorname{Hom}_{S_d}(S^{\beta},M_n^{\otimes d})\cong\operatorname{Hom}_{S_d}(S^{\beta},F_{d-1})\oplus\operatorname{Hom}_{S_d}(S^{\beta},F_d/F_{d-1}). Hom S d ( S β , M n ⊗ d ) ≅ Hom S d ( S β , F d − 1 ) ⊕ Hom S d ( S β , F d / F d − 1 ) . The first addend here consists entirely of S n S_n S n d − 1 d-1 d − 1 S ( n − d , β ) S^{(n-d,\beta)} S ( n − d , β ) < d <d < d Hom S d ( S β , M n ⊗ d ) \operatorname{Hom}_{S_d}(S^{\beta},M_n^{\otimes d}) Hom S d ( S β , M n ⊗ d ) S ( n − d , β ) S^{(n-d,\beta)} S ( n − d , β ) < d <d < d S β ( M n ) = Hom S d ( S β , M n ⊗ d ) S^{\beta}(M_n)=\operatorname{Hom}_{S_d}(S^{\beta},M_n^{\otimes d}) S β ( M n ) = Hom S d ( S β , M n ⊗ d ) (75) .
Lemma A.2 is a small triangular piece of the classical restriction problem from G L n GL_n G L n S n S_n S n S n S_n S n G L n GL_n G L n S n S_n S n S β ( M n ) S^{\beta}(M_n) S β ( M n ) G L n GL_n G L n S β ( Q n ) S^{\beta}(\mathbb{Q}^n) S β ( Q n ) [34 Introduction] discuss this restriction problem and encode such restrictions by evaluating symmetric functions at the eigenvalues of permutation matrices. We only need the top-depth triangular statement above.
A triangular decomposition of T λ T_{\lambda} T λ
We now use the standard Kostka-triangular decomposition of a tensor product of exterior powers. Recall (70) . The symmetric-function identity
e α ′ = ∏ j = 1 r e c j = ∑ β ⊢ d K β ′ , α ′ s β (79) e_{\alpha'}=\prod_{j=1}^{r}e_{c_j}=\sum_{\beta\vdash d}K_{\beta',\alpha'}s_{\beta} \tag*{(79)} e α ′ = j = 1 ∏ r e c j = β ⊢ d ∑ K β ′ , α ′ s β (79) (where K λ , μ K_{\lambda,\mu} K λ , μ 13
T λ ≅ ⨁ β ⊢ d K β ′ , α ′ S β ( M n ) (80) T_{\lambda}\cong\bigoplus_{\beta\vdash d}K_{\beta',\alpha'}S^{\beta}(M_n) \tag*{(80)} T λ ≅ β ⊢ d ⨁ K β ′ , α ′ S β ( M n ) (80) (for an explicit reference, apply [11 §8.3, Corollary 2(b)] to α ′ \alpha' α ′ μ \mu μ S n ⊆ G L n S_n\subseteq GL_n S n ⊆ G L n R ( S n ) R(S_n) R ( S n )
[ T λ ] = ∑ β ⊢ d K β ′ , α ′ [ S β ( M n ) ] . (81) [T_{\lambda}]=\sum_{\beta\vdash d}K_{\beta',\alpha'}[S^{\beta}(M_n)]. \tag*{(81)} [ T λ ] = β ⊢ d ∑ K β ′ , α ′ [ S β ( M n )] . (81) However, it is well-known (see, e.g., [41 Proposition 7.10.5] ) that the Kostka number K β ′ , α ′ K_{\beta',\alpha'} K β ′ , α ′ β ′ ⊵ α ′ \beta'\unrhd\alpha' β ′ ⊵ α ′ ⊵ \unrhd ⊵ β ⊴ α \beta\unlhd\alpha β ⊴ α β ⊴ α \beta\unlhd\alpha β ⊴ α β 1 ≤ α 1 ≤ n − d \beta_1\leq\alpha_1\leq n-d β 1 ≤ α 1 ≤ n − d β = α \beta=\alpha β = α [41 Proposition 7.10.5] ) we have
K α ′ , α ′ = 1. K_{\alpha',\alpha'}=1. K α ′ , α ′ = 1. We obtain the promised triangularity:
Proposition A.3. Let λ = ( n − d , α ) ⊢ n \lambda=(n-d,\alpha)\vdash n λ = ( n − d , α ) ⊢ n
[ T λ ] = [ S λ ] + ∑ β ⊲ α K β ′ , α ′ [ S ( n − d , β ) ] + ∑ μ ⊢ n ; depth ( μ ) < d b λ μ [ S μ ] , (82) [T^\lambda]=[S^\lambda]+\sum_{\beta\lhd\alpha}K_{\beta',\alpha'}[S^{(n-d,\beta)}]+\sum_{\substack{\mu\vdash n;\\ \operatorname{depth}(\mu)<d}}b_{\lambda\mu}[S^\mu], \tag*{(82)} [ T λ ] = [ S λ ] + β ⊲ α ∑ K β ′ , α ′ [ S ( n − d , β ) ] + μ ⊢ n ; depth ( μ ) < d ∑ b λ μ [ S μ ] , (82) where the b λ μ b_{\lambda\mu} b λ μ β ⊲ α \beta\lhd\alpha β ⊲ α
Proof. As we observed, all the nonzero addends in (80) satisfy β ≤ α \beta\leq\alpha β ≤ α β 1 ≤ n − d \beta_1\leq n-d β 1 ≤ n − d Lemma A.2 . This yields
[ T λ ] = ∑ β ⊢ d ; β ⊲ α K β ′ , α ′ ( [ S ( n − d , β ) ] + ∑ μ ⊢ n ; depth ( μ ) < d a β μ [ S μ ] ) [T^\lambda]=\sum_{\substack{\beta\vdash d;\\ \beta\lhd\alpha}}K_{\beta',\alpha'}\left([S^{(n-d,\beta)}]+\sum_{\substack{\mu\vdash n;\\ \operatorname{depth}(\mu)<d}}a_{\beta\mu}[S^\mu]\right) [ T λ ] = β ⊢ d ; β ⊲ α ∑ K β ′ , α ′ [ S ( n − d , β ) ] + μ ⊢ n ; depth ( μ ) < d ∑ a β μ [ S μ ] = ∑ β ⊢ d ; β ≤ α K β ′ , α ′ [ S ( n − d , β ) ] + ∑ μ ⊢ n ; depth ( μ ) < d b λ μ [ S μ ] =\sum_{\substack{\beta\vdash d;\\ \beta\leq\alpha}}K_{\beta',\alpha'}[S^{(n-d,\beta)}]+\sum_{\substack{\mu\vdash n;\\ \operatorname{depth}(\mu)<d}}b_{\lambda\mu}[S^\mu] = β ⊢ d ; β ≤ α ∑ K β ′ , α ′ [ S ( n − d , β ) ] + μ ⊢ n ; depth ( μ ) < d ∑ b λ μ [ S μ ] for some nonnegative integers b λ μ b_{\lambda\mu} b λ μ K α ′ , α ′ = 1 K_{\alpha',\alpha'}=1 K α ′ , α ′ = 1 ∑ β ⊢ d ; β ≤ α K β ′ , α ′ [ S ( n − d , β ) ] \sum_{\substack{\beta\vdash d;\\ \beta\leq\alpha}}K_{\beta',\alpha'}[S^{(n-d,\beta)}] ∑ β ⊢ d ; β ≤ α K β ′ , α ′ [ S ( n − d , β ) ] [ S ( n − d , α ) ] + ∑ β ⊢ d ; β ⊲ α K β ′ , α ′ [ S ( n − d , β ) ] [S^{(n-d,\alpha)}]+\sum_{\substack{\beta\vdash d;\\ \beta\lhd\alpha}}K_{\beta',\alpha'}[S^{(n-d,\beta)}] [ S ( n − d , α ) ] + ∑ β ⊢ d ; β ⊲ α K β ′ , α ′ [ S ( n − d , β ) ] ( n − d , α ) = λ (n-d,\alpha)=\lambda ( n − d , α ) = λ
First proof of Theorem 3.1
First proof of Theorem 3.1. Let A n A_n A n λ \lambda λ R ( S n ) R(S_n) R ( S n ) M n M_n M n [ S λ ] ∈ A n [S^\lambda]\in A_n [ S λ ] ∈ A n λ ⊢ n \lambda\vdash n λ ⊢ n d = depth ( λ ) d=\operatorname{depth}(\lambda) d = depth ( λ ) d d d α \alpha α λ = ( n − d , α ) \lambda=(n-d,\alpha) λ = ( n − d , α )
The element [ T λ ] [T^\lambda] [ T λ ] A n A_n A n A n A_n A n A n A_n A n [ T λ ] [T^\lambda] [ T λ ] [ S λ ] ∈ A n [S^\lambda]\in A_n [ S λ ] ∈ A n
Since the Specht classes [ S λ ] [S^\lambda] [ S λ ] Z \mathbb{Z} Z R ( S n ) R(S_n) R ( S n ) A n = R ( S n ) A_n=R(S_n) A n = R ( S n )
Remark A.4. The two parts of the triangular order can be seen concretely already for n = 6 n=6 n = 6 d = 3 d=3 d = 3 β = ( 3 ) \beta=(3) β = ( 3 ) Sym 3 ( M 6 ) \operatorname{Sym}^3(M_6) Sym 3 ( M 6 ) Lemma A.2 ) gives
S ( 3 ) ( M 6 ) = Sym 3 ( M 6 ) ≅ S ( 3 , 3 ) ⊕ S ( 4 , 1 , 1 ) ⊕ 2 S ( 4 , 2 ) ⊕ 4 S ( 5 , 1 ) ⊕ 3 S ( 6 ) . \mathbb{S}^{(3)}(M_6)=\operatorname{Sym}^3(M_6)\cong S^{(3,3)}\oplus S^{(4,1,1)}\oplus2S^{(4,2)}\oplus4S^{(5,1)}\oplus3S^{(6)}. S ( 3 ) ( M 6 ) = Sym 3 ( M 6 ) ≅ S ( 3 , 3 ) ⊕ S ( 4 , 1 , 1 ) ⊕ 2 S ( 4 , 2 ) ⊕ 4 S ( 5 , 1 ) ⊕ 3 S ( 6 ) . Thus the lower-depth terms need not be lower in dominance order. For instance, ( 3 , 3 ) (3,3) ( 3 , 3 ) ( 4 , 1 , 1 ) (4,1,1) ( 4 , 1 , 1 ) ( 4 , 1 , 1 ) (4,1,1) ( 4 , 1 , 1 ) ( 3 , 3 ) (3,3) ( 3 , 3 )
The second part of the induction—dominance among tails of the same depth—is visible if we take λ = ( 3 , 2 , 1 ) \lambda=(3,2,1) λ = ( 3 , 2 , 1 ) d = 3 d=3 d = 3 α = ( 2 , 1 ) \alpha=(2,1) α = ( 2 , 1 )
T λ = ⋀ 2 M 6 ⊗ M 6 . T_\lambda=\bigwedge\nolimits^2 M_6\otimes M_6. T λ = ⋀ 2 M 6 ⊗ M 6 . Since e 2 e 1 = s ( 2 , 1 ) + s ( 1 , 1 , 1 ) e_2e_1=s_{(2,1)}+s_{(1,1,1)} e 2 e 1 = s ( 2 , 1 ) + s ( 1 , 1 , 1 )
T λ ≅ S ( 2 , 1 ) ( M 6 ) ⊕ S ( 1 , 1 , 1 ) ( M 6 ) . T_\lambda\cong S^{(2,1)}(M_6)\oplus S^{(1,1,1)}(M_6). T λ ≅ S ( 2 , 1 ) ( M 6 ) ⊕ S ( 1 , 1 , 1 ) ( M 6 ) . and the same calculation gives
T ( 3 , 2 , 1 ) ≅ S ( 3 , 2 , 1 ) ⊕ S ( 3 , 1 , 1 , 1 ) ⏟ same depth 3 , ( 1 , 1 , 1 ) ⊲ ( 2 , 1 ) ⊕ 3 S ( 4 , 1 , 1 ) ⊕ 2 S ( 4 , 2 ) ⊕ 3 S ( 5 , 1 ) ⊕ S ( 6 ) ⏟ constituents of depth < 3 . T_{(3,2,1)}\cong S^{(3,2,1)}\oplus\underbrace{S^{(3,1,1,1)}}_{\substack{\text{same depth }3,\\(1,1,1)\lhd(2,1)}}\oplus\underbrace{3S^{(4,1,1)}\oplus2S^{(4,2)}\oplus3S^{(5,1)}\oplus S^{(6)}}_{\text{constituents of depth }<3}. T ( 3 , 2 , 1 ) ≅ S ( 3 , 2 , 1 ) ⊕ same depth 3 , ( 1 , 1 , 1 ) ⊲ ( 2 , 1 ) S ( 3 , 1 , 1 , 1 ) ⊕ constituents of depth < 3 3 S ( 4 , 1 , 1 ) ⊕ 2 S ( 4 , 2 ) ⊕ 3 S ( 5 , 1 ) ⊕ S ( 6 ) . Thus this one example displays exactly the two kinds of already-known terms that occur in (81): smaller-depth constituents, and same-depth constituents whose tails are strictly smaller in dominance order.
Remark A.5 (Relation with Marin’s formulation). Marin states his theorem [30 Theorem 1.1] by saying that the exterior powers of the standard representation V n V_n V n R ( S n ) R(S_n) R ( S n ) λ \lambda λ T λ T_\lambda T λ M n M_n M n M n M_n M n R ( S n ) R(S_n) R ( S n ) M n = 1 ⊕ V n M_n=1\oplus V_n M n = 1 ⊕ V n
⋀ k M n ≅ ⋀ k V n ⊕ ⋀ k − 1 V n . \bigwedge^k M_n\cong\bigwedge^k V_n\oplus\bigwedge^{k-1}V_n. ⋀ k M n ≅ ⋀ k V n ⊕ ⋀ k − 1 V n . This decomposition expresses the exterior powers of M n M_n M n V n V_n V n ⋀ k V n \bigwedge^k V_n ⋀ k V n V n V_n V n M n M_n M n λ \lambda λ V n V_n V n
Two descent-algebra lifts of Boorman–Marin
Theorem 3.1 says that the representation ring R ( S n ) R(S_n) R ( S n ) λ \lambda λ M n M_n M n M n M_n M n V n = M n − 1 V_n=M_n-1 V n = M n − 1 λ \lambda λ
⋀ k V n ≅ S ( n − k , 1 k ) for 0 ≤ k ≤ n − 1. \bigwedge^k V_n\cong S^{(n-k,1^k)}\qquad\text{for }0\leq k\leq n-1. ⋀ k V n ≅ S ( n − k , 1 k ) for 0 ≤ k ≤ n − 1. (see, e.g., [19 Proposition 2] for a proof). Thus the most direct ordinary-ring form of the Boorman–Marin theorem says that the hook representations S ( n − k , 1 k ) S^{(n-k,1^k)} S ( n − k , 1 k ) R ( S n ) R(S_n) R ( S n ) Z \mathbb{Z} Z [39 ] found a remarkably literal lift of this statement to Solomon’s descent algebra: certain exact-descent-class sums attached to hook shapes already generate the whole descent algebra. In fact, his theorem uses only about half of the hooks.
We begin by recalling the noncommutative-symmetric-function language in which this lift is especially transparent. We then translate Schocker’s theorem carefully into our conventions and give an independent proof. First Solomon’s Mackey formula proves generation by all hooks; then multiplication by the longest permutation reduces this family to its upper half. Afterwards we establish a second, different generation theorem: the complete functions H k H n − k H_kH_{n-k} H k H n − k k k k R ( S n ) R(S_n) R ( S n )
Let N S y m Z \mathrm{NSym}_{\mathbb Z} NSym Z Z \mathbb Z Z H 1 , H 2 , … H_1,H_2,\ldots H 1 , H 2 , … H i H_i H i deg H i = i \deg H_i=i deg H i = i H 0 = 1 H_0=1 H 0 = 1 I = ( i 1 , … , i r ) I=(i_1,\ldots,i_r) I = ( i 1 , … , i r )
H I = H i 1 ⋯ H i r . H_I=H_{i_1}\cdots H_{i_r}. H I = H i 1 ⋯ H i r . The elements H I H_I H I I ⊨ n I\vDash n I ⊨ n Z \mathbb Z Z N S y m Z , n \mathrm{NSym}_{\mathbb Z,n} NSym Z , n N S y m Z \mathrm{NSym}_{\mathbb Z} NSym Z noncommutative symmetric functions over Z \mathbb Z Z [14 ] (see [21 §5.4] for a modern introduction).
We shall also use the ribbon basis (see [14 §3.2, §4.4] ). If I ⊨ n I\vDash n I ⊨ n
R I = ∑ J coarsens I ( − 1 ) ℓ ( I ) − ℓ ( J ) H J . (83) R_I=\sum_{J\text{ coarsens }I}(-1)^{\ell(I)-\ell(J)}H_J. \tag*{(83)} R I = J coarsens I ∑ ( − 1 ) ℓ ( I ) − ℓ ( J ) H J . (83) Here a coarsening of I I I
H I = ∑ J coarsens I R J . (84) H_I=\sum_{J\text{ coarsens }I}R_J. \tag*{(84)} H I = J coarsens I ∑ R J . (84) For example,
R ( 2 , 1 , 1 ) = H ( 2 , 1 , 1 ) − H ( 3 , 1 ) − H ( 2 , 2 ) + H ( 4 ) . R_{(2,1,1)}=H_{(2,1,1)}-H_{(3,1)}-H_{(2,2)}+H_{(4)}. R ( 2 , 1 , 1 ) = H ( 2 , 1 , 1 ) − H ( 3 , 1 ) − H ( 2 , 2 ) + H ( 4 ) . For w ∈ S n w\in S_n w ∈ S n
Des ( w ) = { i ∈ [ n − 1 ] : w ( i ) > w ( i + 1 ) } . \operatorname{Des}(w)=\{i\in[n-1]:w(i)>w(i+1)\}. Des ( w ) = { i ∈ [ n − 1 ] : w ( i ) > w ( i + 1 )} . For I = ( i 1 , … , i r ) ⊨ n I=(i_1,\ldots,i_r)\vDash n I = ( i 1 , … , i r ) ⊨ n
D ( I ) = { i 1 , i 1 + i 2 , … , i 1 + ⋯ + i r − 1 } , x I = ∑ w ∈ S n ; Des ( w ) ⊆ D ( I ) w ∈ Z [ S n ] . D(I)=\{i_1,i_1+i_2,\ldots,i_1+\cdots+i_{r-1}\},\qquad
x_I=\sum_{\substack{w\in S_n;\\ \operatorname{Des}(w)\subseteq D(I)}}w\in\mathbb Z[S_n]. D ( I ) = { i 1 , i 1 + i 2 , … , i 1 + ⋯ + i r − 1 } , x I = w ∈ S n ; Des ( w ) ⊆ D ( I ) ∑ w ∈ Z [ S n ] . The (integral) Solomon descent algebra is
D ( S n ) : = ⨁ I ⊨ n Z x I ⊆ Z [ S n ] , (85) \mathcal D(S_n):=\bigoplus_{I\vDash n}\mathbb Zx_I\subseteq\mathbb Z[S_n], \tag*{(85)} D ( S n ) := I ⊨ n ⨁ Z x I ⊆ Z [ S n ] , (85) with the usual group-algebra product. Solomon’s Mackey formula shows that this span is closed under multiplication (see, e.g., [2 Theorem 3.8.1] , [37 proof of Theorem 2.1] , [20 Theorem 4.12] , or many other places). The descent-class correspondence is the Z \mathbb Z Z
ι : N S y m Z , n ⟶ D ( S n ) , H I ⟼ x I . (86) \iota:\mathrm{NSym}_{\mathbb Z,n}\longrightarrow\mathcal D(S_n),\qquad H_I\longmapsto x_I. \tag*{(86)} ι : NSym Z , n ⟶ D ( S n ) , H I ⟼ x I . (86) It is an isomorphism of Z \mathbb{Z} Z
ι ( R I ) = ∑ w ∈ S n ; Des ( w ) = D ( I ) w = : Δ I , \iota(R_I)=\sum_{\substack{w\in S_n;\\ \operatorname{Des}(w)=D(I)}}w=:\Delta_I, ι ( R I ) = w ∈ S n ; Des ( w ) = D ( I ) ∑ w =: Δ I , the so-called exact-descent-class sum corresponding to I I I N S y m \mathrm{NSym} NSym D ( S n ) D(S_n) D ( S n )
Besides its usual product, N S y m Z , n \mathrm{NSym}_{\mathbb{Z},n} NSym Z , n internal product ∗ \ast ∗ D ( S n ) D(S_n) D ( S n ) ( u v ) ( i ) = u ( v ( i ) ) (uv)(i)=u(v(i)) ( uv ) ( i ) = u ( v ( i )) ι \iota ι
ι ( F ∗ G ) = ι ( G ) ι ( F ) . \iota(F\ast G)=\iota(G)\iota(F). ι ( F ∗ G ) = ι ( G ) ι ( F ) . Thus our internal product corresponds to the opposite of the usual group-algebra product. This distinction does not affect generation, but fixes the order in the hook identity below. See, for example, Blessenohl–Schocker [5 Introduction and Chapter 14] .
We shall use the standard matrix form of Solomon’s Mackey formula, due in this language to Garsia–Remmel ([13, Proposition 1.1], [37, Corollary 2.4], [4, Proposition 4.3], [44, Theorem 2], [5, Remark B.5]). If I = ( i 1 , … , i p ) I=(i_1,\ldots,i_p) I = ( i 1 , … , i p ) J = ( j 1 , … , j q ) J=(j_1,\ldots,j_q) J = ( j 1 , … , j q ) n n n
H I ∗ H J = ∑ M H comp ( M ) , (87) H_I\ast H_J=\sum_M H_{\operatorname{comp}(M)}, \tag*{(87)} H I ∗ H J = M ∑ H comp ( M ) , (87) where M = ( m u v ) M=(m_{uv}) M = ( m uv ) p × q p\times q p × q I I I J J J
∑ v = 1 q m u v = i u for all 1 ≤ u ≤ p , ∑ u = 1 p m u v = j v for all 1 ≤ v ≤ q . \sum_{v=1}^{q}m_{uv}=i_u\qquad\text{for all }1\leq u\leq p,\qquad
\sum_{u=1}^{p}m_{uv}=j_v\qquad\text{for all }1\leq v\leq q. v = 1 ∑ q m uv = i u for all 1 ≤ u ≤ p , u = 1 ∑ p m uv = j v for all 1 ≤ v ≤ q . The composition comp ( M ) \operatorname{comp}(M) comp ( M ) M M M row-reading composition of M M M
Example B.1. Take I = ( 3 , 1 ) I=(3,1) I = ( 3 , 1 ) J = ( 2 , 2 ) J=(2,2) J = ( 2 , 2 ) 2 × 2 2\times2 2 × 2 I I I J J J
( 1 2 1 0 ) , ( 2 1 0 1 ) . \begin{pmatrix}1&2\\1&0\end{pmatrix},
\qquad
\begin{pmatrix}2&1\\0&1\end{pmatrix}. ( 1 1 2 0 ) , ( 2 0 1 1 ) . Their row-reading compositions are ( 1 , 2 , 1 ) (1,2,1) ( 1 , 2 , 1 ) ( 2 , 1 , 1 ) (2,1,1) ( 2 , 1 , 1 )
H ( 3 , 1 ) ∗ H ( 2 , 2 ) = H ( 1 , 2 , 1 ) + H ( 2 , 1 , 1 ) . H_{(3,1)}\ast H_{(2,2)}=H_{(1,2,1)}+H_{(2,1,1)}. H ( 3 , 1 ) ∗ H ( 2 , 2 ) = H ( 1 , 2 , 1 ) + H ( 2 , 1 , 1 ) . Finally, recall the abelianization map
π : N S y m Z ⟶ S y m m Z , H r ⟼ h r . (88) \pi:\mathrm{NSym}_{\mathbb{Z}}\longrightarrow\mathrm{Symm}_{\mathbb{Z}},\qquad H_r\longmapsto h_r. \tag*{(88)} π : NSym Z ⟶ Symm Z , H r ⟼ h r . (88) It satisfies
π ( R I ) = r I , (89) \pi(R_I)=r_I, \tag*{(89)} π ( R I ) = r I , (89) where r I r_I r I I I I π \pi π π \pi π [28 §I.7, Example 23 (e)] or [5 Appendix B, proof of the $\xi^{\xi}$ formula] ); this is essentially an application of the Mackey formula for tensor products of permutation characters.
In these conventions, Solomon’s epimorphism is the surjective ring homomorphism
θ : D ( S n ) ⟶ R ( S n ) , x I ⟼ [ Ind S I S n 1 ] , (90) \theta: \mathcal{D}(S_n) \longrightarrow R(S_n), \qquad x_I \longmapsto[\operatorname{Ind}_{S_I}^{S_n} 1], \tag*{(90)} θ : D ( S n ) ⟶ R ( S n ) , x I ⟼ [ Ind S I S n 1 ] , (90) where (for any composition I = ( i 1 , i 2 , … , i r ) I = (i_1,i_2,\ldots,i_r) I = ( i 1 , i 2 , … , i r ) n n n S I = S i 1 × ⋯ × S i r S_I = S_{i_1} \times\cdots\times S_{i_r} S I = S i 1 × ⋯ × S i r i 1 , … , i r i_1,\ldots,i_r i 1 , … , i r Ind S I S n 1 \operatorname{Ind}_{S_I}^{S_n} 1 Ind S I S n 1 i 1 , … , i r i_1,\ldots,i_r i 1 , … , i r π n \pi_n π n n n n π \pi π
θ ∘ ι = ch − 1 ∘ π n . (91) \theta\circ\iota= \operatorname{ch}^{-1} \circ\pi_n. \tag*{(91)} θ ∘ ι = ch − 1 ∘ π n . (91) Indeed, ch ( [ Ind S I S n 1 ] ) = h I = π ( H I ) \operatorname{ch}([\operatorname{Ind}_{S_I}^{S_n} 1]) = h_I = \pi(H_I) ch ([ Ind S I S n 1 ]) = h I = π ( H I ) ch ∘ θ ∘ ι = π n \operatorname{ch} \circ\theta\circ\iota= \pi_n ch ∘ θ ∘ ι = π n π n \pi_n π n R ( S n ) R(S_n) R ( S n ) θ \theta θ ι \iota ι h λ h_\lambda h λ λ ⊢ n \lambda\vdash n λ ⊢ n Z \mathbb{Z} Z S y m m Z , n \mathrm{Symm}_{\mathbb{Z},n} Symm Z , n
θ ( ι ( R I ) ) = ch − 1 ( r I ) . \theta(\iota(R_I)) = \operatorname{ch}^{-1}(r_I). θ ( ι ( R I )) = ch − 1 ( r I ) . Schocker’s hook generators: the direct lift
For 0 ≤ k ≤ n − 1 0 \leq k \leq n-1 0 ≤ k ≤ n − 1
Q n , k : = R ( n − k , 1 k ) ∈ N S y m Z , n . (92) Q_{n,k} := R_{(n-k,1^k)} \in\mathrm{NSym}_{\mathbb{Z},n}. \tag*{(92)} Q n , k := R ( n − k , 1 k ) ∈ NSym Z , n . (92) Thus, under the correspondence ι \iota ι N S y m Z , n \mathrm{NSym}_{\mathbb{Z},n} NSym Z , n Q n , k Q_{n,k} Q n , k
{ n − k , n − k + 1 , … , n − 1 } \{n-k,n-k+1,\ldots,n-1\} { n − k , n − k + 1 , … , n − 1 } (this is the empty set when k = 0 k=0 k = 0 180 ∘ 180^\circ 18 0 ∘ ( n − k , 1 k ) (n-k,1^k) ( n − k , 1 k )
π ( Q n , k ) = s ( n − k , 1 k ) (93) \pi(Q_{n,k}) = s_{(n-k,1^k)} \tag*{(93)} π ( Q n , k ) = s ( n − k , 1 k ) (93) (since 180 ∘ 180^\circ 18 0 ∘ ⋀ k V n \bigwedge^k V_n ⋀ k V n Q n , k Q_{n,k} Q n , k λ \lambda λ
Schocker’s theorem [39 p. 152, Theorem] is stated using the exact-descent-class sums Δ { 1 , … , k } \Delta_{\{1,\ldots,k\}} Δ { 1 , … , k } ( n − 1 ) / 2 ≤ k ≤ n − 1 (n-1)/2 \leq k \leq n-1 ( n − 1 ) /2 ≤ k ≤ n − 1 w 0 w_0 w 0 D D D { n − i : i ∈ D } \{n-i : i \in D\} { n − i : i ∈ D } Δ { 1 , … , k } \Delta_{\{1,\ldots,k\}} Δ { 1 , … , k } { n − k , … , n − 1 } \{n-k,\ldots,n-1\} { n − k , … , n − 1 } ι ( Q n , k ) \iota(Q_{n,k}) ι ( Q n , k )
Theorem B.2 (Schocker). Fix n ≥ 1 n \ge1 n ≥ 1 D ( S n ) \mathcal{D}(S_n) D ( S n )
ι ( Q n , k ) for all k satisfying ⌊ n − 1 2 ⌋ ≤ k ≤ n − 1. \iota(Q_{n,k}) \quad\text{for all } k \text{ satisfying } \left\lfloor\frac{n-1}{2}\right\rfloor\le k \le n-1. ι ( Q n , k ) for all k satisfying ⌊ 2 n − 1 ⌋ ≤ k ≤ n − 1. Under Solomon’s epimorphism, these generators map to the irreducible hook characters χ ( n − k , 1 k ) \chi^{(n-k,1^k)} χ ( n − k , 1 k ) [39 p. 152, Corollary 1] .
We first prove the weaker statement that all the Q n , k Q_{n,k} Q n , k
Proposition B.3. Fix n ≥ 1 n \ge1 n ≥ 1 Z \mathbb{Z} Z N S y m Z , n \mathrm{NSym}_{\mathbb{Z},n} NSym Z , n
Q n , 0 , Q n , 1 , … , Q n , n − 1 . Q_{n,0}, Q_{n,1}, \ldots, Q_{n,n-1}. Q n , 0 , Q n , 1 , … , Q n , n − 1 . Proof. Let B n B_n B n Z \mathbb{Z} Z Q n , k Q_{n,k} Q n , k H I H_I H I B n B_n B n I I I ( n ) (n) ( n ) H ( n ) = R ( n ) = Q n , 0 H_{(n)} = R_{(n)} = Q_{n,0} H ( n ) = R ( n ) = Q n , 0
I = ( m , a , a 3 , … , a r ) , r ≥ 2 , I=(m,a,a_3,\ldots,a_r), \qquad r \ge2, I = ( m , a , a 3 , … , a r ) , r ≥ 2 , and assume that H L ∈ B n H_L \in B_n H L ∈ B n 14 L 1 > m L_1 > m L 1 > m L 1 = m L_1=m L 1 = m ℓ ( L ) > r \ell(L)>r ℓ ( L ) > r
I ′ = ( m + a , a 3 , … , a r ) . I'=(m+a,a_3,\ldots,a_r). I ′ = ( m + a , a 3 , … , a r ) . Since the first part of I ′ I' I ′ m m m H I ′ ∈ B n H_{I'} \in B_n H I ′ ∈ B n
Q n , a = R ( n − a , 1 a ) . Q_{n,a}=R_{(n-a,1^a)}. Q n , a = R ( n − a , 1 a ) . Step 1: locate the possible leading parts. By (82), Q n , a Q_{n,a} Q n , a H J H_J H J J = ( j 1 , … , j q ) J=(j_1,\ldots,j_q) J = ( j 1 , … , j q ) ( n − a , 1 a ) (n-a,1^a) ( n − a , 1 a ) J J J
j 1 ≥ n − a , hence ∑ v = 2 q j v ≤ a . (94) j_1 \ge n-a, \qquad\text{hence} \qquad\sum_{v=2}^{q}j_v \le a. \tag*{(94)} j 1 ≥ n − a , hence v = 2 ∑ q j v ≤ a . (94) Consider a matrix M = ( m u v ) M=(m_{uv}) M = ( m uv ) H I ′ ∗ H J H_{I'} * H_J H I ′ ∗ H J m + a m+a m + a
m 11 = m + a − ∑ v = 2 q m 1 v ≥ m + a − ∑ v = 2 q j v ( since each v ≥ 2 satisfies m 1 v ≤ ∑ u = 1 r − 1 m u v = j v ) ≥ m ( since ∑ v = 2 q j v ≤ a ) . \begin{aligned}
m_{11} &= m+a-\sum_{v=2}^{q}m_{1v}\\
&\ge m+a-\sum_{v=2}^{q}j_v
\qquad\left(\text{since each }v\ge2\text{ satisfies }m_{1v}\le\sum_{u=1}^{r-1}m_{uv}=j_v\right)\\
&\ge m
\qquad\left(\text{since }\sum_{v=2}^{q}j_v\le a\right).
\end{aligned} m 11 = m + a − v = 2 ∑ q m 1 v ≥ m + a − v = 2 ∑ q j v ( since each v ≥ 2 satisfies m 1 v ≤ u = 1 ∑ r − 1 m uv = j v ) ≥ m ( since v = 2 ∑ q j v ≤ a ) . In particular, m 11 > 0 m_{11} > 0 m 11 > 0 m 11 m_{11} m 11 comp ( M ) \operatorname{comp}(M) comp ( M ) H I ′ ∗ Q n , a H_{I'} * Q_{n,a} H I ′ ∗ Q n , a m m m
Step 2: analyze the equality case. Suppose that m 11 = m m_{11} = m m 11 = m
j 1 = n − a , ∑ v = 2 q j v = a . j_1 = n-a,\qquad\sum_{v=2}^{q} j_v = a. j 1 = n − a , v = 2 ∑ q j v = a . and every entry below the first row in columns 2 , … , q 2,\ldots,q 2 , … , q J J J
J = ( n − a , b 1 , … , b s ) with b 1 + ⋯ + b s = a . J=(n-a,b_1,\ldots,b_s)\qquad\text{with}\qquad b_1+\cdots+b_s=a. J = ( n − a , b 1 , … , b s ) with b 1 + ⋯ + b s = a . and there is exactly one matrix M M M m m m
( m b 1 b 2 ⋯ b s a 3 0 0 ⋯ 0 a 4 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ a r 0 0 ⋯ 0 ) . \begin{pmatrix}
m & b_1 & b_2 & \cdots& b_s\\
a_3 & 0 & 0 & \cdots& 0\\
a_4 & 0 & 0 & \cdots& 0\\
\vdots& \vdots& \vdots& \ddots& \vdots\\
a_r & 0 & 0 & \cdots& 0
\end{pmatrix}. m a 3 a 4 ⋮ a r b 1 0 0 ⋮ 0 b 2 0 0 ⋮ 0 ⋯ ⋯ ⋯ ⋱ ⋯ b s 0 0 ⋮ 0 . Its row-reading composition is
( m , b 1 , … , b s , a 3 , … , a r ) , (m,b_1,\ldots,b_s,a_3,\ldots,a_r), ( m , b 1 , … , b s , a 3 , … , a r ) , which has length r + s − 1 ≥ r r+s-1\ge r r + s − 1 ≥ r s = 1 s=1 s = 1 J = ( n − a , a ) J=(n-a,a) J = ( n − a , a ) I I I H ( n − a , a ) H_{(n-a,a)} H ( n − a , a ) Q n , a = R ( n − a , 1 a ) Q_{n,a}=R_{(n-a,1^a)} Q n , a = R ( n − a , 1 a )
( − 1 ) ( a + 1 ) − 2 = ( − 1 ) a − 1 . (-1)^{(a+1)-2}=(-1)^{a-1}. ( − 1 ) ( a + 1 ) − 2 = ( − 1 ) a − 1 . Therefore
H I ′ ∗ Q n , a = ( − 1 ) a − 1 H I + ∑ L c L H L , (95) H_{I'} * Q_{n,a}=(-1)^{a-1}H_I+\sum_L c_LH_L, \tag*{(95)} H I ′ ∗ Q n , a = ( − 1 ) a − 1 H I + L ∑ c L H L , (95) where c L ∈ Z c_L\in\mathbb{Z} c L ∈ Z L L L L 1 > m L_1>m L 1 > m L 1 = m L_1=m L 1 = m ℓ ( L ) > r \ell(L)>r ℓ ( L ) > r B n B_n B n H I H_I H I ± 1 \pm1 ± 1 H I ∈ B n H_I\in B_n H I ∈ B n
Remark B.4. The proof is an integral triangular elimination. The relevant order on compositions is deliberately simple: first compare the first parts, and if they agree, compare the lengths. Formula (94) expresses H I H_I H I
Proof of Theorem B.2. Let w 0 ∈ S n w_0\in S_n w 0 ∈ S n w 0 ( i ) = n + 1 − i w_0(i)=n+1-i w 0 ( i ) = n + 1 − i w 0 w_0 w 0
ι ( Q n , n − 1 ) = w 0 . \iota(Q_{n,n-1})=w_0. ι ( Q n , n − 1 ) = w 0 . For any w ∈ S n w\in S_n w ∈ S n w 0 w_0 w 0
Des ( w w 0 ) = { n − i : i ∈ [ n − 1 ] ∖ Des ( w ) } . \operatorname{Des}(ww_0)=\{n-i:i\in[n-1]\setminus\operatorname{Des}(w)\}. Des ( w w 0 ) = { n − i : i ∈ [ n − 1 ] ∖ Des ( w )} . If Des ( w ) = { n − k , … , n − 1 } \operatorname{Des}(w)=\{n-k,\ldots,n-1\} Des ( w ) = { n − k , … , n − 1 } { k + 1 , … , n − 1 } \{k+1,\ldots,n-1\} { k + 1 , … , n − 1 } Q n , n − 1 − k Q_{n,n-1-k} Q n , n − 1 − k w ↦ w w 0 w\mapsto ww_0 w ↦ w w 0
ι ( Q n , k ) w 0 = ι ( Q n , n − 1 − k ) . \iota(Q_{n,k})w_0=\iota(Q_{n,n-1-k}). ι ( Q n , k ) w 0 = ι ( Q n , n − 1 − k ) . That is, ι ( Q n , k ) ι ( Q n , n − 1 ) = ι ( Q n , n − 1 − k ) \iota(Q_{n,k})\iota(Q_{n,n-1})=\iota(Q_{n,n-1-k}) ι ( Q n , k ) ι ( Q n , n − 1 ) = ι ( Q n , n − 1 − k ) ι ( Q n , n − 1 ) = w 0 \iota(Q_{n,n-1})=w_0 ι ( Q n , n − 1 ) = w 0
Q n , n − 1 ∗ Q n , k = Q n , n − 1 − k for all 0 ≤ k ≤ n − 1. (96) Q_{n,n-1}*Q_{n,k}=Q_{n,n-1-k}\qquad\text{for all }0\le k\le n-1. \tag*{(96)} Q n , n − 1 ∗ Q n , k = Q n , n − 1 − k for all 0 ≤ k ≤ n − 1. (96) Now let B n + B_n^+ B n + Z \mathbb{Z} Z Q n , k Q_{n,k} Q n , k ⌈ ( n − 1 ) / 2 ⌉ ≤ k ≤ n − 1 \lceil(n-1)/2\rceil\le k\le n-1 ⌈( n − 1 ) /2 ⌉ ≤ k ≤ n − 1 j < ⌈ ( n − 1 ) / 2 ⌉ j<\lceil(n-1)/2\rceil j < ⌈( n − 1 ) /2 ⌉ k = n − 1 − j k=n-1-j k = n − 1 − j k ≥ ⌈ ( n − 1 ) / 2 ⌉ k\ge\lceil(n-1)/2\rceil k ≥ ⌈( n − 1 ) /2 ⌉ Q n , k Q_{n,k} Q n , k Q n , n − 1 Q_{n,n-1} Q n , n − 1 B n + B_n^+ B n + Q n , j Q_{n,j} Q n , j B n + B_n^+ B n + j = n − 1 − k j=n-1-k j = n − 1 − k B n + B_n^+ B n + Q n , 0 , Q n , 1 , … , Q n , n − 1 Q_{n,0},Q_{n,1},\ldots,Q_{n,n-1} Q n , 0 , Q n , 1 , … , Q n , n − 1 NSym Z , n \operatorname{NSym}_{\mathbb{Z},n} NSym Z , n NSym Z , n \operatorname{NSym}_{\mathbb{Z},n} NSym Z , n Q n , k Q_{n,k} Q n , k ⌈ ( n − 1 ) / 2 ⌉ ≤ k ≤ n − 1 \lceil(n-1)/2\rceil\le k\le n-1 ⌈( n − 1 ) /2 ⌉ ≤ k ≤ n − 1 ι \iota ι D ( S n ) D(S_n) D ( S n ) ι ( Q n , k ) \iota(Q_{n,k}) ι ( Q n , k ) ⌈ ( n − 1 ) / 2 ⌉ ≤ k ≤ n − 1 \lceil(n-1)/2\rceil\le k\le n-1 ⌈( n − 1 ) /2 ⌉ ≤ k ≤ n − 1 Z \mathbb{Z} Z
Second proof of Theorem 3.1
Proposition B.3 now gives a direct integral descent-algebra proof of Boorman–Marin.
Second proof of Theorem 3.1. Applying the surjective ring morphism π \pi π
s ( n − k , 1 k ) for all 0 ≤ k ≤ n − 1 s_{(n-k,1^k)}\qquad\text{for all }0\le k\le n-1 s ( n − k , 1 k ) for all 0 ≤ k ≤ n − 1 generate ( Symm Z , n , ∗ ) (\operatorname{Symm}_{\mathbb{Z},n},*) ( Symm Z , n , ∗ ) ch − 1 \operatorname{ch}^{-1} ch − 1 ⋀ k V n \bigwedge^k V_n ⋀ k V n R ( S n ) R(S_n) R ( S n ) V n = M n − 1 V_n=M_n-1 V n = M n − 1 λ \lambda λ M n M_n M n λ k ( V n ) = ⋀ k V n \lambda^k(V_n)=\bigwedge^k V_n λ k ( V n ) = ⋀ k V n k k k λ \lambda λ R ( S n ) R(S_n) R ( S n ) R ( S n ) R(S_n) R ( S n )
A second lift: two-block complete functions
The hook ribbons above are the natural lifts of the exterior powers appearing in Boorman–Marin. There is, however, another pleasantly small generating family upstairs. The following theorem is independent of Schocker’s theorem and will lead to a different family of generators downstairs, as well as eventually to a different – third – proof of Theorem 3.1.
Theorem B.5. Fix n ≥ 1 n\ge1 n ≥ 1 Z \mathbb{Z} Z NSym Z , n \operatorname{NSym}_{\mathbb{Z},n} NSym Z , n n n n
H ( k , n − k ) = H k H n − k with 1 ≤ k ≤ n , (97) H_{(k,n-k)}=H_kH_{n-k}\qquad\text{with }1\le k\le n, \tag*{(97)} H ( k , n − k ) = H k H n − k with 1 ≤ k ≤ n , (97) where a zero part in a composition is understood to be omitted by default.
Proof. Let A n A_n A n Z \mathbb{Z} Z N S y m Z , n \mathrm{NSym}_{\mathbb{Z},n} NSym Z , n H I ∈ A n H_I \in A_n H I ∈ A n I ⊨ n I \vDash n I ⊨ n I I I I = ( n ) I=(n) I = ( n ) H I = H ( n , 0 ) H_I=H_{(n,0)} H I = H ( n , 0 )
Now let
I = ( m , a 2 , a 3 , … , a r ) , r ≥ 2 , I=(m,a_2,a_3,\ldots,a_r), \qquad r\ge2, I = ( m , a 2 , a 3 , … , a r ) , r ≥ 2 , and assume that H K ∈ A n H_K\in A_n H K ∈ A n K ⊨ n K\vDash n K ⊨ n m m m
I ′ = ( m + a 2 , a 3 , … , a r ) and J = ( n − a 2 , a 2 ) . I'=(m+a_2,a_3,\ldots,a_r) \qquad\text{and}\qquad J=(n-a_2,a_2). I ′ = ( m + a 2 , a 3 , … , a r ) and J = ( n − a 2 , a 2 ) . The first part of I ′ I' I ′ m + a 2 > m m+a_2>m m + a 2 > m H I ′ ∈ A n H_{I'}\in A_n H I ′ ∈ A n H J H_J H J A n A_n A n H I ′ ∗ H J ∈ A n H_{I'}*H_J\in A_n H I ′ ∗ H J ∈ A n
Apply (86) to H I ′ ∗ H J H_{I'}*H_J H I ′ ∗ H J M M M
( m + a 2 − d , d ) . (m+a_2-d,d). ( m + a 2 − d , d ) . Since the second column has total sum a 2 a_2 a 2 0 ≤ d ≤ a 2 0\le d\le a_2 0 ≤ d ≤ a 2 d < a 2 d<a_2 d < a 2 comp ( M ) \operatorname{comp}(M) comp ( M ) m + a 2 − d > m m+a_2-d>m m + a 2 − d > m d = a 2 d=a_2 d = a 2
( m a 2 a 3 0 a 4 0 ⋮ ⋮ a r 0 ) . \begin{pmatrix}
m & a_2\\
a_3 & 0\\
a_4 & 0\\
\vdots& \vdots\\
a_r & 0
\end{pmatrix}. m a 3 a 4 ⋮ a r a 2 0 0 ⋮ 0 . Its row-reading composition is I I I
H I ′ ∗ H J = H I + ∑ K ⊨ n K 1 > m c K H K (98) H_{I'}*H_J=H_I+\sum_{\substack{K\vDash n\\K_1>m}}c_KH_K \tag*{(98)} H I ′ ∗ H J = H I + K ⊨ n K 1 > m ∑ c K H K (98) for some c K ∈ N c_K\in\mathbb{N} c K ∈ N K 1 K_1 K 1 K K K A n A_n A n H I ′ ∗ H J H_{I'}*H_J H I ′ ∗ H J H I ∈ A n H_I\in A_n H I ∈ A n □ \square □
Remark B.6. The proof is triangular in a precise elementary sense: when (97) is solved for H I H_I H I I 1 I_1 I 1
Under the descent-algebra identification ι \iota ι H ( k , n − k ) ∈ N S y m Z , n H_{(k,n-k)}\in\mathrm{NSym}_{\mathbb{Z},n} H ( k , n − k ) ∈ NSym Z , n
a k = x ( k , n − k ) = ∑ w ∈ S n Des ( w ) ⊆ { k } w = ∑ w ∈ S n w ( 1 ) < ⋯ < w ( k ) w ( k + 1 ) < ⋯ < w ( n ) w ∈ D ( S n ) , a_k=x_{(k,n-k)}=
\sum_{\substack{w\in S_n\\ \operatorname{Des}(w)\subseteq\{k\}}}w
=
\sum_{\substack{w\in S_n\\ w(1)<\cdots<w(k)\\w(k+1)<\cdots<w(n)}}w
\in D(S_n), a k = x ( k , n − k ) = w ∈ S n Des ( w ) ⊆ { k } ∑ w = w ∈ S n w ( 1 ) < ⋯ < w ( k ) w ( k + 1 ) < ⋯ < w ( n ) ∑ w ∈ D ( S n ) , with empty chains omitted; this is a sum over the so-called Grassmannian permutations (or inverse riffle shuffles ) with their only possible descent (if any) at k k k Theorem B.5 takes the following equivalent form:
Corollary B.7. For every n ≥ 1 n \ge1 n ≥ 1 D ( S n ) \mathcal{D}(S_n) D ( S n ) a 1 , a 2 , … , a n a_1,a_2,\ldots,a_n a 1 , a 2 , … , a n
This is somewhat similar, but not identical, to Theorem B.2. Schocker’s generators are exact descent-class sums and map to irreducible hook characters. The elements a k a_k a k
The two-block generators downstairs
Applying the abelianization map π \pi π
Corollary B.8. For every n ≥ 1 n \ge1 n ≥ 1 ( S y m m Z , n , ∗ ) (\mathrm{Symm}_{\mathbb{Z},n},*) ( Symm Z , n , ∗ ) n n n
with 1 ≤ k ≤ n 1 \le k \le n 1 ≤ k ≤ n
Equivalently, R ( S n ) R(S_n) R ( S n ) n n n
P n , k : = Q [ ( [ n ] k ) ] (99) P_{n,k}:=\mathbb{Q}\left[\binom{[n]}{k}\right] \tag*{(99)} P n , k := Q [ ( k [ n ] ) ] (99) with 1 ≤ k ≤ n 1 \le k \le n 1 ≤ k ≤ n k k k [ n ] [n] [ n ]
Proof. Under the Frobenius characteristic,
ch n ( P n , k ) = h k h n − k , \operatorname{ch}_n(P_{n,k})=h_kh_{n-k}, ch n ( P n , k ) = h k h n − k , since P n , k ≅ Ind S k × S n − k S n 1 P_{n,k}\cong\operatorname{Ind}_{S_k\times S_{n-k}}^{S_n}\mathbf{1} P n , k ≅ Ind S k × S n − k S n 1 ( k , n − k ) (k,n-k) ( k , n − k ) [5 Chapter 12 and Appendix B] , this is ξ ( k , n − k ) \xi^{(k,n-k)} ξ ( k , n − k ) ( k , n − k ) (k,n-k) ( k , n − k ) ( A , A c ) (A,A^c) ( A , A c ) k k k A A A [ n ] [n] [ n ]
Third proof of Theorem 3.1
For curiosity’s sake, we shall finally outline how these permutation modules P n , k P_{n,k} P n , k λ \lambda λ M n = P n , 1 M_n=P_{n,1} M n = P n , 1
We extend the definition (98) to k = 0 k=0 k = 0 P n , 0 P_{n,0} P n , 0 P n , n P_{n,n} P n , n S n S_n S n
For positive integers c 1 , … , c r c_1,\ldots,c_r c 1 , … , c r d = c 1 + ⋯ + c r ≤ n d=c_1+\cdots+c_r\le n d = c 1 + ⋯ + c r ≤ n
Y c 1 , … , c r = Q [ { ( A 1 , … , A r ) : A i ⊆ [ n ] , ∣ A i ∣ = c i and the A i are pairwise disjoint } ] . (100) Y_{c_1,\ldots,c_r}=\mathbb{Q}\left[\{(A_1,\ldots,A_r):A_i\subseteq[n],\ |A_i|=c_i\text{ and the }A_i\text{ are pairwise disjoint}\}\right]. \tag*{(100)} Y c 1 , … , c r = Q [ {( A 1 , … , A r ) : A i ⊆ [ n ] , ∣ A i ∣ = c i and the A i are pairwise disjoint } ] . (100) This is isomorphic to a Young permutation module of tabloids (encoding each A i A_i A i i i i [ n ] ∖ ( A 1 ∪ A 2 ∪ ⋯ ∪ A r ) [n]\setminus(A_1\cup A_2\cup\cdots\cup A_r) [ n ] ∖ ( A 1 ∪ A 2 ∪ ⋯ ∪ A r ) ( r + 1 ) (r+1) ( r + 1 ) d d d support size of this permutation module. Note that Y k = P n , k Y_k=P_{n,k} Y k = P n , k
Lemma B.9. Let L n L_n L n λ \lambda λ R ( S n ) R(S_n) R ( S n ) M n M_n M n P n , j ∈ L n P_{n,j} \in L_n P n , j ∈ L n j < d j<d j < d Y c 1 , … , c r Y_{c_1,\ldots,c_r} Y c 1 , … , c r d d d r ≥ 2 r \ge2 r ≥ 2 L n L_n L n
Proof sketch. We prove the assertion simultaneously for all support sizes d d d d d d d = 1 d=1 d = 1 r ≥ 2 r \ge2 r ≥ 2 d ≥ 2 d \ge2 d ≥ 2 P n , j ∈ L n P_{n,j} \in L_n P n , j ∈ L n j < d j<d j < d
Since r ≥ 2 r \ge2 r ≥ 2 c i < d c_i<d c i < d P n , c i P_{n,c_i} P n , c i L n L_n L n P n , c : = P n , c 1 ⊗ P n , c 2 ⊗ ⋯ ⊗ P n , c r P_{n,\mathbf{c}} := P_{n,c_1} \otimes P_{n,c_2} \otimes\cdots\otimes P_{n,c_r} P n , c := P n , c 1 ⊗ P n , c 2 ⊗ ⋯ ⊗ P n , c r ( A 1 , … , A r ) (A_1,\ldots,A_r) ( A 1 , … , A r ) [ n ] [n] [ n ] ∣ A i ∣ = c i |A_i|=c_i ∣ A i ∣ = c i P n , c \mathfrak{P}_{n,\mathbf{c}} P n , c S n S_n S n B ⊆ [ r ] B \subseteq[r] B ⊆ [ r ]
R B = ( ⋂ i ∈ B A i ) ∖ ( ⋃ i ∉ B A i ) . R_B=\left(\bigcap_{i\in B} A_i\right)\setminus\left(\bigcup_{i\notin B} A_i\right). R B = ( i ∈ B ⋂ A i ) ∖ i ∈ / B ⋃ A i . (so that the R B R_B R B B B B A 1 ∪ A 2 ∪ ⋯ ∪ A r A_1 \cup A_2 \cup\cdots\cup A_r A 1 ∪ A 2 ∪ ⋯ ∪ A r A 1 , A 2 , … , A r A_1,A_2,\ldots,A_r A 1 , A 2 , … , A r S n S_n S n P n , c \mathfrak{P}_{n,\mathbf{c}} P n , c ∣ R B ∣ |R_B| ∣ R B ∣ (100) . Each of the other orbits has union of size strictly smaller than d d d Y b 1 , … , b s Y_{b_1,\ldots,b_s} Y b 1 , … , b s < d <d < d
For every such smaller-support orbit module, let e < d e<d e < d P n , j P_{n,j} P n , j j < e j<e j < e L n L_n L n j < e < d j<e<d j < e < d L n L_n L n Y e = P n , e Y_e=P_{n,e} Y e = P n , e L n L_n L n e < d e<d e < d L n L_n L n P n , c 1 ⊗ ⋯ ⊗ P n , c r P_{n,c_1} \otimes\cdots\otimes P_{n,c_r} P n , c 1 ⊗ ⋯ ⊗ P n , c r Y c 1 , … , c r Y_{c_1,\ldots,c_r} Y c 1 , … , c r
Proposition B.10. For every 0 ≤ k ≤ n 0 \le k \le n 0 ≤ k ≤ n P n , k ∈ L n P_{n,k} \in L_n P n , k ∈ L n
Proof sketch. We use strong induction on k k k k = 0 k=0 k = 0 k = 1 k=1 k = 1 k ≥ 2 k \ge2 k ≥ 2 P n , j ∈ L n P_{n,j} \in L_n P n , j ∈ L n j < k j<k j < k
Step 1: decompose the symmetric power into permutation orbits. The symmetric power Sym k ( M n ) \operatorname{Sym}^k(M_n) Sym k ( M n ) k k k e 1 , … , e n e_1,\ldots,e_n e 1 , … , e n S n S_n S n λ ⊢ k \lambda\vdash k λ ⊢ k m r ( λ ) m_r(\lambda) m r ( λ ) λ \lambda λ r r r λ \lambda λ
Y m 1 ( λ ) , m 2 ( λ ) , … Y_{m_1(\lambda),m_2(\lambda),\ldots} Y m 1 ( λ ) , m 2 ( λ ) , … with zero entries omitted. Its support size is ℓ ( λ ) = ∑ r m r ( λ ) \ell(\lambda)=\sum_r m_r(\lambda) ℓ ( λ ) = ∑ r m r ( λ )
For λ = ( 1 k ) \lambda=(1^k) λ = ( 1 k ) P n , k P_{n,k} P n , k λ ⊢ k \lambda\vdash k λ ⊢ k ℓ ( λ ) < k \ell(\lambda)<k ℓ ( λ ) < k L n L_n L n m r ( λ ) m_r(\lambda) m r ( λ ) P n , ℓ ( λ ) P_{n,\ell(\lambda)} P n , ℓ ( λ ) L n L_n L n
[ Sym k ( M n ) ] = [ P n , k ] + ∑ λ ⊢ k ; λ ≠ ( 1 k ) [ Y m 1 ( λ ) , m 2 ( λ ) , … ] , (101) [\operatorname{Sym}^k(M_n)] = [P_{n,k}] + \sum_{\substack{\lambda\vdash k;\\\lambda\ne(1^k)}} [Y_{m_1(\lambda),m_2(\lambda),\ldots}], \tag*{(101)} [ Sym k ( M n )] = [ P n , k ] + λ ⊢ k ; λ = ( 1 k ) ∑ [ Y m 1 ( λ ) , m 2 ( λ ) , … ] , (101) and all addends in the sum belong to L n L_n L n
Step 2: isolate the k k k The left-hand side belongs to L n L_n L n R ( S n ) [ [ t ] ] R(S_n)[[t]] R ( S n ) [[ t ]] [10 Lemma 3.29 and the preceding derivation] for an elementary proof using eigenvalues)15
∑ j ≥ 0 [ Sym j ( M n ) ] t j = 1 λ − t ( [ M n ] ) = λ − t ( − [ M n ] ) , \sum_{j\geq0} [\operatorname{Sym}^{j}(M_n)]t^j = \frac{1}{\lambda_{-t}([M_n])} = \lambda_{-t}(-[M_n]), j ≥ 0 ∑ [ Sym j ( M n )] t j = λ − t ([ M n ]) 1 = λ − t ( − [ M n ]) , so each symmetric-power class [ Sym j ( M n ) ] [\operatorname{Sym}^{j}(M_n)] [ Sym j ( M n )] λ \lambda λ L n L_n L n (101) belong to L n L_n L n [ P n , k ] ∈ L n [P_{n,k}] \in L_n [ P n , k ] ∈ L n
Third proof of Theorem 3.1 (sketched). By Corollary B.8 , the classes [ P n , k ] [P_{n,k}] [ P n , k ] R ( S n ) R(S_n) R ( S n ) λ \lambda λ L n L_n L n M n M_n M n R ( S n ) ⊆ L n R(S_n) \subseteq L_n R ( S n ) ⊆ L n L n = R ( S n ) L_n = R(S_n) L n = R ( S n )
