Introduction
Lusztig constructed the canonical basis of a half of a quantum group in finite type A , D , E A,D,E A , D , E [14 ] . His geometric construction used intersection cohomology on varieties of quiver representations. In [15 ] he used induction and restriction of equivariant sheaves to construct the basis for arbitrary symmetric Cartan data. In that construction, the coefficients of multiplication and twisted comultiplication record the multiplicities of shifted simple perverse sheaves. Consequently they are Laurent polynomials with nonnegative integer coefficients [15 11.5] . Kashiwara constructed global crystal bases using the crystal at parameter zero and its lift to the quantum group [11 ] . Grojnowski and Lusztig compared the global and geometric bases in symmetric type [9 ] .
The two halves act together on integrable weight modules. In finite type A A A [1 ] . Lusztig constructed the canonical basis of the modified quantum group in general by studying tensor products of a simple integrable lowest-weight module and a simple integrable highest-weight module [16 ] . The quasi-R R R
In [16 §11] Lusztig asked whether the multiplication and comultiplication coefficients for this basis have the same positivity as those of the half algebra. The question appears as Conjecture 25.4.2 in [17 ] . Write v v v U ˙ \dot{U} U ˙ Q ( v ) \mathbb{Q}(v) Q ( v ) B ˙ \dot{B} B ˙ 1 ν 1_\nu 1 ν ν \nu ν U ˙ 1 ν \dot{U}1_\nu U ˙ 1 ν ν \nu ν B ˙ \dot{B} B ˙ N [ v , v − 1 ] \mathbb{N}[v,v^{-1}] N [ v , v − 1 ]
The flag realization supplies geometric models for multiplication in type A A A A A A [20 ] . Fu and Shoji used the affine Schur algebras and the positive half of a larger affine quantum group to prove multiplication positivity in the original modified algebra [8 ] . Fan and Li proved positivity for its modified comultiplication [7 ] .
A categorical approach begins with the quiver Hecke algebras of Khovanov and Lauda and Rouquier [12 , 19 ] . For geometric parameters in characteristic zero, Varagnolo and Vasserot identified the classes of normalized indecomposable projectives with the half-algebra canonical basis [21 ] . Induction makes the product coefficients graded direct-summand multiplicities. For the modified algebra, Li constructed a geometric realization using double-framed quivers and formulated the identification of its geometric labels with the canonical basis as a separate conjecture [13 Conjecture 6.11] .
The diagrammatic category of the modified algebra also has an indecomposable basis. Webster identified it with Lusztig’s basis for finite type A A A D D D E E E [22 Corollary B and Corollary 8.10] . For arbitrary symmetric Cartan matrices his tensor-product identification applies to tensor products of highest-weight modules [22 Theorem 8.7] . The finite-type identification of the modified basis uses the longest Weyl group element to pass from lowest–highest tensor products to that setting.
Fang and Lan used framed diagrams to recover tensor-product canonical bases of highest-weight modules from the canonical basis for a larger quantum group [6 ] . Fang and He adjoin one framing vertex at each original vertex and realize the canonical bases of lowest–highest Demazure tensor subspaces using the canonical basis of the negative half of the enlarged quantum group [5 Theorems 3.5 and 3.7] . They use this realization to prove the comultiplication part of the conjecture and the related positivity of the bilinear form for every symmetric Cartan datum. Their multiplication theorem applies when one factor belongs to a parabolic subalgebra with a finite-type positive part, or to its image under exchange of the two halves [5 Theorem 6.6] .
Theorem 1.1. Let U ˙ \dot{U} U ˙ b , b ′ ∈ B ˙ b,b' \in\dot{B} b , b ′ ∈ B ˙
b b ′ = ∑ c ∈ B ˙ m b , b ′ c ( v ) c , m b , b ′ c ( v ) ∈ N [ v , v − 1 ] . bb'=\sum_{c\in\dot{B}}m^c_{b,b'}(v)c,\qquad m^c_{b,b'}(v)\in\mathbb{N}[v,v^{-1}]. b b ′ = c ∈ B ˙ ∑ m b , b ′ c ( v ) c , m b , b ′ c ( v ) ∈ N [ v , v − 1 ] . Corollary 1.2. Lusztig’s Conjecture 25.4.2 holds for every symmetric Cartan datum and every root datum of that type.
Proof. Theorem 1.1 gives multiplication positivity. Fang and He’s comultiplication theorem gives the other assertion for the same Cartan and root data [5 Theorem 6.3 and §6.6] . □
For a given simple root, the raising functor acts at the Demazure stages where that root is a left descent. These stages are cofinal in Bruhat order. We construct the functor on a quotient by an ideal of morphisms at each such stage and prove compatibility with transition functors. This gives the action of the full diagrammatic quantum group on the colimit. Specializing the symmetric dot polynomials on the framing strands identifies the Hom spaces in its positive-word corner with cyclotomic Hom spaces. This identification gives full evaluation, so an indecomposable in the diagrammatic category evaluates to a single shifted canonical label or to zero. The fixed-weight-difference comparison with Lusztig’s tensor bases then identifies the diagrammatic basis in every right column.
Weights and quiver Hecke categories
Let I I I A = ( a i j ) i , j ∈ I A=(a_{ij})_{i,j\in I} A = ( a ij ) i , j ∈ I a i i = 2 a_{ii}=2 a ii = 2 a i j a_{ij} a ij i ≠ j i\ne j i = j X X X Y = Hom ( X , Z ) Y=\operatorname{Hom}(X,\mathbb{Z}) Y = Hom ( X , Z ) α i ∈ X \alpha_i\in X α i ∈ X h i ∈ Y h_i\in Y h i ∈ Y ⟨ h i , α j ⟩ = a i j \langle h_i,\alpha_j\rangle=a_{ij} ⟨ h i , α j ⟩ = a ij ⟨ , ⟩ \langle\ ,\ \rangle ⟨ , ⟩ W W W s i ( λ ) = λ − ⟨ h i , λ ⟩ α i s_i(\lambda)=\lambda-\langle h_i,\lambda\rangle\alpha_i s i ( λ ) = λ − ⟨ h i , λ ⟩ α i w ≤ v 0 w\le v_0 w ≤ v 0 v 0 v_0 v 0 w w w
Set A = Z [ v , v − 1 ] \mathcal{A}=\mathbb{Z}[v,v^{-1}] A = Z [ v , v − 1 ] h h h
[ h ] = v h − v − h v − v − 1 , [ a ] ! = ∏ r = 1 a [ r ] ( a ≥ 0 ) . [h]=\frac{v^h-v^{-h}}{v-v^{-1}},\qquad[a]!=\prod_{r=1}^{a}[r]\quad(a\ge0). [ h ] = v − v − 1 v h − v − h , [ a ]! = r = 1 ∏ a [ r ] ( a ≥ 0 ) . The empty product is 1. Let f \mathfrak{f} f Q ( v ) \mathbb{Q}(v) Q ( v ) θ i \theta_i θ i
∑ r = 0 1 − a i j ( − 1 ) r θ i ( r ) θ j θ i ( 1 − a i j − r ) = 0 ( i ≠ j ) , θ i ( r ) = θ i r [ r ] ! . \sum_{r=0}^{1-a_{ij}}(-1)^r\theta_i^{(r)}\theta_j\theta_i^{(1-a_{ij}-r)}=0\quad(i\ne j),\qquad\theta_i^{(r)}=\frac{\theta_i^r}{[r]!}. r = 0 ∑ 1 − a ij ( − 1 ) r θ i ( r ) θ j θ i ( 1 − a ij − r ) = 0 ( i = j ) , θ i ( r ) = [ r ]! θ i r . Its integral form A f \mathcal{A}\mathfrak{f} A f B \mathbf{B} B U + U^+ U + U − U^- U − f \mathfrak{f} f E i E_i E i F i F_i F i y ∈ f y\in\mathfrak{f} y ∈ f y − y^- y − U − U^- U − 1 ν 1_\nu 1 ν ν \nu ν 1 μ U ˙ 1 ν 1_\mu\dot{U}1_\nu 1 μ U ˙ 1 ν 1 μ 1_\mu 1 μ A U ˙ \mathcal{A}\dot{U} A U ˙ E i ( a ) = E i a / [ a ] ! E_i^{(a)}=E_i^a/[a]! E i ( a ) = E i a / [ a ]! F i ( a ) = F i a / [ a ] ! F_i^{(a)} = F_i^a/[a]! F i ( a ) = F i a / [ a ]! a ≥ 0 a \ge0 a ≥ 0 E i , F i E_i,F_i E i , F i α i , − α i \alpha_i,-\alpha_i α i , − α i
( E i F j − F j E i ) 1 ν = δ i j [ ⟨ h i , ν ⟩ ] 1 ν . (E_iF_j-F_jE_i)1_\nu=\delta_{ij}[\langle h_i,\nu\rangle]1_\nu. ( E i F j − F j E i ) 1 ν = δ ij [⟨ h i , ν ⟩] 1 ν . We use the free abelian group on the root symbols α i \alpha_i α i X X X E i , F i E_i,F_i E i , F i α i , − α i \alpha_i,-\alpha_i α i , − α i y ∈ f y \in\mathfrak{f} y ∈ f ∣ y ∣ |y| ∣ y ∣ y − y^{-} y − ∣ θ i ∣ = α i |\theta_i|=\alpha_i ∣ θ i ∣ = α i
Framing and a fixed right weight
Adjoin I ′ = { i ′ : i ∈ I } I'=\{i':i\in I\} I ′ = { i ′ : i ∈ I } i ′ i' i ′ i i i n = ∣ I ∣ n=|I| n = ∣ I ∣
A ~ = ( A − I n − I n 2 I n ) , P = Z I ⊔ I ′ . \widetilde{A}=\begin{pmatrix}A&-I_n\\-I_n&2I_n\end{pmatrix},\qquad P=\mathbb{Z}^{I\sqcup I'}. A = ( A − I n − I n 2 I n ) , P = Z I ⊔ I ′ . The simple coroots on P P P A ~ \widetilde{A} A 2 I n 2I_n 2 I n det A ~ = 2 n det ( A − 1 2 I n ) = det ( 2 A − I n ) ≡ 1 ( m o d 2 ) \det\widetilde{A}=2^n\det(A-\frac{1}{2}I_n)=\det(2A-I_n)\equiv1\pmod2 det A = 2 n det ( A − 2 1 I n ) = det ( 2 A − I n ) ≡ 1 ( mod 2 ) P P P P P P A ~ − 1 \widetilde{A}^{-1} A − 1 ( α i , ν ) = ⟨ h i , ν ⟩ (\alpha_i,\nu)=\langle h_i,\nu\rangle ( α i , ν ) = ⟨ h i , ν ⟩ P P P I I I
Write f ~ \widetilde{\mathfrak f} f B ~ \widetilde{\mathbf B} B ζ ∈ P \zeta\in P ζ ∈ P m = ( m i ) i ∈ I m=(m_i)_{i\in I} m = ( m i ) i ∈ I
θ m = ∏ i ∈ I θ i ′ ( m i ) , γ m = ∑ i ∈ I m i α i ′ , η m = ( 0 , m ) ∈ P . \theta_m=\prod_{i\in I}\theta_{i'}^{(m_i)},\qquad\gamma_m=\sum_{i\in I}m_i\alpha_{i'},\qquad\eta_m=(0,m)\in P. θ m = i ∈ I ∏ θ i ′ ( m i ) , γ m = i ∈ I ∑ m i α i ′ , η m = ( 0 , m ) ∈ P . The factors in θ m \theta_m θ m m i ≥ max ( 0 , ⟨ h i , ζ ⟩ ) m_i\ge\max(0,\langle h_i,\zeta\rangle) m i ≥ max ( 0 , ⟨ h i , ζ ⟩) i ∈ I i\in I i ∈ I λ 2 = − γ m \lambda_2=-\gamma_m λ 2 = − γ m λ 1 = − ζ − γ m \lambda_1=-\zeta-\gamma_m λ 1 = − ζ − γ m I I I m i m_i m i m i − ⟨ h i , ζ ⟩ m_i-\langle h_i,\zeta\rangle m i − ⟨ h i , ζ ⟩ λ 2 − λ 1 = ζ \lambda_2-\lambda_1=\zeta λ 2 − λ 1 = ζ w ∈ W w\in W w ∈ W μ w = − w λ 1 \mu_w=-w\lambda_1 μ w = − w λ 1
F m = f θ m f ⊆ f ~ , F_m=\mathfrak f\theta_m\mathfrak f\subseteq\widetilde{\mathfrak f}, F m = f θ m f ⊆ f , where the right side denotes the linear span of the indicated products.
Geometric projectives
All categories are over k = C k=\mathbb{C} k = C I ⊔ I ′ I\sqcup I' I ⊔ I ′ − a i j -a_{ij} − a ij i , j i,j i , j A ~ \widetilde{A} A R ( β ) R(\beta) R ( β ) β \beta β β \beta β β \beta β i = ( i 1 , … , i N ) \mathbf{i}=(i_1,\ldots,i_N) i = ( i 1 , … , i N ) e ( i ) e(\mathbf{i}) e ( i ) x r x_r x r 1 ≤ r ≤ N 1\le r\le N 1 ≤ r ≤ N τ r \tau_r τ r 1 ≤ r < N 1\le r<N 1 ≤ r < N r , r + 1 r,r+1 r , r + 1 0 0 0 2 2 2 − ( α i r , α i r + 1 ) -(\alpha_{i_r},\alpha_{i_{r+1}}) − ( α i r , α i r + 1 ) [21 Remark 3.3(c)] , with
τ r 2 e ( i ) = Q i r i r + 1 ( x r , x r + 1 ) e ( i ) , \tau_r^2 e(\mathbf{i})=Q_{i_r i_{r+1}}(x_r,x_{r+1})e(\mathbf{i}), τ r 2 e ( i ) = Q i r i r + 1 ( x r , x r + 1 ) e ( i ) , Q i j ( u 1 , u 2 ) = ( − 1 ) # { arrows i → j } ( u 1 − u 2 ) − a i j ( i ≠ j ) , Q i i = 0. Q_{ij}(u_1,u_2)=(-1)^{\#\{\text{arrows }i\to j\}}(u_1-u_2)^{-a_{ij}}\quad(i\ne j),\qquad Q_{ii}=0. Q ij ( u 1 , u 2 ) = ( − 1 ) # { arrows i → j } ( u 1 − u 2 ) − a ij ( i = j ) , Q ii = 0. Thus Q i j ( u 1 , u 2 ) = Q j i ( u 2 , u 1 ) Q_{ij}(u_1,u_2)=Q_{ji}(u_2,u_1) Q ij ( u 1 , u 2 ) = Q j i ( u 2 , u 1 )
For contents β 1 , β 2 \beta_1,\beta_2 β 1 , β 2 e ( β 1 , β 2 ) e(\beta_1,\beta_2) e ( β 1 , β 2 ) β 1 \beta_1 β 1 β 2 \beta_2 β 2 R ( β 1 ) ⊗ k R ( β 2 ) R(\beta_1)\otimes_kR(\beta_2) R ( β 1 ) ⊗ k R ( β 2 ) R ( β 1 + β 2 ) R(\beta_1+\beta_2) R ( β 1 + β 2 ) P 1 , P 2 P_1,P_2 P 1 , P 2 P 1 P 2 P_1P_2 P 1 P 2
R ( β 1 + β 2 ) e ( β 1 , β 2 ) ⊗ R ( β 1 ) ⊗ k R ( β 2 ) ( P 1 ⊗ k P 2 ) . R(\beta_1+\beta_2)e(\beta_1,\beta_2)\otimes_{R(\beta_1)\otimes_kR(\beta_2)}(P_1\otimes_kP_2). R ( β 1 + β 2 ) e ( β 1 , β 2 ) ⊗ R ( β 1 ) ⊗ k R ( β 2 ) ( P 1 ⊗ k P 2 ) . The restriction Res β 1 , β 2 ( M ) \operatorname{Res}_{\beta_1,\beta_2}(M) Res β 1 , β 2 ( M ) e ( β 1 , β 2 ) M e(\beta_1,\beta_2)M e ( β 1 , β 2 ) M P I ~ \mathscr P_{\widetilde I} P I β \beta β P I \mathscr P_I P I ∣ P 1 ∣ |P_1| ∣ P 1 ∣ P 1 P_1 P 1 F P 1 FP_1 F P 1 P 1 P_1 P 1 F i F_i F i F R ( α i ) F_{R(\alpha_i)} F R ( α i ) F P 1 FP_1 F P 1 2 2 2 f : P 1 → P 2 f:P_1\to P_2 f : P 1 → P 2 F f : F P 1 → F P 2 F_f:FP_1\to FP_2 F f : F P 1 → F P 2
For d , j ∈ Z d,j\in\mathbb Z d , j ∈ Z ( M ⟨ d ⟩ ) j = M j − d (M\langle d\rangle)_j=M_{j-d} ( M ⟨ d ⟩ ) j = M j − d HOM d ( M , N ) = Hom ( M ⟨ d ⟩ , N ) \operatorname{HOM}^d(M,N)=\operatorname{Hom}(M\langle d\rangle,N) HOM d ( M , N ) = Hom ( M ⟨ d ⟩ , N ) d d d HOM \operatorname{HOM} HOM d d d K 0 K_0 K 0 v v v ⟨ 1 ⟩ \langle1\rangle ⟨ 1 ⟩
The geometric categorification gives normalized projectives P b P_b P b b ∈ B ~ b\in\widetilde B b ∈ B
K 0 ( P I ~ ) ⟶ A ~ f , [ P b ] ⟼ b . K_0(\mathscr P_{\widetilde I})\longrightarrow\widetilde{\mathcal A}_{\!f},\qquad[P_b]\longmapsto b. K 0 ( P I ) ⟶ A f , [ P b ] ⟼ b . For each β \beta β [21 Theorem 3.6] identifies the algebra with the Ext algebra of the sum S β S_\beta S β M [ r ] j = M j + r M[r]_j=M_{j+r} M [ r ] j = M j + r v v v ⟨ − 1 ⟩ \langle-1\rangle ⟨ − 1 ⟩ v ↦ v − 1 v\mapsto v^{-1} v ↦ v − 1 θ i \theta_i θ i A A A b b b b b b P b ⟨ d ⟩ P_b\langle d\rangle P b ⟨ d ⟩
The projective P b P_b P b Ext ∙ ( S β , I C b ) \operatorname{Ext}^{\bullet}(S_\beta,IC_b) Ext ∙ ( S β , I C b ) I C b IC_b I C b [21 §§4.6–4.7] . Each such perverse sheaf is a summand of a shift of S β S_\beta S β
HOM d ( P b , P c ) ≅ Ext d ( I C b , I C c ) . \operatorname{HOM}^{d}(P_b,P_c) \cong\operatorname{Ext}^{d}(IC_b,IC_c). HOM d ( P b , P c ) ≅ Ext d ( I C b , I C c ) . The right side is zero for d < 0 d<0 d < 0 d = 0 d=0 d = 0 k k k b = c b=c b = c [21 Lemma 3.8(b)] , a torus weight is counted once per occurrence among the arrow-space summands. The inversion-set identity used there matches multiplicities for each arrow separately, hence matches the full products of weights. For ϵ = 1 , − 1 \epsilon=1,-1 ϵ = 1 , − 1 r ϵ r_\epsilon r ϵ θ i \theta_i θ i θ i ⊗ 1 + 1 ⊗ θ i \theta_i\otimes1+1\otimes\theta_i θ i ⊗ 1 + 1 ⊗ θ i v ϵ ( ∣ x 2 ∣ , ∣ y 1 ∣ ) v^{\epsilon(\lvert x_2\rvert,\lvert y_1\rvert)} v ϵ (∣ x 2 ∣ , ∣ y 1 ∣) ( x 1 ⊗ x 2 ) ( y 1 ⊗ y 2 ) (x_1\otimes x_2)(y_1\otimes y_2) ( x 1 ⊗ x 2 ) ( y 1 ⊗ y 2 ) r − 1 r_{-1} r − 1 ∣ x 2 ∣ \lvert x_2\rvert ∣ x 2 ∣ ∣ y 1 ∣ \lvert y_1\rvert ∣ y 1 ∣ − ( ∣ x 2 ∣ , ∣ y 1 ∣ ) -(\lvert x_2\rvert,\lvert y_1\rvert) − (∣ x 2 ∣ , ∣ y 1 ∣) θ i \theta_i θ i a ∈ e F a\in e\mathcal{F} a ∈ e F a 1 , a 2 a_1,a_2 a 1 , a 2
r − 1 ( a ) = τ ( r 1 ( a ) ) , τ ( a 1 ⊗ a 2 ) = v − ( ∣ a 1 ∣ , ∣ a 2 ∣ ) a 2 ⊗ a 1 . r_{-1}(a)=\tau\left(r_1(a)\right),\qquad\tau(a_1\otimes a_2)=v^{-(\lvert a_1\rvert,\lvert a_2\rvert)}a_2\otimes a_1. r − 1 ( a ) = τ ( r 1 ( a ) ) , τ ( a 1 ⊗ a 2 ) = v − (∣ a 1 ∣ , ∣ a 2 ∣) a 2 ⊗ a 1 . For homogeneous a , b , c , d a,b,c,d a , b , c , d v v v τ \tau τ r 1 r_1 r 1 a ⊗ b a\otimes b a ⊗ b c ⊗ d c\otimes d c ⊗ d
( ∣ b ∣ , ∣ c ∣ ) − ( ∣ a ∣ + ∣ c ∣ , ∣ b ∣ + ∣ d ∣ ) = − ( ∣ a ∣ , ∣ b ∣ ) − ( ∣ c ∣ , ∣ d ∣ ) − ( ∣ a ∣ , ∣ d ∣ ) . (\lvert b\rvert,\lvert c\rvert)-(\lvert a\rvert+\lvert c\rvert,\lvert b\rvert+\lvert d\rvert)=-(\lvert a\rvert,\lvert b\rvert)-(\lvert c\rvert,\lvert d\rvert)-(\lvert a\rvert,\lvert d\rvert). (∣ b ∣ , ∣ c ∣) − (∣ a ∣ + ∣ c ∣ , ∣ b ∣ + ∣ d ∣) = − (∣ a ∣ , ∣ b ∣) − (∣ c ∣ , ∣ d ∣) − (∣ a ∣ , ∣ d ∣) . This is the exponent of the r − 1 r_{-1} r − 1 τ ( a ⊗ b ) \tau(a\otimes b) τ ( a ⊗ b ) τ ( c ⊗ d ) \tau(c\otimes d) τ ( c ⊗ d ) τ \tau τ θ i \theta_i θ i
Lemma 2.1. Suppose a graded k k k P b P_b P b HOM d ( P b , P c ) = 0 \operatorname{HOM}^{d}(P_b,P_c)=0 HOM d ( P b , P c ) = 0 d < 0 d<0 d < 0 HOM 0 ( P b , P c ) = δ b c k \operatorname{HOM}^{0}(P_b,P_c)=\delta_{bc}k HOM 0 ( P b , P c ) = δ b c k P b ⟨ d ⟩ P_b\langle d\rangle P b ⟨ d ⟩
Proof. An object whose identity belongs to the ideal is zero in the quotient. Delete such summands from a given finite sum and order the remaining summands by shift. A degree-zero map from P b ⟨ d ⟩ P_b\langle d\rangle P b ⟨ d ⟩ P c ⟨ e ⟩ P_c\langle e\rangle P c ⟨ e ⟩ d − e d-e d − e
For an idempotent e e e p p p e − p e-p e − p u = e p + ( 1 − e ) ( 1 − p ) u=ep+(1-e)(1-p) u = e p + ( 1 − e ) ( 1 − p ) e u = u p eu=up e u = u p e e e k k k
Cyclotomic projectives
For a dominant weight λ \lambda λ Λ λ \Lambda_\lambda Λ λ ω Λ λ {}^{\omega}\Lambda_\lambda ω Λ λ E i E_i E i F i F_i F i R λ ( β ) R^\lambda(\beta) R λ ( β ) R ( β ) R(\beta) R ( β ) x N ⟨ h i , λ ⟩ e ( i ) = 0 x_N^{\langle h_i,\lambda\rangle}e(\mathbf{i})=0 x N ⟨ h i , λ ⟩ e ( i ) = 0 i \mathbf{i} i N > 0 N>0 N > 0 i i i R λ ( 0 ) = R ( 0 ) = k R^\lambda(0)=R(0)=k R λ ( 0 ) = R ( 0 ) = k Λ λ \Lambda_\lambda Λ λ [10 Theorem 6.2] . The lowest-weight action is obtained by composing the two diagram reflections in [18 Propositions 2.6.4–2.6.5] . The two reversals of composition cancel, the weights change sign, and the lowest-weight induction operators are the E i E_i E i ω Λ λ {}^{\omega}\Lambda_\lambda ω Λ λ [4 Remark 4.22] .
The normalizations follow from the half-algebra projectives. Base change along R ( β ) → R λ ( β ) R(\beta)\to R^\lambda(\beta) R ( β ) → R λ ( β ) R ( β ) R(\beta) R ( β ) b ↦ b − η λ b\mapsto b^{-\eta_\lambda} b ↦ b − η λ η λ \eta_\lambda η λ [17 Theorem 14.4.11] identifies the nonzero images with the distinct normalized basis labels of Λ λ \Lambda_\lambda Λ λ
Write L ( − λ 1 ) \mathcal{L}(-\lambda_1) L ( − λ 1 ) L w L_w L w ξ μ w \xi_{\mu_w} ξ μ w μ w \mu_w μ w L η m + L^+_{\eta_m} L η m + Λ ^ η m \widehat{\Lambda}_{\eta_m} Λ η m e e e W W W L e L_e L e R ( 0 ) = k R(0)=k R ( 0 ) = k
Along a length-increasing step u ↦ s i u u\mapsto s_i u u ↦ s i u μ u \mu_u μ u i i i − r -r − r r ≥ 0 r\geq0 r ≥ 0 E i ( r ) E_i^{(r)} E i ( r ) F i ( r ) F_i^{(r)} F i ( r ) L u L_u L u L s i u L_{s_i u} L s i u r ( r + ⟨ h i , μ u ⟩ ) r(r+\langle h_i,\mu_u\rangle) r ( r + ⟨ h i , μ u ⟩) F i ( r ) F_i^{(r)} F i ( r ) E i ( r ) E_i^{(r)} E i ( r ) [18 Lemma 2.6.14] . Therefore
HOM ( L s i u , L s i u ) ≅ HOM ( F i ( r ) L s i u , L u ) ≅ HOM ( L u , L u ) . \operatorname{HOM}(L_{s_i u},L_{s_i u})\cong\operatorname{HOM}(F_i^{(r)}L_{s_i u},L_u)\cong\operatorname{HOM}(L_u,L_u). HOM ( L s i u , L s i u ) ≅ HOM ( F i ( r ) L s i u , L u ) ≅ HOM ( L u , L u ) . Induction gives HOM ( L w , L w ) = k \operatorname{HOM}(L_w,L_w)=k HOM ( L w , L w ) = k s i w < w s_iw<w s i w < w μ w \mu_w μ w i i i ⟨ h i , μ w ⟩ ≥ 0 \langle h_i,\mu_w\rangle\ge0 ⟨ h i , μ w ⟩ ≥ 0 E i L w = 0 E_iL_w=0 E i L w = 0
The thickening maps
For w ∈ W w\in W w ∈ W ω V w ( λ 1 ) = U − ξ μ w {}^{\omega}V_w(\lambda_1)=U^{-\xi_{\mu_w}} ω V w ( λ 1 ) = U − ξ μ w ω Λ λ 1 {}^{\omega}\Lambda_{\lambda_1} ω Λ λ 1 E i E_i E i s i w < w s_iw<w s i w < w E i E_i E i η λ 2 \eta_{\lambda_2} η λ 2 Λ λ 2 \Lambda_{\lambda_2} Λ λ 2
Δ ( F i ) = F i ⊗ K − h i + 1 ⊗ F i , Δ ( E i ) = E i ⊗ 1 + K h i ⊗ E i , \Delta(F_i)=F_i\otimes K_{-h_i}+1\otimes F_i,\qquad\Delta(E_i)=E_i\otimes1+K_{h_i}\otimes E_i, Δ ( F i ) = F i ⊗ K − h i + 1 ⊗ F i , Δ ( E i ) = E i ⊗ 1 + K h i ⊗ E i , where K h K_h K h ν \nu ν v ⟨ h , ν ⟩ v^{\langle h,\nu\rangle} v ⟨ h , ν ⟩ R R R [17 §24.1] . The latter is the intertwiner from the factorwise bar of the coproduct action to the action with the algebra bar, normalized to have constant term 1 ⊗ 1 1\otimes1 1 ⊗ 1 v − 1 v^{-1} v − 1 Z [ v − 1 ] \mathbb{Z}[v^{-1}] Z [ v − 1 ]
To apply Fang–He’s thickening results with the lattice P P P E i , F i E_i,F_i E i , F i E i , F i E_i,F_i E i , F i P P P
The subspace F m F_m F m B ∩ F m \mathbf{B}\cap F_m B ∩ F m [6 Proposition 4.3] . By [5 Proposition 1.5 and Theorems 3.5, 3.7] , the maps
a w : f ⟶ ω V w ( λ 1 ) , y ⟼ y − ξ μ w , Φ w : F m ⟶ ω V w ( λ 1 ) ⊗ Λ λ 2 a_w:f\longrightarrow{}^{\omega}V_w(\lambda_1),\quad y\longmapsto y^{-\xi_{\mu_w}},\qquad\Phi_w:F_m\longrightarrow{}^{\omega}V_w(\lambda_1)\otimes\Lambda_{\lambda_2} a w : f ⟶ ω V w ( λ 1 ) , y ⟼ y − ξ μ w , Φ w : F m ⟶ ω V w ( λ 1 ) ⊗ Λ λ 2 send canonical basis elements to canonical basis elements or zero. In both cases the maps are surjective and the killed labels span the kernel. Thus each target label has a unique preimage label: two distinct surviving labels with the same image would have a difference in the kernel. Let M μ w M_{\mu_w} M μ w u u u μ w \mu_w μ w f f f y ↦ y − u y\mapsto y^{-u} y ↦ y − u F m → M μ w ⊗ Λ λ 2 F_m\to M_{\mu_w}\otimes\Lambda_{\lambda_2} F m → M μ w ⊗ Λ λ 2 [5 Lemma 2.2] . Its inverse ψ \psi ψ wt ( x ) \operatorname{wt}(x) wt ( x ) x ∈ Λ λ 2 x \in\Lambda_{\lambda_2} x ∈ Λ λ 2
ψ ( y − u ⊗ η λ 2 ) = θ m y , (1) \psi(y^{-u} \otimes\eta_{\lambda_2})=\theta_m y, \tag*{(1)} ψ ( y − u ⊗ η λ 2 ) = θ m y , (1) ψ ( y − u ⊗ F i x ) = F i ψ ( y − u ⊗ x ) − v − ⟨ h i , wt ( x ) ⟩ ψ ( F i y − u ⊗ x ) . (2) \psi(y^{-u} \otimes F_i x)=F_i\psi(y^{-u} \otimes x)-v^{-\langle h_i,\operatorname{wt}(x)\rangle}\psi(F_i y^{-u} \otimes x). \tag*{(2)} ψ ( y − u ⊗ F i x ) = F i ψ ( y − u ⊗ x ) − v − ⟨ h i , wt ( x )⟩ ψ ( F i y − u ⊗ x ) . (2) On the right F i F_i F i θ i \theta_i θ i F i F_i F i y − u ⊗ x y^{-u} \otimes x y − u ⊗ x F i y − u ⊗ K − h i x F_i y^{-u} \otimes K_{-h_i}x F i y − u ⊗ K − h i x Φ w \Phi_w Φ w θ m \theta_m θ m ξ μ w ⊗ η λ 2 \xi_{\mu_w} \otimes\eta_{\lambda_2} ξ μ w ⊗ η λ 2 x θ m y x\theta_m y x θ m y x , y ∈ f x,y \in\mathbf f x , y ∈ f x − x^{-} x − y − ξ μ w ⊗ η λ 2 y^{-}\xi_{\mu_w} \otimes\eta_{\lambda_2} y − ξ μ w ⊗ η λ 2
Write u η m u_{\eta_m} u η m θ m − u η m \theta_m^{-}u_{\eta_m} θ m − u η m u η m u_{\eta_m} u η m m i m_i m i θ m − \theta_m^{-} θ m − E i E_i E i F j F_j F j θ m − u η m \theta_m^{-}u_{\eta_m} θ m − u η m I I I η m − γ m \eta_m-\gamma_m η m − γ m m i m_i m i η λ 2 \eta_{\lambda_2} η λ 2 θ m − u η m \theta_m^{-}u_{\eta_m} θ m − u η m ι : Λ λ 2 → Λ ~ η m \iota:\Lambda_{\lambda_2}\to\widetilde{\Lambda}_{\eta_m} ι : Λ λ 2 → Λ η m E i , F i E_i,F_i E i , F i η m \eta_m η m
Lemma 2.2. The labels in B ~ ∩ F m \widetilde{\mathbf B}\cap F_m B ∩ F m u η m u_{\eta_m} u η m b θ m b\theta_m b θ m b ∈ B b\in\mathbf B b ∈ B b − η λ 2 ≠ 0 b^{-}\eta_{\lambda_2}\ne0 b − η λ 2 = 0
Proof. For b ∈ B b\in\mathbf B b ∈ B θ i ′ ( m i ) \theta_{i'}^{(m_i)} θ i ′ ( m i ) i ′ i' i ′ [17 §14.4] gives a canonical label of right string length m i m_i m i i ′ i' i ′ m i m_i m i i ′ i' i ′ r r r f ~ θ i ′ ( r ) \widetilde{\mathbf f}\theta_{i'}^{(r)} f θ i ′ ( r )
Every original right factor of positive content annihilates u η m u_{\eta_m} u η m F m F_m F m U − θ m − u η m = ι ( Λ λ 2 ) U^{-}\theta_m^{-}u_{\eta_m}=\iota(\Lambda_{\lambda_2}) U − θ m − u η m = ι ( Λ λ 2 ) b θ m b\theta_m b θ m Λ λ 2 \Lambda_{\lambda_2} Λ λ 2
In the framing content of F m F_m F m f ~ θ i ′ ( m i + 1 ) \widetilde{\mathbf f}\theta_{i'}^{(m_i+1)} f θ i ′ ( m i + 1 ) i ′ i' i ′ m i m_i m i f ~ θ i \widetilde{\mathbf f}\theta_i f θ i i ∈ I i\in I i ∈ I [5 Equation (3.4)] .
For an original content α \alpha α γ \gamma γ ( r 1 ( z ) ) γ , α (r_1(z))_{\gamma,\alpha} ( r 1 ( z ) ) γ , α
a ⊗ b ⟼ b − ξ μ w ⊗ a − u η m . a\otimes b\longmapsto b^{-}\xi_{\mu_w}\otimes a^{-}u_{\eta_m}. a ⊗ b ⟼ b − ξ μ w ⊗ a − u η m . Summing its values on ( r 1 ( z ) ) γ , α (r_1(z))_{\gamma,\alpha} ( r 1 ( z ) ) γ , α ( 1 ⊗ ι ) Φ w ( z ) (1\otimes\iota)\Phi_w(z) ( 1 ⊗ ι ) Φ w ( z ) [5 Lemma 2.1] . The powers in this formula follow from the map b ⊗ a ↦ b − ⊗ K − ∣ b ∣ a − b\otimes a\mapsto b^{-}\otimes K_{-|b|}a^{-} b ⊗ a ↦ b − ⊗ K − ∣ b ∣ a − K − α = K − ∑ i n i h i K_{-\alpha}=K_{-\sum_i n_i h_i} K − α = K − ∑ i n i h i α = ∑ i n i α i \alpha=\sum_i n_i\alpha_i α = ∑ i n i α i Δ ( F i ) \Delta(F_i) Δ ( F i ) a , b , c , d a,b,c,d a , b , c , d
( K − ∣ b ∣ a − ) ( K − ∣ c ∣ d − ) = v − ( ∣ a ∣ , ∣ c ∣ ) K − ( ∣ b ∣ + ∣ c ∣ ) ( a d ) − (K_{-|b|}a^-)(K_{-|c|}d^-)=v^{-(|a|,|c|)}K_{-(|b|+|c|)}(ad)^- ( K − ∣ b ∣ a − ) ( K − ∣ c ∣ d − ) = v − ( ∣ a ∣ , ∣ c ∣ ) K − ( ∣ b ∣ + ∣ c ∣ ) ( a d ) − shows that it intertwines the r − 1 r_{-1} r − 1 τ \tau τ r 1 r_1 r 1 r − 1 r_{-1} r − 1 v − ( α , γ ) v^{-(\alpha,\gamma)} v − ( α , γ ) v ( α , γ ) v^{(\alpha,\gamma)} v ( α , γ ) K − α K_{-\alpha} K − α a − υ η m a^-\upsilon_{\eta_m} a − υ η m η m \eta_m η m x θ m y x\theta_m y x θ m y y y y θ m \theta_m θ m x − x^- x −
The Demazure quotients
Define ev w ( P ) = F P L w \operatorname{ev}_w(P)=F_P L_w ev w ( P ) = F P L w P ∈ P I P\in\mathcal{P}_I P ∈ P I F f L w F_fL_w F f L w f f f I w I_w I w I ~ w \widetilde{I}_w I w P I ~ \mathcal{P}_{\widetilde{I}} P I id Z f \operatorname{id}_Z f id Z f f ∈ I w f\in I_w f ∈ I w Z Z Z P I ~ \mathcal{P}_{\widetilde{I}} P I
A w = Kar ( P I ~ / I ~ w ) . \mathcal{A}_w=\operatorname{Kar}(\mathcal{P}_{\widetilde{I}}/\widetilde{I}_w). A w = Kar ( P I / I w ) . Here Kar \operatorname{Kar} Kar β \beta β μ w − β \mu_w-\beta μ w − β C w \mathcal{C}_w C w P b P_b P b b ∈ B ‾ ∩ F m b\in\overline{\mathbf{B}}\cap F_m b ∈ B ∩ F m
Proposition 3.1. The normalized indecomposable objects of C w \mathcal{C}_w C w ω V w ( λ 1 ) ⊗ Λ λ 2 {}^{\omega}V_w(\lambda_1)\otimes\Lambda_{\lambda_2} ω V w ( λ 1 ) ⊗ Λ λ 2 Φ w \Phi_w Φ w
Proof. The kernel ker a w \ker a_w ker a w f \mathfrak{f} f M μ w ≅ f M_{\mu_w}\cong\mathfrak{f} M μ w ≅ f ( ker a w ) ⊗ Λ λ 2 (\ker a_w)\otimes\Lambda_{\lambda_2} ( ker a w ) ⊗ Λ λ 2 y ∈ ker a w y\in\ker a_w y ∈ ker a w (1) gives θ m y ∈ f ~ ker a w \theta_m y\in\widetilde{\mathfrak{f}}\ker a_w θ m y ∈ f ker a w (2) , the first term stays in this left ideal and the second does as well because θ i y ∈ ker a w \theta_i y\in\ker a_w θ i y ∈ ker a w η λ 2 \eta_{\lambda_2} η λ 2 ker Φ w ⊆ f ~ ker a w \ker\Phi_w\subseteq\widetilde{\mathfrak{f}}\ker a_w ker Φ w ⊆ f ker a w b b b ker Φ w \ker\Phi_w ker Φ w z c zc z c z ∈ B ‾ z\in\overline{\mathbf{B}} z ∈ B c ∈ B ∩ ker a w c\in\mathbf{B}\cap\ker a_w c ∈ B ∩ ker a w P b P_b P b P z P c P_zP_c P z P c F P c L w = 0 F_{P_c}L_w=0 F P c L w = 0 P z P c P_zP_c P z P c P b P_b P b
For the converse, for each pair of output contents ( α , γ ) (\alpha,\gamma) ( α , γ ) α \alpha α
D w ; α , γ = ( ev w ⊠ ev η m ) Res α , γ . D_{w;\alpha,\gamma}=(\operatorname{ev}_w\boxtimes\operatorname{ev}_{\eta_m})\operatorname{Res}_{\alpha,\gamma}. D w ; α , γ = ( ev w ⊠ ev η m ) Res α , γ . where ev η m ( Q ) = F Q L η m \operatorname{ev}_{\eta_m}(Q)=F_QL_{\eta_m} ev η m ( Q ) = F Q L η m D w ; α , γ D_{w;\alpha,\gamma} D w ; α , γ α + γ \alpha+\gamma α + γ μ w − α \mu_w-\alpha μ w − α η m − γ \eta_m-\gamma η m − γ ⊠ \boxtimes ⊠ Lemma 2.1 , with zero ideal, shows that these labels and their shifts account for all summands. The split group therefore has the basis of ordinary tensor labels. Write D w D_w D w
Take f : P 1 → P 2 f:P_1 \to P_2 f : P 1 → P 2 I w I_w I w ρ \rho ρ Res α , γ ( Z P j ) \operatorname{Res}_{\alpha,\gamma}(ZP_j) Res α , γ ( Z P j ) ∣ Z ∣ , ρ |Z|,\rho ∣ Z ∣ , ρ α , γ \alpha,\gamma α , γ [18 Proposition 2.4.4] . If any of ρ \rho ρ L η m + L_{\eta_m}^{+} L η m +
( α − ρ γ ρ 0 ) \begin{pmatrix}
\alpha-\rho& \gamma\\
\rho& 0
\end{pmatrix} ( α − ρ ρ γ 0 ) This matrix occurs only if α − ρ \alpha-\rho α − ρ P j P_j P j F f L w = 0 F_fL_w=0 F f L w = 0 D w D_w D w I ~ w \widetilde I_w I w
The class of D w ; α , γ ( P b ) D_{w;\alpha,\gamma}(P_b) D w ; α , γ ( P b ) ( r − 1 ( b ) ) α , γ (r_{-1}(b))_{\alpha,\gamma} ( r − 1 ( b ) ) α , γ a ⊗ b ′ ↦ a − ξ μ w ⊗ ( b ′ − u η m a\otimes b' \mapsto a-\xi_{\mu_w}\otimes(b' - u_{\eta_m} a ⊗ b ′ ↦ a − ξ μ w ⊗ ( b ′ − u η m r − 1 = τ r 1 r_{-1}=\tau r_1 r − 1 = τ r 1
[ D w ; α , γ ( P b ) ] = v − ( α , γ ) ( ( 1 ⊗ ι ) Φ w ( b ) ) α , γ [D_{w;\alpha,\gamma}(P_b)] = v^{-(\alpha,\gamma)}\bigl((1\otimes\iota)\Phi_w(b)\bigr)_{\alpha,\gamma} [ D w ; α , γ ( P b )] = v − ( α , γ ) ( ( 1 ⊗ ι ) Φ w ( b ) ) α , γ The two factor weights distinguish the formal contents because the enlarged simple roots are independent in P P P Φ w ( b ) ≠ 0 \Phi_w(b)\ne0 Φ w ( b ) = 0 D w ( P b ) ≠ 0 D_w(P_b)\ne0 D w ( P b ) = 0 P b P_b P b Lemma 2.1 now identifies the indecomposable objects. □
Transition functors
The restriction of I ~ w \widetilde I_w I w P I \mathcal P_I P I I w I_w I w I w I_w I w A w \mathcal A_w A w A w I \mathcal A_w^I A w I L ( − λ 1 ) \mathcal L(-\lambda_1) L ( − λ 1 )
For w ≤ v 0 w\leq v_0 w ≤ v 0 W W W ξ μ w \xi_{\mu_w} ξ μ w W V v 0 ( λ 1 ) {}^{W}V_{v_0}(\lambda_1) W V v 0 ( λ 1 ) a v 0 a_{v_0} a v 0 d v 0 w ∈ B d_{v_0w}\in B d v 0 w ∈ B ξ μ v 0 \xi_{\mu_{v_0}} ξ μ v 0 ξ μ w \xi_{\mu_w} ξ μ w μ v 0 − μ w \mu_{v_0}-\mu_w μ v 0 − μ w
τ v 0 w : F P d v 0 w L v 0 → ∼ L w . \tau_{v_0w}:F_{P_{d_{v_0w}}}L_{v_0}\xrightarrow{\sim}L_w. τ v 0 w : F P d v 0 w L v 0 ∼ L w . Choose one such isomorphism, taking the identity for v 0 = w v_0=w v 0 = w f ∈ I w f\in I_w f ∈ I w f id P d v 0 w ∈ I v 0 f\,\operatorname{id}_{P_{d_{v_0w}}}\in I_{v_0} f id P d v 0 w ∈ I v 0 F f F_f F f τ v 0 w \tau_{v_0w} τ v 0 w id Z f \operatorname{id}_{Z} f id Z f I ~ w \widetilde{I}_{w} I w id Z ( f id P d v 0 w ) ∈ I ~ v 0 \operatorname{id}_{Z}(f\operatorname{id}_{P_{d_{v_{0}w}}})\in\widetilde{I}_{v_{0}} id Z ( f id P d v 0 w ) ∈ I v 0 T v 0 w : A w → A v 0 T_{v_{0}w}:\mathcal{A}_{w}\to\mathcal{A}_{v_{0}} T v 0 w : A w → A v 0 ∣ P d v 0 w ∣ = μ v 0 − μ w |P_{d_{v_{0}w}}|=\mu_{v_{0}}-\mu_{w} ∣ P d v 0 w ∣ = μ v 0 − μ w F m F_{m} F m
Φ v 0 ( z d v 0 w ) = inc ( Φ w ( z ) ) , \Phi_{v_{0}}(zd_{v_{0}w})=\operatorname{inc}(\Phi_{w}(z)), Φ v 0 ( z d v 0 w ) = inc ( Φ w ( z )) , where inc is the tensor of Demazure inclusion with the identity. It suffices to check this on z = x θ m y z=x\theta_{m}y z = x θ m y d v 0 w d_{v_{0}w} d v 0 w z d v 0 w zd_{v_{0}w} z d v 0 w F m F_{m} F m F m F_{m} F m [5 Proposition 1.6] . For a surviving label this image is a single normalized label, so equality in the split group gives the corresponding degree-zero isomorphism in C v 0 \mathcal{C}_{v_{0}} C v 0
For w ≤ v 0 ≤ u w\leq v_{0}\leq u w ≤ v 0 ≤ u P d v 0 w P d u v 0 P_{d_{v_{0}w}}P_{d_{uv_{0}}} P d v 0 w P d u v 0 P d u w P_{d_{uw}} P d u w A u \mathcal{A}_{u} A u L u L_{u} L u L w L_{w} L w τ \tau τ L u L_{u} L u a u a_{u} a u A u \mathcal{A}_{u} A u
c u , v 0 , w : P d v 0 w P d u v 0 ⟶ P d u w c_{u,v_{0},w}:P_{d_{v_{0}w}}P_{d_{uv_{0}}}\longrightarrow P_{d_{uw}} c u , v 0 , w : P d v 0 w P d u v 0 ⟶ P d u w such that
τ u w ∘ ev u ( c u , v 0 , w ) = τ v 0 w ∘ F P d v 0 w ( τ u v 0 ) . \tau_{uw}\circ\operatorname{ev}_{u}(c_{u,v_{0},w})=\tau_{v_{0}w}\circ F_{P_{d_{v_{0}w}}}(\tau_{uv_{0}}). τ u w ∘ ev u ( c u , v 0 , w ) = τ v 0 w ∘ F P d v 0 w ( τ u v 0 ) . Indeed the degree-zero isomorphisms of the evaluated objects form a k × k^{\times} k × L w L_{w} L w k k k A w \mathcal{A}_{w} A w c u , v 0 , w c_{u,v_{0},w} c u , v 0 , w T u v 0 T v 0 w T_{uv_{0}}T_{v_{0}w} T u v 0 T v 0 w T u w T_{uw} T u w τ w w \tau_{ww} τ w w
For w ≤ v 0 ≤ u ≤ t w\leq v_{0}\leq u\leq t w ≤ v 0 ≤ u ≤ t P d v 0 w P d u v 0 P d t u P_{d_{v_{0}w}}P_{d_{uv_{0}}}P_{d_{tu}} P d v 0 w P d u v 0 P d t u P d t w P_{d_{tw}} P d tw
c t , u , w ∘ ( c u , v 0 , w id P d t u ) , c_{t,u,w}\circ(c_{u,v_{0},w}\operatorname{id}_{P_{d_{tu}}}), c t , u , w ∘ ( c u , v 0 , w id P d t u ) , c t , v 0 , w ∘ ( id P d v 0 w c t , u , v 0 ) . c_{t,v_{0},w}\circ(\operatorname{id}_{P_{d_{v_{0}w}}}c_{t,u,v_{0}}). c t , v 0 , w ∘ ( id P d v 0 w c t , u , v 0 ) . After evaluating at L t L_{t} L t τ t w \tau_{tw} τ tw
τ v 0 w ∘ F P d v 0 w ( τ u w ) ∘ F P d v 0 w F P d u v 0 ( τ t u ) . \tau_{v_{0}w}\circ F_{P_{d_{v_{0}w}}}(\tau_{uw})\circ F_{P_{d_{v_{0}w}}}F_{P_{d_{uv_{0}}}}(\tau_{tu}). τ v 0 w ∘ F P d v 0 w ( τ u w ) ∘ F P d v 0 w F P d u v 0 ( τ t u ) . For the first comparison, use naturality of the map on the first two factors with respect to τ t u \tau_{tu} τ t u c u , v 0 , w c_{u,v_{0},w} c u , v 0 , w c t , u , v 0 c_{t,u,v_{0}} c t , u , v 0 F P d v 0 w F_{P_{d_{v_{0}w}}} F P d v 0 w
Single-root actions on the quotients
Fix a raising color i ∈ I i \in I i ∈ I X X X λ \lambda λ h = ⟨ h i , λ ⟩ h = \langle h_i,\lambda\rangle h = ⟨ h i , λ ⟩ F i F_i F i λ \lambda λ E i ⟨ 1 + h ⟩ E_i\langle1+h\rangle E i ⟨ 1 + h ⟩
η X : X ⟶ E i F i X , ϵ X : F i E i X ⟶ X \eta_X : X \longrightarrow E_iF_iX,\qquad\epsilon_X : F_iE_iX \longrightarrow X η X : X ⟶ E i F i X , ϵ X : F i E i X ⟶ X of degrees 1 − h 1-h 1 − h 1 + h 1+h 1 + h F i X F_iX F i X ⟨ h − 1 ⟩ \langle h-1\rangle ⟨ h − 1 ⟩
Write F = F i F=F_i F = F i E = E i E=E_i E = E i T = F j T=F_j T = F j j j j τ : F T ⟶ T F \tau: FT \longrightarrow TF τ : F T ⟶ T F
σ X = ( E T ϵ X ) ∘ ( E τ E X ) ∘ η T E X : T E X ⟶ E T X . (3) \sigma_X=(ET\epsilon_X)\circ(E\tau E_X)\circ\eta_{TEX}: TEX\longrightarrow ETX. \tag*{(3)} σ X = ( E T ϵ X ) ∘ ( E τ E X ) ∘ η T E X : T E X ⟶ E T X . (3) At T E X TEX T E X h + 2 − a i j h+2-a_{ij} h + 2 − a ij (3) are − 1 − h + a i j -1-h+a_{ij} − 1 − h + a ij − a i j -a_{ij} − a ij 1 + h 1+h 1 + h σ X \sigma_X σ X i ≠ j i\ne j i = j n ≥ 0 n\ge0 n ≥ 0
[ n ] X = ⨁ r = 0 n − 1 X ⟨ n − 1 − 2 r ⟩ , [ 0 ] X = 0. [n]X=\bigoplus_{r=0}^{n-1}X\langle n-1-2r\rangle,\qquad[0]X=0. [ n ] X = r = 0 ⨁ n − 1 X ⟨ n − 1 − 2 r ⟩ , [ 0 ] X = 0. For i = j i=j i = j
{ M X : F E X ⊕ [ h ] X ⟶ E F X , h ≥ 0 , N X : F E X ⟶ E F X ⊕ [ − h ] X , h < 0. (4) \begin{cases}
M_X:FEX\oplus[h]X\longrightarrow EFX,&h\ge0,\\
N_X:FEX\longrightarrow EFX\oplus[-h]X,&h<0. \tag*{(4)}
\end{cases} { M X : F E X ⊕ [ h ] X ⟶ E F X , N X : F E X ⟶ E F X ⊕ [ − h ] X , h ≥ 0 , h < 0. (4) Their F E X ⟶ E F X FEX\longrightarrow EFX F E X ⟶ E F X σ X \sigma_X σ X x : F ⟶ F x:F\longrightarrow F x : F ⟶ F M X M_X M X 0 ≤ r < h 0\le r<h 0 ≤ r < h E ( x X h − 1 − r ) ∘ η X E(x_X^{h-1-r})\circ\eta_X E ( x X h − 1 − r ) ∘ η X h − 1 − 2 r h-1-2r h − 1 − 2 r N X N_X N X 0 ≤ r < − h 0\le r<-h 0 ≤ r < − h ϵ X ∘ x E X r \epsilon_X\circ x_{E X}^{r} ϵ X ∘ x E X r 1 + h + 2 r 1+h+2r 1 + h + 2 r [18 Theorem 2.6.7(2)] .
Lemma 4.1. Let two graded additive categories have negative quiver Hecke actions with common weights and parameters, and the shifted adjunction and isomorphisms (3) –(4) for a fixed color i i i F j ′ , E ′ F'_j,E' F j ′ , E ′ η ′ , ϵ ′ \eta',\epsilon' η ′ , ϵ ′ H H H k k k a j : H F j ⟶ F j ′ H a_j:HF_j\longrightarrow F'_jH a j : H F j ⟶ F j ′ H V V V E V = E ′ H V = 0 EV=E'HV=0 E V = E ′ H V = 0 H E ⟶ E ′ H HE\longrightarrow E'H H E ⟶ E ′ H a i − 1 a_i^{-1} a i − 1 H H H
Proof. Write a = a i a=a_i a = a i F ′ = F i ′ F'=F'_i F ′ = F i ′ X X X
ρ X = E ′ H ( ϵ X ) ∘ E ′ ( a E X − 1 ) ∘ η H E X ′ . \rho_X=E'H(\epsilon_X)\circ E'(a^{-1}_{EX})\circ\eta'_{HEX}. ρ X = E ′ H ( ϵ X ) ∘ E ′ ( a E X − 1 ) ∘ η H E X ′ . Its two adjunction maps have degrees 1 + h 1+h 1 + h − 1 − h -1-h − 1 − h E X EX E X λ + α i \lambda+\alpha_i λ + α i
ε H X ∘ F ′ ( ρ X ) ∘ a E X = H ( ε X ) , E ′ ( a X ) ∘ ρ F X ∘ H ( η X ) = η H X ′ . (5) \begin{aligned}
\varepsilon_{HX}\circ F'(\rho_X)\circ a_{EX}&=H(\varepsilon_X),\\
E'(a_X)\circ\rho_{FX}\circ H(\eta_X)&=\eta'_{HX}. \tag*{(5)}
\end{aligned} ε H X ∘ F ′ ( ρ X ) ∘ a E X E ′ ( a X ) ∘ ρ F X ∘ H ( η X ) = H ( ε X ) , = η H X ′ . (5) The first identity is the defining transpose of ρ X \rho_X ρ X ρ F X \rho_{FX} ρ F X η ′ \eta' η ′ H ( η X ) H(\eta_X) H ( η X ) a a a
H ( ε F X ) ∘ a F ′ X − 1 ∘ F ′ H ( η X ) = H ( ε F X ∘ F ( η X ) ) ∘ a X − 1 = a X − 1 . H(\varepsilon_{FX})\circ a^{-1}_{F'X}\circ F'H(\eta_X)=H(\varepsilon_{FX}\circ F(\eta_X))\circ a^{-1}_X=a^{-1}_X. H ( ε F X ) ∘ a F ′ X − 1 ∘ F ′ H ( η X ) = H ( ε F X ∘ F ( η X )) ∘ a X − 1 = a X − 1 . Applying E ′ E' E ′ E ′ ( a X ) E'(a_X) E ′ ( a X )
Put T ′ = F j ′ T'=F'_j T ′ = F j ′ b = a j b=a_j b = a j σ ′ \sigma' σ ′
E ′ ( b X ) ∘ ρ T X ∘ H ( σ X ) = σ H X ′ ∘ T ′ ( ρ X ) ∘ b E X . (6) E'(b_X)\circ\rho_{TX}\circ H(\sigma_X)=\sigma'_{HX}\circ T'(\rho_X)\circ b_{EX}. \tag*{(6)} E ′ ( b X ) ∘ ρ T X ∘ H ( σ X ) = σ H X ′ ∘ T ′ ( ρ X ) ∘ b E X . (6) To see this, transpose the left-hand side by F ′ ⊣ E ′ F'\dashv E' F ′ ⊣ E ′ (5) , and naturality of a a a
b X ∘ H ( ε T X ∘ F ( σ X ) ) ∘ a T E X − 1 . b_X\circ H(\varepsilon_{TX}\circ F(\sigma_X))\circ a^{-1}_{TEX}. b X ∘ H ( ε T X ∘ F ( σ X )) ∘ a T E X − 1 . Since ε T X ∘ F ( σ X ) = T ( ε X ) ∘ τ E X \varepsilon_{TX}\circ F(\sigma_X)=T(\varepsilon_X)\circ\tau_{EX} ε T X ∘ F ( σ X ) = T ( ε X ) ∘ τ E X b b b
T ′ H ( ε X ) ∘ b F E X ∘ H ( τ E X ) ∘ a T E X − 1 . T'H(\varepsilon_X)\circ b_{FEX}\circ H(\tau_{EX})\circ a^{-1}_{TEX}. T ′ H ( ε X ) ∘ b F E X ∘ H ( τ E X ) ∘ a T E X − 1 . Negative crossing compatibility says
T ′ ( a E X ) ∘ b F E X ∘ H ( τ E X ) = τ H E X ′ ∘ F ′ ( b E X ) ∘ a T E X . T'(a_{EX})\circ b_{FEX}\circ H(\tau_{EX})=\tau'_{HEX}\circ F'(b_{EX})\circ a_{TEX}. T ′ ( a E X ) ∘ b F E X ∘ H ( τ E X ) = τ H E X ′ ∘ F ′ ( b E X ) ∘ a T E X . Substituting it and then (5) gives
T ′ ( ε H X ′ ) ∘ T ′ F ′ ( ρ X ) ∘ τ H E X ′ ∘ F ′ ( b E X ) = T ′ ( ε H X ′ ) ∘ τ E ′ H X ′ ∘ F ′ T ′ ( ρ X ) ∘ F ′ ( b E X ) . T'(\varepsilon'_{HX})\circ T'F'(\rho_X)\circ\tau'_{HEX}\circ F'(b_{EX})=T'(\varepsilon'_{HX})\circ\tau'_{E'HX}\circ F'T'(\rho_X)\circ F'(b_{EX}). T ′ ( ε H X ′ ) ∘ T ′ F ′ ( ρ X ) ∘ τ H E X ′ ∘ F ′ ( b E X ) = T ′ ( ε H X ′ ) ∘ τ E ′ H X ′ ∘ F ′ T ′ ( ρ X ) ∘ F ′ ( b E X ) . The last equality is naturality of τ ′ \tau' τ ′ (6) .
For j ≠ i j\ne i j = i (6) gives invertibility of ρ F j X \rho_{F_jX} ρ F j X ρ X \rho_X ρ X j = i j=i j = i
χ X = E ′ ( a X ) ∘ ρ F X , ψ X = F ′ ( ρ X ) ∘ a E X . \chi_X=E'(a_X)\circ\rho_{FX},\qquad\psi_X=F'(\rho_X)\circ a_{EX}. χ X = E ′ ( a X ) ∘ ρ F X , ψ X = F ′ ( ρ X ) ∘ a E X . Equation (6) compares the F E X → E ′ F X FEX\to E'FX F E X → E ′ F X ρ \rho ρ x X r x^r_X x X r a X ∘ H ( x X ) = x H X ′ ∘ a X a_X\circ H(x_X)=x'_{HX}\circ a_X a X ∘ H ( x X ) = x H X ′ ∘ a X
χ X ∘ H ( E ( x X r ) ∘ η X ) = E ′ ( ( x H X ′ ) r ) ∘ E ′ ( a X ) ∘ ρ F X ∘ H ( η X ) = E ′ ( ( x H X ′ ) r ) ∘ η H X ′ . \begin{aligned}
\chi_X\circ H(E(x^r_X)\circ\eta_X)&=E'((x'_{HX})^r)\circ E'(a_X)\circ\rho_{FX}\circ H(\eta_X)\\
&=E'((x'_{HX})^r)\circ\eta'_{HX}.
\end{aligned} χ X ∘ H ( E ( x X r ) ∘ η X ) = E ′ (( x H X ′ ) r ) ∘ E ′ ( a X ) ∘ ρ F X ∘ H ( η X ) = E ′ (( x H X ′ ) r ) ∘ η H X ′ . For a counit row, the same dot relation at E X EX E X x ′ x' x ′ ρ X \rho_X ρ X (5) give
ε H X ′ ∘ ( x E ′ H X ′ ) r ∘ ψ X = ε H X ′ ∘ ψ X ∘ H ( x E X r ) = H ( ε X ∘ x E X r ) . \varepsilon'_{HX}\circ(x'_{E'HX})^r\circ\psi_X=\varepsilon'_{HX}\circ\psi_X\circ H(x^r_{EX})=H(\varepsilon_X\circ x^r_{EX}). ε H X ′ ∘ ( x E ′ H X ′ ) r ∘ ψ X = ε H X ′ ∘ ψ X ∘ H ( x E X r ) = H ( ε X ∘ x E X r ) . Consequently
χ X ∘ H ( M X ) = M H X ′ ∘ ( ψ X ⊕ 1 ) ( h ≥ 0 ) , \chi_X\circ H(M_X)=M'_{HX}\circ(\psi_X\oplus1)\qquad(h\ge0), χ X ∘ H ( M X ) = M H X ′ ∘ ( ψ X ⊕ 1 ) ( h ≥ 0 ) , ( χ X ⊕ 1 ) ∘ H ( N X ) = N H X ′ ∘ ψ X ( h < 0 ) . (\chi_X\oplus1)\circ H(N_X)=N'_{HX}\circ\psi_X\qquad(h<0). ( χ X ⊕ 1 ) ∘ H ( N X ) = N H X ′ ∘ ψ X ( h < 0 ) . These identities imply invertibility of χ X \chi_X χ X ρ X \rho_X ρ X χ X ⊕ 1 \chi_X \oplus1 χ X ⊕ 1 χ X \chi_X χ X ρ F X \rho_F X ρ F X
At V V V ρ V \rho_V ρ V
If H , K H,K H , K a H , a K a^H,a^K a H , a K ξ : H → K \xi:H\to K ξ : H → K ( E ′ ξ X ) ∘ ρ X H = ρ X K ∘ ξ E X (E'\xi_X)\circ\rho_X^H=\rho_X^K\circ\xi_{E X} ( E ′ ξ X ) ∘ ρ X H = ρ X K ∘ ξ E X ξ X ∘ H ( ϵ X ) ∘ ( a E X H ) − 1 \xi_X\circ H(\epsilon_X)\circ(a^H_{E X})^{-1} ξ X ∘ H ( ϵ X ) ∘ ( a E X H ) − 1 ξ \xi ξ ϵ X \epsilon_X ϵ X a E X K ∘ ξ F E X = F ′ ( ξ E X ) ∘ a E X H a^K_{E X}\circ\xi_{F E X}=F'(\xi_{E X})\circ a^H_{E X} a E X K ∘ ξ F E X = F ′ ( ξ E X ) ∘ a E X H K ( ϵ X ) ∘ ( a E X K ) − 1 ∘ F ′ ( ξ E X ) K(\epsilon_X)\circ(a^K_{E X})^{-1}\circ F'(\xi_{E X}) K ( ϵ X ) ∘ ( a E X K ) − 1 ∘ F ′ ( ξ E X ) (5) twice. They therefore agree with the mate of the composite negative comparison. □
Fix w ∈ W w\in W w ∈ W i ∈ I i\in I i ∈ I s i w < w s_iw<w s i w < w n = ⟨ h i , μ w ⟩ n=\langle h_i,\mu_w\rangle n = ⟨ h i , μ w ⟩ n ≥ 0 n\ge0 n ≥ 0 E i L w = 0 E_iL_w=0 E i L w = 0 HOM ( L w , L w ) \operatorname{HOM}(L_w,L_w) HOM ( L w , L w ) k k k (4) , E i F i L w ≃ [ n ] L w E_iF_iL_w\simeq[n]L_w E i F i L w ≃ [ n ] L w
HOM ( F i L w , F i L w ) ≅ HOM ( L w , [ n ] L w ⟨ n − 1 ⟩ ) . \operatorname{HOM}(F_iL_w,F_iL_w)\cong\operatorname{HOM}(L_w,[n]L_w\langle n-1\rangle). HOM ( F i L w , F i L w ) ≅ HOM ( L w , [ n ] L w ⟨ n − 1 ⟩) . For n = 0 n=0 n = 0 F i L w = 0 F_iL_w=0 F i L w = 0 2 n − 2 − 2 r 2n-2-2r 2 n − 2 − 2 r 0 ≤ r < n 0\le r<n 0 ≤ r < n HOM ( L w , L w ) \operatorname{HOM}(L_w,L_w) HOM ( L w , L w ) x i x_i x i F i L w F_iL_w F i L w n n n
For either the pure or the thickened quiver Hecke algebra, write A β A_\beta A β
A β = R ( β ) / ( R ( β ) x N n e ( β − α i , i ) R ( β ) ) ( N = ht β ≥ 1 ) . (7) A_\beta=R(\beta)\big/\big(R(\beta)x_N^n e(\beta-\alpha_i,i)R(\beta)\big)\qquad(N=\operatorname{ht}\beta\ge1). \tag*{(7)} A β = R ( β ) / ( R ( β ) x N n e ( β − α i , i ) R ( β ) ) ( N = ht β ≥ 1 ) . (7) Here ht β \operatorname{ht}\beta ht β x N x_N x N e ( β − α i , i ) e(\beta-\alpha_i,i) e ( β − α i , i ) β \beta β i i i A 0 = k A_0=k A 0 = k F j F_j F j j j j X X X β \beta β μ w − β \mu_w-\beta μ w − β
E i X = e ( i , β − α i ) X ⟨ − 1 − ⟨ h i , μ w − β ⟩ ⟩ . (8) E_iX=e(i,\beta-\alpha_i)X\langle-1-\langle h_i,\mu_w-\beta\rangle\rangle. \tag*{(8)} E i X = e ( i , β − α i ) X ⟨ − 1 − ⟨ h i , μ w − β ⟩⟩ . (8) The idempotent selects words beginning in i i i A β − α i A_{\beta-\alpha_i} A β − α i
For this choice of parameters, [18 Definition 2.8.2 and Theorems 2.8.16–2.8.18] give the indicated functors on finitely generated graded projectives, the negative diagrams by right multiplication, the induction–restriction adjunction with the shift in (8) , and the invertibility in (3)–(4). The cyclotomic exponent is prescribed only at i i i j ≠ i j\ne i j = i j j j [18 Proposition 2.8.14 and Theorems 2.8.15–2.8.16] . The displayed maps of Theorem 2.8.18 are the counit F E → 1 FE \to1 F E → 1 1 → E F 1 \to EF 1 → E F
Lemma 4.2. The adjunction and (3) –(4) descend to A w A_w A w E i E_i E i C w C_w C w s i w < w s_iw < w s i w < w
Proof. On pure words, evaluation at L w L_w L w (7) . The generating relation is the n n n F i L w F_iL_w F i L w A β A_\beta A β k k k E i E_i E i L w L_w L w E i E_i E i Lemma 4.1 gives the E i E_i E i
For a pure map f ∈ I w f \in I_w f ∈ I w E i E_i E i A β A_\beta A β R ( β ) R(\beta) R ( β ) A β A_\beta A β A β ⊗ R ( β ) Q A_\beta\otimes_{R(\beta)} Q A β ⊗ R ( β ) Q Q Q Q Lemma 2.1 gives a lift for each of its objects. Consequently E i f E_if E i f E i E_i E i I w I_w I w
In the thickened A β A_\beta A β J J J I ~ w \widetilde I_w I w A β A_\beta A β I ~ w \widetilde I_w I w x i n x_i^n x i n E i E_i E i J J J J J J f ∈ J f \in J f ∈ J E i f ∈ J E_if \in J E i f ∈ J F j F_j F j j ≠ i j \ne i j = i E i F j f E_iF_jf E i F j f F j E i f F_jE_if F j E i f j = i j=i j = i h ≥ 0 h \ge0 h ≥ 0 E i F i f E_iF_if E i F i f M M M F i E i f ⊕ [ h ] f F_iE_if \oplus[h]f F i E i f ⊕ [ h ] f h < 0 h < 0 h < 0 N N N E i F i f E_iF_if E i F i f F i E i f F_iE_if F i E i f J J J F j F_j F j J J J J J J E i E_i E i
Write [ k ] [k] [ k ] μ w \mu_w μ w j 1 , … , j N j_1,\ldots,j_N j 1 , … , j N (3) gives commutation on classes with the letters j r ≠ i j_r \ne i j r = i (4) give E i F i − F i E i = [ ⟨ h i , λ ⟩ ] E_iF_i-F_iE_i = [\langle h_i,\lambda\rangle] E i F i − F i E i = [⟨ h i , λ ⟩] λ \lambda λ E i E_i E i [ k ] [k] [ k ]
E i ( F j 1 ⋯ F j N [ k ] ) = ∑ 1 ≤ r ≤ N j r = i [ ⟨ h i , μ w − ∑ s > r α j s ⟩ ] F j 1 ⋯ F j r ^ ⋯ F j N [ k ] . E_i(F_{j_1}\cdots F_{j_N}[k])=\sum_{\substack{1\le r\le N\\ j_r=i}}\left[\left\langle h_i,\mu_w-\sum_{s>r}\alpha_{j_s}\right\rangle\right]F_{j_1}\cdots\widehat{F_{j_r}}\cdots F_{j_N}[k]. E i ( F j 1 ⋯ F j N [ k ]) = 1 ≤ r ≤ N j r = i ∑ [ ⟨ h i , μ w − s > r ∑ α j s ⟩ ] F j 1 ⋯ F j r ⋯ F j N [ k ] . A hat omits that factor. This is also the deletion formula on the negative half of the enlarged Verma module at μ w \mu_w μ w x θ m y x\theta_m y x θ m y x x x y y y F m F_m F m y ∈ f y\in\mathfrak{f} y ∈ f θ m y \theta_m y θ m y y y y y − ξ μ w y^{-}\xi_{\mu_w} y − ξ μ w Φ w \Phi_w Φ w E i E_i E i Φ w ( θ m y ) = y − ξ μ w ⊗ η λ 2 \Phi_w(\theta_m y)=y^{-}\xi_{\mu_w}\otimes\eta_{\lambda_2} Φ w ( θ m y ) = y − ξ μ w ⊗ η λ 2 E i η λ 2 = 0 E_i\eta_{\lambda_2}=0 E i η λ 2 = 0 μ w − γ m − ∣ y ∣ \mu_w-\gamma_m-|y| μ w − γ m − ∣ y ∣ Φ w \Phi_w Φ w F j F_j F j E i E_i E i E i E_i E i F m F_m F m F m F_m F m C w C_w C w F j F_j F j j ∈ I j\in I j ∈ I F m F_m F m
Suppose w ≤ v 0 w\le v_0 w ≤ v 0 s i w < w s_iw<w s i w < w s i v 0 < v 0 s_iv_0<v_0 s i v 0 < v 0 T v 0 w T_{v_0w} T v 0 w P d v 0 w P_{dv_0w} P d v 0 w L w L_w L w v 0 v_0 v 0 E i E_i E i E i L w = 0 E_iL_w=0 E i L w = 0 E i P d v 0 w E_iP_{dv_0w} E i P d v 0 w E i P d v 0 w = 0 E_iP_{dv_0w}=0 E i P d v 0 w = 0 Lemma 4.1 now holds, and the transition preserves E i E_i E i μ w − β \mu_w-\beta μ w − β
The action on the Bruhat colimit
The Bruhat order on W W W w ∗ v 0 w*v_0 w ∗ v 0 w w w v 0 v_0 v 0 w w w v 0 v_0 v 0 i ∈ I i\in I i ∈ I W i − = { w ∈ W : s i w < w } W_i^-=\{w\in W:s_iw<w\} W i − = { w ∈ W : s i w < w } u u u s i u < u s_iu<u s i u < u s i u s_iu s i u s i u > u s_iu>u s i u > u
Form A ∞ \mathcal{A}_\infty A ∞ ( w , P ) (w,P) ( w , P ) P ∈ A w P\in\mathcal{A}_w P ∈ A w
Hom A ∞ ( ( w , P ) , ( v 0 , Q ) ) = l i m → u ≥ w , v 0 Hom A u ( T u w P , T u v 0 Q ) . \operatorname{Hom}_{\mathcal{A}_\infty}((w,P),(v_0,Q))=\underset{u\ge w,v_0}{\varinjlim}\operatorname{Hom}_{\mathcal{A}_u}(T_{uw}P,T_{uv_0}Q). Hom A ∞ (( w , P ) , ( v 0 , Q )) = u ≥ w , v 0 lim Hom A u ( T u w P , T u v 0 Q ) . A map f : T u w P → T u v 0 Q f:T_{uw}P\to T_{uv_0}Q f : T u w P → T u v 0 Q t ≥ u t\ge u t ≥ u
( id Q c t , u , v 0 ) ∘ T t u ( f ) ∘ ( id P c t , u , w ) − 1 . (\operatorname{id}_Q c_{t,u,v_0})\circ T_{tu}(f)\circ(\operatorname{id}_P c_{t,u,w})^{-1}. ( id Q c t , u , v 0 ) ∘ T t u ( f ) ∘ ( id P c t , u , w ) − 1 . The associativity of the comparisons c c c A ∞ I \mathcal{A}_{\infty}^{I} A ∞ I C ∞ \mathcal{C}_{\infty} C ∞ C w C_w C w ( w , P ) (w,P) ( w , P ) P P P A w A_w A w
The negative diagrams act on the colimit because the transitions intertwine all of them. To define E i E_i E i W i − W_i^{-} W i − E i E_i E i W i − W_i^{-} W i − E i E_i E i Lemma 4.2 identify the resulting objects and transport the morphism back to the chosen representatives. Their composition compatibility makes the result independent of the stage. It also ensures preservation of identities and composition by using a single common stage for any finite diagram. Units, counits and mixed and same-color inverses pass to the colimit by the same argument. In particular the mixed maps with the primed lowering colors remain invertible.
For the original colors use the dual presentation of the categorified quantum group U \mathcal{U} U F i ⊣ E i F_i \dashv E_i F i ⊣ E i [18 Theorem 2.6.7(2)] , compared with the Rouquier–Brundan and cyclic presentations in [3 , 2 ] . We use the native adjunctions of that presentation throughout. In particular the upward dot and crossing operators mean the right mates of the downward dot and crossing, with the adjunction shifts removed. Each inverse requirement involves one raising color i i i W i − W_i^{-} W i − Lemma 4.2 . Their coherence gives a functor A ∞ I → L ( − λ 1 ) \mathcal{A}_{\infty}^{I} \to\mathcal{L}(-\lambda_1) A ∞ I → L ( − λ 1 ) I w I_w I w
The subcategory C ∞ \mathcal{C}_{\infty} C ∞ ω V w ( λ 1 ) ⊗ Λ λ 2 {}^{\omega}V_w(\lambda_1)\otimes\Lambda_{\lambda_2} ω V w ( λ 1 ) ⊗ Λ λ 2 ω Λ λ 1 ⊗ Λ λ 2 {}^{\omega}\Lambda_{\lambda_1}\otimes\Lambda_{\lambda_2} ω Λ λ 1 ⊗ Λ λ 2 k k k A A A Lemma 4.2 identifies its original E i , F i E_i,F_i E i , F i
Specialization of the framing center
Fix the framing multiplicities m i m_i m i Θ = P θ m \Theta= P_{\theta_m} Θ = P θ m P P P [ Θ P ] = θ m [ P ] [\Theta P] = \theta_m[P] [ Θ P ] = θ m [ P ] F m F_m F m Θ P \Theta P Θ P F m F_m F m Θ P \Theta P Θ P C w \mathcal C_w C w
S m = ⨂ i ∈ I k [ z i , 1 , … , z i , m i ] S m i , S m , + = ⨁ d > 0 ( S m ) d . S_m = \bigotimes_{i\in I} k[z_{i,1},\ldots,z_{i,m_i}]^{\mathfrak S_{m_i}}, \qquad S_{m,+} = \bigoplus_{d>0}(S_m)_d. S m = i ∈ I ⨂ k [ z i , 1 , … , z i , m i ] S m i , S m , + = d > 0 ⨁ ( S m ) d . Here z i , a z_{i,a} z i , a a a a i ′ i' i ′ 2 2 2 S m i \mathfrak S_{m_i} S m i m i = 0 m_i=0 m i = 0 k k k S m , + S_{m,+} S m , + S m S_m S m S m S_m S m
The transition functors append only original strands, so they preserve this central action. The induction bimodule for F i F_i F i E i E_i E i E i E_i E i (8) , since the removed strand has original color i i i F i , E i F_i,E_i F i , E i S m , + HOM ( X , Y ) S_{m,+}\operatorname{HOM}(X,Y) S m , + HOM ( X , Y ) 2 2 2
C 0 = C ∞ / ( S m , + ) . \mathcal C^0 = \mathcal C_\infty/(S_{m,+}). C 0 = C ∞ / ( S m , + ) . Write C w 0 \mathcal C_w^0 C w 0 w w w X X X S m , + HOM ( X , X ) S_{m,+}\operatorname{HOM}(X,X) S m , + HOM ( X , X ) X X X Lemma 2.1 implies that these quotients are idempotent complete and retain the same normalized labels, with the same degree-zero orthogonality and negative-degree vanishing.
Lemma 6.1. For P , Q ∈ P I P,Q\in\mathcal P_I P , Q ∈ P I
HOM P I ( P , Q ) / I w ( P , Q ) → ∼ HOM C w 0 ( Θ P , Θ Q ) , f ↦ id Θ f . \operatorname{HOM}_{\mathcal P_I}(P,Q)/I_w(P,Q) \xrightarrow{\sim} \operatorname{HOM}_{\mathcal C_w^0}(\Theta P,\Theta Q), \qquad f\mapsto\operatorname{id}_\Theta f. HOM P I ( P , Q ) / I w ( P , Q ) ∼ HOM C w 0 ( Θ P , Θ Q ) , f ↦ id Θ f . These isomorphisms commute with the transition functors and give a fully faithful functor Θ : A ∞ I → C 0 \Theta:\mathcal A_\infty^I\to\mathcal C^0 Θ : A ∞ I → C 0
Proof. Both Hom spaces vanish if P , Q P,Q P , Q β \beta β
HOM P I ( Θ P , Θ Q ) ≅ S m ⊗ k HOM P I ( P , Q ) . \operatorname{HOM}_{\mathcal P_I}(\Theta P,\Theta Q)\cong S_m\otimes_k\operatorname{HOM}_{\mathcal P_I}(P,Q). HOM P I ( Θ P , Θ Q ) ≅ S m ⊗ k HOM P I ( P , Q ) . In fact, for word projectives with the primed block on the left, every permutation matching the boundary colors preserves the two blocks. Choose reduced expressions for these permutations inside the blocks. The KLR PBW basis then consists of products of the two block bases. Cutting by the primed divided-power idempotents replaces the primed nilHecke algebra by its symmetric polynomial corner S m S_m S m
Apply the Mackey filtration of D w D_w D w Θ P \Theta P Θ P L η m + L_{\eta_m}^+ L η m + P P P Θ \Theta Θ γ m \gamma_m γ m β \beta β
( F P L w ⊠ F Θ L η m + ) ⟨ − ( γ m , β ) ⟩ (F_P L_w \boxtimes F_\Theta L_{\eta_m}^+)\langle-(\gamma_m,\beta)\rangle ( F P L w ⊠ F Θ L η m + ) ⟨ − ( γ m , β )⟩ The analogous piece for Q Q Q m i m_i m i m i m_i m i k k k § 2 . Hence F Θ L η m + F_\Theta L_{\eta_m}^+ F Θ L η m + k k k S m S_m S m s ⊗ f s\otimes f s ⊗ f s s s ev w ( f ) \operatorname{ev}_w(f) ev w ( f )
By construction, S m ⊗ I w ( P , Q ) S_m\otimes I_w(P,Q) S m ⊗ I w ( P , Q ) I ~ w ( Θ P , Θ Q ) \widetilde I_w(\Theta P,\Theta Q) I w ( Θ P , Θ Q ) D w D_w D w I ~ w \widetilde I_w I w I ~ w \widetilde I_w I w ev w \operatorname{ev}_w ev w
S m , + ⊗ k HOM P I ( P , Q ) + S m ⊗ k I w ( P , Q ) . S_{m,+}\otimes_k \operatorname{HOM}_{\mathcal P_I}(P,Q)+S_m\otimes_k I_w(P,Q). S m , + ⊗ k HOM P I ( P , Q ) + S m ⊗ k I w ( P , Q ) . Specialization adds the first summand to the ideal. Thus the ideal in the specialized corner is exactly this sum, which proves the formula. Left induction commutes with right induction by each transition tail, and the isomorphism is induced by left induction. Passing to the colimit proves full faithfulness. □
Pure evaluation gives an equivalence A ∞ I → L ( − λ 1 ) A^I_\infty\to\mathcal{L}(-\lambda_1) A ∞ I → L ( − λ 1 ) I w I_w I w L e L_e L e [4 Remark 4.22] . The presentation comparison rescales the dot and crossing generators by units, so the mates of § 5 also generate these spaces. The action comparison lifts them to the pure colimit. The induced isomorphism on endomorphisms of word sums lifts idempotents, which split in the colimit. Composing lifts on word sums with the lifted idempotents gives fullness between the summands. We also write L e L_e L e e e e
Fullness of the evaluation
We compare the upward diagrams on L e L_e L e X 0 = Θ L e X_0=\Theta L_e X 0 = Θ L e A ∞ \mathcal{A}_\infty A ∞ X 0 X_0 X 0 μ e − γ m = ζ \mu_e-\gamma_m=\zeta μ e − γ m = ζ P = F i 1 ⋯ F i ℓ P=F_{i_1}\cdots F_{i_\ell} P = F i 1 ⋯ F i ℓ β = ∑ a = 1 ℓ α i a \beta=\sum_{a=1}^{\ell}\alpha_{i_a} β = ∑ a = 1 ℓ α i a T T T γ \gamma γ
b P , T : P T ⟶ T P , deg b P , T = − ( β , γ ) . b_{P,T}:PT\longrightarrow TP,\qquad\deg b_{P,T}=-(\beta,\gamma). b P , T : P T ⟶ T P , deg b P , T = − ( β , γ ) . For an original negative KLR diagram f : P → Q f:P\to Q f : P → Q g : T → T ′ g:T\to T' g : T → T ′
b Q , T ( f T ) = ( T f ) b P , T , ( g P ) b P , T = b P , T ′ ( P g ) . (9) b_{Q,T}(fT)=(Tf)b_{P,T},\qquad(gP)b_{P,T}=b_{P,T'}(Pg). \tag*{(9)} b Q , T ( f T ) = ( T f ) b P , T , ( g P ) b P , T = b P , T ′ ( P g ) . (9) For a generating crossing, the correction in the braid relation vanishes because its first and last colors lie in disjoint color sets. This applies also when the two strands of the generating crossing have the same color.
Let R P R_P R P P P P η P \eta_P η P ϵ P \epsilon_P ϵ P
b ˇ P , T = ( R P T ϵ P ) ( R P b P , T R P ) ( η P T R P ) : T R P ⟶ R P T . \check{b}_{P,T}=(R_PT\epsilon_P)(R_Pb_{P,T}R_P)(\eta_PTR_P):TR_P\longrightarrow R_PT. b ˇ P , T = ( R P T ϵ P ) ( R P b P , T R P ) ( η P T R P ) : T R P ⟶ R P T . Equivalently its transpose under P ⊣ R P P\dashv R_P P ⊣ R P
( ϵ P T ) ( P b ˇ P , T ) = ( T ϵ P ) ( b P , T R P ) . (10) (\epsilon_PT)(P\check{b}_{P,T})=(T\epsilon_P)(b_{P,T}R_P). \tag*{(10)} ( ϵ P T ) ( P b ˇ P , T ) = ( T ϵ P ) ( b P , T R P ) . (10) For f : P → Q f:P\to Q f : P → Q
f r = ( R P ϵ Q ) ( R P f R Q ) ( η P R Q ) : R Q ⟶ R P . f^r=(R_P\epsilon_Q)(R_PfR_Q)(\eta_PR_Q):R_Q\longrightarrow R_P. f r = ( R P ϵ Q ) ( R P f R Q ) ( η P R Q ) : R Q ⟶ R P . Its transpose satisfies ϵ P ( P f r ) = ϵ Q ( f R Q ) \epsilon_P(Pf^r)=\epsilon_Q(fR_Q) ϵ P ( P f r ) = ϵ Q ( f R Q )
b ˇ P , T ( T f r ) = ( f r T ) b ˇ Q , T , ( R P g ) b ˇ P , T = b ˇ P , T ′ ( g R P ) . (11) \check{b}_{P,T}(Tf^r)=(f^rT)\check{b}_{Q,T},\qquad(R_Pg)\check{b}_{P,T}=\check{b}_{P,T'}(gR_P). \tag*{(11)} b ˇ P , T ( T f r ) = ( f r T ) b ˇ Q , T , ( R P g ) b ˇ P , T = b ˇ P , T ′ ( g R P ) . (11) Under P ⊣ R P P\dashv R_P P ⊣ R P
( T ϵ P ) ( b P , T R P ) ( P T f r ) = ( T ϵ Q ) ( T f R Q ) ( b P , T R Q ) = ( T ϵ Q ) ( b Q , T R Q ) ( f T R Q ) . \begin{aligned}
(T\epsilon_P)(b_{P,T}R_P)(PTf^r)=(T\epsilon_Q)(TfR_Q)(b_{P,T}R_Q) \\
&=(T\epsilon_Q)(b_{Q,T}R_Q)(fTR_Q).
\end{aligned} ( T ϵ P ) ( b P , T R P ) ( P T f r ) = ( T ϵ Q ) ( T f R Q ) ( b P , T R Q ) = ( T ϵ Q ) ( b Q , T R Q ) ( f T R Q ) . The first equality uses naturality of the crossing in its argument and the transpose identity for f r f^r f r (9) . The first right-hand side transposes to
( ϵ Q T ) ( f R Q T ) ( P b ˇ Q , T ) = ( ϵ Q T ) ( Q b ˇ Q , T ) ( f T R Q ) , (\epsilon_QT)(fR_QT)(P\check{b}_{Q,T})=(\epsilon_QT)(Q\check{b}_{Q,T})(fTR_Q), ( ϵ Q T ) ( f R Q T ) ( P b ˇ Q , T ) = ( ϵ Q T ) ( Q b ˇ Q , T ) ( f T R Q ) , which is the same by (10) . The two sides for g g g g ( T ϵ P ) ( b P , T R P ) g(T\epsilon_P)(b_{P,T}R_P) g ( T ϵ P ) ( b P , T R P ) ( T ′ ϵ P ) ( b P , T ′ R P ) ( P g R P ) (T'\epsilon_P)(b_{P,T'}R_P)(PgR_P) ( T ′ ϵ P ) ( b P , T ′ R P ) ( P g R P ) (9) and naturality of g g g
For an original negative word Q Q Q U U U
b P Q , T = ( b P , T Q ) ( P b Q , T ) , b P , T U = ( T b P , U ) ( b P , T U ) . b_{PQ,T}=(b_{P,T}Q)(Pb_{Q,T}),\qquad b_{P,TU}=(Tb_{P,U})(b_{P,T}U). b P Q , T = ( b P , T Q ) ( P b Q , T ) , b P , T U = ( T b P , U ) ( b P , T U ) . The composite right adjoint is R P Q = R Q R P R_{PQ}=R_QR_P R P Q = R Q R P ε P ( P ε Q R P ) \varepsilon_P(P\varepsilon_QR_P) ε P ( P ε Q R P )
b ~ P Q , T = ( R Q b ~ P , T ) ( b ~ Q , T R P ) , b ~ P , T U = ( b ~ P , T U ) ( T b ~ P , U ) . (12) \begin{aligned}
\widetilde b_{PQ,T}&=(R_Q\widetilde b_{P,T})(\widetilde b_{Q,T}R_P),&
\widetilde b_{P,TU}&=(\widetilde b_{P,T}U)(T\widetilde b_{P,U}). \tag*{(12)}
\end{aligned} b P Q , T = ( R Q b P , T ) ( b Q , T R P ) , b P , T U = ( b P , T U ) ( T b P , U ) . (12) For the first identity, transposing its right-hand side under P Q ⊣ R Q R P PQ\dashv R_QR_P P Q ⊣ R Q R P
( ε P T ) ( P ε Q R P T ) ( P Q R Q b ~ P , T ) ( P Q b ~ Q , T R P ) = ( ε P T ) ( P b ~ P , T ) ( P ε Q T R P ) ( P Q b ~ Q , T R P ) = ( T ε P ) ( b P , T R P ) ( P T ε Q R P ) ( P b Q , T R Q R P ) = ( T ε P Q ) ( b P Q , T R Q R P ) . \begin{aligned}
&(\varepsilon_PT)(P\varepsilon_QR_PT)(PQR_Q\widetilde b_{P,T})(PQ\widetilde b_{Q,T}R_P)\\
={}&(\varepsilon_PT)(P\widetilde b_{P,T})(P\varepsilon_QTR_P)(PQ\widetilde b_{Q,T}R_P)\\
={}&(T\varepsilon_P)(b_{P,T}R_P)(PT\varepsilon_QR_P)(Pb_{Q,T}R_QR_P)\\
={}&(T\varepsilon_{PQ})(b_{PQ,T}R_QR_P).
\end{aligned} = = = ( ε P T ) ( P ε Q R P T ) ( P Q R Q b P , T ) ( P Q b Q , T R P ) ( ε P T ) ( P b P , T ) ( P ε Q T R P ) ( P Q b Q , T R P ) ( T ε P ) ( b P , T R P ) ( P T ε Q R P ) ( P b Q , T R Q R P ) ( T ε P Q ) ( b P Q , T R Q R P ) . The second equality uses (10) for P , T P,T P , T Q , T Q,T Q , T P ⊣ R P P\dashv R_P P ⊣ R P
( ε P T U ) ( P b ~ P , T U ) ( P T b ~ P , U ) = ( T ε P U ) ( b P , T R P U ) ( P T b ~ P , U ) = ( T ε P U ) ( T P b ~ P , U ) ( b P , T U R P ) = ( T U ε P ) ( b P , T U R P ) . \begin{aligned}
&(\varepsilon_PTU)(P\widetilde b_{P,TU})(PT\widetilde b_{P,U})=(T\varepsilon_{PU})(b_{P,T}R_{PU})(PT\widetilde b_{P,U})\\
={}&(T\varepsilon_{PU})(TP\widetilde b_{P,U})(b_{P,TU}R_P)\\
={}&(TU\varepsilon_P)(b_{P,TU}R_P).
\end{aligned} = = ( ε P T U ) ( P b P , T U ) ( P T b P , U ) = ( T ε P U ) ( b P , T R P U ) ( P T b P , U ) ( T ε P U ) ( T P b P , U ) ( b P , T U R P ) ( T U ε P ) ( b P , T U R P ) . This proves both identities.
Put P ∨ = E i ℓ ⋯ E i 1 P^\vee=E_{i_\ell}\cdots E_{i_1} P ∨ = E i ℓ ⋯ E i 1 λ \lambda λ F i F_i F i E i ⟨ 1 + ⟨ h i , λ ⟩ ⟩ E_i\langle1+\langle h_i,\lambda\rangle\rangle E i ⟨ 1 + ⟨ h i , λ ⟩⟩
R P 1 λ = P ∨ 1 λ ⟨ c β ( λ ) ⟩ , R_P1_\lambda=P^\vee1_\lambda\langle c_\beta(\lambda)\rangle, R P 1 λ = P ∨ 1 λ ⟨ c β ( λ )⟩ , c β ( λ ) = ∑ a = 1 ℓ ( 1 + ( α i a , λ + ∑ b < a α i b ) ) = ( β , β ) 2 + ( β , λ ) . c_\beta(\lambda)=\sum_{a=1}^{\ell}\left(1+\left(\alpha_{i_a},\lambda+\sum_{b<a}\alpha_{i_b}\right)\right)=\frac{(\beta,\beta)}{2}+(\beta,\lambda). c β ( λ ) = a = 1 ∑ ℓ ( 1 + ( α i a , λ + b < a ∑ α i b ) ) = 2 ( β , β ) + ( β , λ ) . The unit and counit used in the mate have degree zero, so b ~ P , T \widetilde b_{P,T} b P , T − ( β , γ ) -(\beta,\gamma) − ( β , γ ) c β ( λ − γ ) − c β ( λ ) = − ( β , γ ) c_\beta(\lambda-\gamma)-c_\beta(\lambda)=-(\beta,\gamma) c β ( λ − γ ) − c β ( λ ) = − ( β , γ )
s P , T : T P ∨ ⟶ P ∨ T . s_{P,T}:TP^\vee\longrightarrow P^\vee T. s P , T : T P ∨ ⟶ P ∨ T . For original i i i r r r τ i r : F i F r ⟶ F r F i \tau_{ir}:F_iF_r\longrightarrow F_rF_i τ i r : F i F r ⟶ F r F i η i , ε i \eta_i,\varepsilon_i η i , ε i i i i § 4 . Then
s F i , F r = ( E i F r ε i ) ( E i τ i r E i ) ( η i F r E i ) . s_{F_i,F_r}=(E_iF_r\varepsilon_i)(E_i\tau_{ir}E_i)(\eta_iF_rE_i). s F i , F r = ( E i F r ε i ) ( E i τ i r E i ) ( η i F r E i ) . This is the mixed isomorphism (3) . The two factorizations in (12) also hold for s s s β , δ \beta,\delta β , δ c β + δ ( λ ) = c β ( λ ) + c δ ( λ + β ) c_{\beta+\delta}(\lambda)=c_\beta(\lambda)+c_\delta(\lambda+\beta) c β + δ ( λ ) = c β ( λ ) + c δ ( λ + β ) s P , T s_{P,T} s P , T
For a negative diagram f : P → Q f:P\to Q f : P → Q f ∨ f^\vee f ∨ f ∨ : Q ∨ → P ∨ f^\vee:Q^\vee\to P^\vee f ∨ : Q ∨ → P ∨ (11) becomes
s P , T ( T f ∨ ) = ( f ∨ T ) s Q , T . s_{P,T}(Tf^\vee)=(f^\vee T)s_{Q,T}. s P , T ( T f ∨ ) = ( f ∨ T ) s Q , T . We use the upward dots and crossings in the native mate normalization of § 5 . They are the f ∨ f^{\vee} f ∨ (11) to the primed divided-power idempotents, and applying their normalization shifts, gives
κ E α : Θ E α ⟶ E α Θ \kappa_{E_\alpha}:\Theta E_\alpha\longrightarrow E_\alpha\Theta κ E α : Θ E α ⟶ E α Θ for every positive word E α E_\alpha E α α \alpha α
Lemma 7.1. For positive words E α , E β E_\alpha,E_\beta E α , E β X 0 X_0 X 0 HOM C 0 ( E α X 0 , E β X 0 ) \operatorname{HOM}_{\mathcal C_0}(E_\alpha X_0,E_\beta X_0) HOM C 0 ( E α X 0 , E β X 0 )
Proof. Lemma 6.1 and the pure evaluation identify this Hom space with HOM L ( − λ 1 ) ( E α L e , E β L e ) \operatorname{HOM}_{\mathcal L_{(-\lambda_1)}}(E_\alpha L_e,E_\beta L_e) HOM L ( − λ 1 ) ( E α L e , E β L e )
The map from the latter space sends f f f κ E β Θ ( f ) κ E α − 1 \kappa_{E_\beta}\Theta(f)\kappa_{E_\alpha}^{-1} κ E β Θ ( f ) κ E α − 1 X 0 X_0 X 0
Proposition 7.2. Evaluation at X 0 X_0 X 0
ev ζ , m : U ˙ 1 ζ ⟶ C 0 . \operatorname{ev}_{\zeta,m}:\dot{\mathcal U}_1^\zeta\longrightarrow\mathcal C^0. ev ζ , m : U ˙ 1 ζ ⟶ C 0 . Here U ˙ 1 ζ \dot{\mathcal U}_1^\zeta U ˙ 1 ζ ζ \zeta ζ
Proof. Consider two words F α E γ F_\alpha E_\gamma F α E γ F β E δ F_\beta E_\delta F β E δ R F α R_{F_\alpha} R F α F α F_\alpha F α
HOM C 0 ( E γ X 0 , R F α F β E δ X 0 ) . \operatorname{HOM}_{\mathcal C_0}(E_\gamma X_0,R_{F_\alpha}F_\beta E_\delta X_0). HOM C 0 ( E γ X 0 , R F α F β E δ X 0 ) . In the source, R F α F β E δ R_{F_\alpha}F_\beta E_\delta R F α F β E δ F u E v F_uE_v F u E v h ≥ 0 h\geq0 h ≥ 0 E i F i E_iF_i E i F i F i E i F_iE_i F i E i h < 0 h<0 h < 0 E i F i E_iF_i E i F i F i E i F_iE_i F i E i
For a term F u E v ⟨ d ⟩ F_uE_v\langle d\rangle F u E v ⟨ d ⟩ L F u L_{F_u} L F u F u F_u F u F i F_i F i λ \lambda λ E i ⟨ − 1 − ⟨ h i , λ ⟩ ⟩ E_i\langle-1-\langle h_i,\lambda\rangle\rangle E i ⟨ − 1 − ⟨ h i , λ ⟩⟩ [18 ] , Lemma
2.6.14]. For a word, compose the left adjoints in reverse order, taking each shift at the weight at which its functor is applied. Adjunction gives
Hom C 0 ( E γ X 0 , F μ E ν X 0 ⟨ d ⟩ ) ≅ Hom C 0 ( L F μ E γ X 0 , E ν X 0 ⟨ d ⟩ ) . \operatorname{Hom}_{\mathcal{C}_0}(E_{\gamma}X_0,F_{\mu}E_{\nu}X_0\langle d\rangle)\cong\operatorname{Hom}_{\mathcal{C}_0}(L_{F_{\mu}}E_{\gamma}X_0,E_{\nu}X_0\langle d\rangle). Hom C 0 ( E γ X 0 , F μ E ν X 0 ⟨ d ⟩) ≅ Hom C 0 ( L F μ E γ X 0 , E ν X 0 ⟨ d ⟩) . Both words on the right are positive up to shift. The shifts are represented in the source, so Lemma 7.1 lifts every homogeneous map in these spaces. Composing with the source adjunctions and the source projections and inclusions gives a lift between the original words. Arbitrary words are summands of finite sums of words in the same order F α E γ F_{\alpha}E_{\gamma} F α E γ
Identification of the canonical basis
The diagrammatic Grothendieck group surjection is
γ : A ˙ U ⟶ K 0 ( U ˙ ) . \gamma:\dot{\mathcal{A}}\mathcal{U}\longrightarrow K_0(\dot{\mathcal{U}}). γ : A ˙ U ⟶ K 0 ( U ˙ ) . It sends divided powers and weight idempotents to their diagrammatic classes and uses the shift ⟨ 1 ⟩ \langle1\rangle ⟨ 1 ⟩ v v v [4 §3.5, (3.16)] . The split Grothendieck group is the direct sum over pairs of boundary weights, with product induced by composition and zero for incompatible weights.
Proposition 8.1. The map γ \gamma γ
Proof. The normalized labels of C 0 \mathcal{C}^0 C 0 Λ Λ λ 1 ⊗ Λ λ 2 {}_{\Lambda}\Lambda_{\lambda_1}\otimes\Lambda_{\lambda_2} Λ Λ λ 1 ⊗ Λ λ 2 γ \gamma γ A U ˙ 1 ζ \mathcal{A}\dot{\mathcal{U}}_{1\zeta} A U ˙ 1 ζ π ζ , m \pi_{\zeta,m} π ζ , m
π ζ , m ( x ) = x ( ξ − λ 1 ⊗ η λ 2 ) . \pi_{\zeta,m}(x)=x(\xi_{-\lambda_1}\otimes\eta_{\lambda_2}). π ζ , m ( x ) = x ( ξ − λ 1 ⊗ η λ 2 ) . The action of E i , F i E_i,F_i E i , F i X 0 X_0 X 0 Q ( v ) \mathbb{Q}(v) Q ( v ) [ r ] ! [r]! [ r ]! A \mathcal{A} A
For a nonzero a ∈ A U ˙ 1 ζ a\in\mathcal{A}\dot{\mathcal{U}}_{1\zeta} a ∈ A U ˙ 1 ζ m i m_i m i λ 2 − λ 1 = ζ \lambda_2-\lambda_1=\zeta λ 2 − λ 1 = ζ m i m_i m i [17 Theorem 25.2.1] . These images are linearly independent, so π ζ , m ( a ) ≠ 0 \pi_{\zeta,m}(a)\ne0 π ζ , m ( a ) = 0 γ ( a ) ≠ 0 \gamma(a)\ne0 γ ( a ) = 0
The diagrammatic spanning set consists of finitely many reduced boundary diagrams decorated with dots and positive-degree bubble monomials [12 Proposition 3.11] . Dots and bubble variables have positive degrees, with finitely many bubble variables in each degree. This gives a lower degree bound and finite dimension in each degree for fixed boundaries. The same bounds hold for summands. The degree-zero endomorphism algebra of each object is finite-dimensional, so a decomposition of its identity into primitive orthogonal idempotents gives a finite decomposition into objects with local endomorphism rings. The source is therefore Krull–Schmidt. For an indecomposable object, fullness makes the degree-zero endomorphism algebra of its image a quotient of its local endomorphism algebra. If the image is nonzero, the quotient is local and the image is indecomposable.
Now let P P P
γ − 1 [ P ] = ∑ b ∈ T a b ( v ) b \gamma^{-1}[P] = \sum_{b\in T} a_b(v)b γ − 1 [ P ] = b ∈ T ∑ a b ( v ) b with T T T a b ( v ) ∈ A a_b(v)\in\mathcal{A} a b ( v ) ∈ A b ∈ T b\in T b ∈ T m m m T T T P P P c c c
∑ b ∈ T a b ( v ) π ζ , m ( b ) = v s c \sum_{b\in T} a_b(v)\pi_{\zeta,m}(b)=v^s c b ∈ T ∑ a b ( v ) π ζ , m ( b ) = v s c for some s ∈ Z s\in\mathbb{Z} s ∈ Z T = { b } T=\{b\} T = { b } a b ( v ) = v s a_b(v)=v^s a b ( v ) = v s A \mathcal{A} A − s -s − s
For composable canonical basis elements b , b ′ b,b' b , b ′ P b , P b ′ P_b,P_{b'} P b , P b ′
P b P b ′ ≅ ⨁ c , s P c ⟨ s ⟩ ⊕ n c , s , n c , s ∈ N P_bP_{b'}\cong\bigoplus_{c,s}P_c\langle s\rangle^{\oplus n_{c,s}},\qquad n_{c,s}\in\mathbb{N} P b P b ′ ≅ c , s ⨁ P c ⟨ s ⟩ ⊕ n c , s , n c , s ∈ N where c c c s s s ∑ s n c , s v s \sum_s n_{c,s}v^s ∑ s n c , s v s c c c b b ′ bb' b b ′ b b ′ = 0 bb'=0 b b ′ = 0
Change of root datum
Let X X X A A A δ = ∑ i ∈ I n i α i \delta=\sum_{i\in I}n_i\alpha_i δ = ∑ i ∈ I n i α i ρ X ( δ ) \rho_X(\delta) ρ X ( δ ) ρ P ( δ ) \rho_P(\delta) ρ P ( δ ) X X X P P P ν ∈ X \nu\in X ν ∈ X ν ^ ∈ P \hat{\nu} \in P ν ^ ∈ P ⟨ h i , ν ^ ⟩ = ⟨ h i , ν ⟩ \langle h_i,\hat{\nu}\rangle= \langle h_i,\nu\rangle ⟨ h i , ν ^ ⟩ = ⟨ h i , ν ⟩ i ∈ I i \in I i ∈ I P P P
The simply connected weight lattice for A A A Z I \mathbb{Z}^I Z I A A A ν \nu ν ν ^ \hat{\nu} ν ^ [17 23.2.5 and 25.2.2–25.2.4] sends each element u + u − 1 ν u^+u^-1_\nu u + u − 1 ν ν \nu ν [5 §6.6] . Here u + u^+ u + u − u^- u − A A A X X X P P P
T ν , ν ^ : U ˙ 1 ν ⟶ U ˙ P 1 ν ^ . T_{\nu,\hat{\nu}}:\dot{U}1_\nu\longrightarrow\dot{U}_P1_{\hat{\nu}}. T ν , ν ^ : U ˙ 1 ν ⟶ U ˙ P 1 ν ^ . The subscript P P P P P P T ν , ν ^ ( u + u − 1 ν ) = u + u − 1 ν ^ T_{\nu,\hat{\nu}}(u^+u^-1_\nu)=u^+u^-1_{\hat{\nu}} T ν , ν ^ ( u + u − 1 ν ) = u + u − 1 ν ^
This isomorphism intertwines left multiplication by each E i , F i E_i,F_i E i , F i F i , E j F_i,E_j F i , E j δ \delta δ ν + ρ X ( δ ) \nu+\rho_X(\delta) ν + ρ X ( δ ) ν ^ + ρ P ( δ ) \hat{\nu}+\rho_P(\delta) ν ^ + ρ P ( δ ) E i , F i E_i,F_i E i , F i
Proof of Theorem 1.1. Section 8 proves the theorem for P P P b b b b ′ b' b ′ b ′ b' b ′ ν \nu ν λ \lambda λ δ ′ \delta' δ ′ [17 25.2.1(e) and 25.2.4] , so λ = ν + ρ X ( δ ′ ) \lambda=\nu+\rho_X(\delta') λ = ν + ρ X ( δ ′ ) λ ^ = ν ^ + ρ P ( δ ′ ) \hat{\lambda}=\hat{\nu}+\rho_P(\delta') λ ^ = ν ^ + ρ P ( δ ′ ) b = u 1 λ b=u1_\lambda b = u 1 λ u u u E i , F i E_i,F_i E i , F i T λ , λ ^ ( b ) = u 1 λ ^ T_{\lambda,\hat{\lambda}}(b)=u1_{\hat{\lambda}} T λ , λ ^ ( b ) = u 1 λ ^ T ν , ν ^ ( b ′ ) T_{\nu,\hat{\nu}}(b') T ν , ν ^ ( b ′ ) λ ^ \hat{\lambda} λ ^
T ν , ν ^ ( b b ′ ) = T λ , λ ^ ( b ) T ν , ν ^ ( b ′ ) . T_{\nu,\hat{\nu}}(bb')=T_{\lambda,\hat{\lambda}}(b)\,T_{\nu,\hat{\nu}}(b'). T ν , ν ^ ( b b ′ ) = T λ , λ ^ ( b ) T ν , ν ^ ( b ′ ) . The column isomorphism sends distinct canonical labels to distinct canonical labels. The product coefficients therefore agree with those for P P P
For a symmetric Cartan matrix presented with a positive integral symmetrizing integer d d d v d v^d v d v ↦ v d v\mapsto v^d v ↦ v d R R R X X X
Zhili College, Tsinghua University, Beijing 100084, China
Email address: tangqy24@mails.tsinghua.edu.cn
