aa r X i v : . [ m a t h . QA ] J u l THE DARK SIDE OF GENERALIZED DEMAZURE CRYSTALS
JONAH BLASIAK
Abstract.
Naoi [20] showed that tensor products of perfect Kirillov-Reshetikhin crys-tals are isomorphic to certain generalized Demazure crystals. We extend Naoi’s resultsto address distinguished subsets of these tensor products. In type A, these are naturallydescribed in terms of katabolizable tableaux which was key to resolving conjectures ofShimozono-Weyman [25] and Chen-Haiman [2] in [1]. Introduction
Naoi [20] showed that tensor products of perfect Kirillov-Reshetikhin (KR) crystals areisomorphic to certain generalized Demazure crystals introduced by Lakshmibai-Littelmann-Magyar [14]. From this he obtained a Demazure operator formula for their charactersusing the well-developed theory of Demazure crystals [5, 11, 14, 17]. This formed a keystep in his resolution of the X = M conjecture [19] in type D (1) n .We extend Naoi’s result to match a larger class of generalized Demazure crystals withcertain subsets of tensor products of perfect KR crystals, termed Kirillov-Reshetikhinaffine Demazure (DARK) crystals. Our result follows directly from techniques of [20], butthe deep combinatorial consequences shown for type A in [1] motivate this presentationof the results in the full generality of any nonexceptional type.Naoi’s work encompasses several earlier results connecting Demazure and KR crys-tals [4, 23, 24]. The emphasis in these works is on providing a model for KR crystalsusing the well-developed theory of Demazure crystals, whereas here we are interested inusing KR crystals to understand generalized Demazure crystals. While there are combi-natorial models of highest weight crystals for affine type [7, 8, 9, 16, 18] which lead toexplicit descriptions of generalized Demazure crystals, our explorations suggest that thecombinatorics afforded by DARK crystals is simpler. This is possible because the isomor-phism between generalized Demazure and DARK crystals is combinatorially nontrivial,roughly analogous to the different models for the U q ( gl n )-highest weight crystal B ( ν )afforded by semistandard tableaux of shape ν versus those provided by an embedding B ( ν ) ֒ → B ( λ ) ⊗ B ( µ ). 2. Background on crystals
We follow [20] almost completely and review the notation we will need, emphasizingconventions which may not be well known.
Key words and phrases.
Kirillov-Reshetikhin crystals, energy, Demazure crystals, katabolism.This work was supported by NSF Grant DMS-1855784.
Affine Kac-Moody Lie algebras.
Let g be a complex affine Kac-Moody Lie al-gebra of nonexceptional type (i.e., of type A (1) n , B (1) n , C (1) n , D (1) n , A (2)2 n − , A (2)2 n , or D (2) n +1 ). Let I = { , , . . . , n } be the Dynkin nodes and A = ( a ij ) i,j ∈ I the Cartan matrix. Let h ⊂ g bethe Cartan subalgebra, which has a basis consisting of the simple coroots { α ∨ i | i ∈ I } ⊂ h together with the scaling element d ∈ h . We have the linearly independent simple roots { α i | i ∈ I } ⊂ h ∗ , with pairings h α ∨ i , α j i = a ij and h d, α i i = δ i ( i, j ∈ I ). Let ( a , . . . , a n )(resp. ( a ∨ , . . . , a ∨ n )) be the unique tuple of relatively prime positive integers that give alinear dependence relation among the columns (resp. rows) of A .Choose N ∈ Z ≥ and fundamental weights { Λ i | i ∈ I } ⊂ h ∗ such that h α ∨ i , Λ j i = δ ij and h d, Λ j i ∈ N − Z for i, j ∈ I , the choices to be discussed further below. Let δ = P i ∈ I a i α i be the null root. Note that { Λ i | i ∈ I } ⊔ { δ } is a basis of h ∗ and h α ∨ i , δ i = 0for i ∈ I and h d, δ i = a . Let P = (cid:8) µ ∈ h ∗ | h α ∨ i , µ i ∈ Z for i ∈ I, h d, µ i ∈ N − Z (cid:9) = L i ∈ I Z Λ i ⊕ Z δa N ⊂ h ∗ be the weight lattice and P + = P i ∈ I Z ≥ Λ i + Z δa N the dominantweights.Let cl : h ∗ → h ∗ / C δ be the canonical projection, and set P cl = cl( P ) = L i ∈ I Z cl(Λ i ).Let aff : h ∗ / C δ → h ∗ be the section of cl satisfying h d, aff( µ ) i = 0 for all µ ∈ h ∗ / C δ . Set ̟ i = aff(cl(Λ i − a ∨ i Λ )) for i ∈ I := I \ { } .The affine Weyl group W can be realized as the subgroup of GL ( h ∗ ) generated by thesimple reflections s i ( i ∈ I ), where s i acts by s i ( µ ) = µ − h α ∨ i , µ i α i . Let W be thesubgroup generated by s i for i ∈ I .Let c i = max { , a i /a ∨ i } for i ∈ I , and define f M = L i ∈ I Z c i ̟ i ⊂ P . For µ ∈ f M , definethe translation t µ ∈ GL ( h ∗ ) as in [6, Equation 6.5.2] and set T = { t µ | µ ∈ f M } . Thesesatisfy t µ t λ = t µ + λ and wt µ w − = t w ( µ ) for w ∈ W and µ, λ ∈ f M . Thus f W = W ⋉ T isa subgroup of GL ( h ∗ ), called the extended affine Weyl group .Let Σ ⊂ f W denote the subgroup which takes the set { α i | i ∈ I } to itself. Thus eachelement τ ∈ Σ yields a permutation of I , which we also denote by τ ; it is an automorphismof the Dynkin diagram, meaning that a ij = a τ ( i ) τ ( j ) for all i, j ∈ I . (See [20, § § f W ∼ = W ⋊ Σ. As discussed in [20, § N and h d, Λ i i so that for all τ ∈ Σ, τ (Λ j ) = Λ τ ( j ) for all j ∈ I and τ ( δ ) = δ . Note that this implies f W preserves P .2.2. Crystals.
Let U q ( g ) be the quantized enveloping algebra (as in [10]) specified bythe above data and the symmetric bilinear form ( · , · ) : P × P → Q defined by ( α i , α j ) = a ∨ i a − i a ij , ( α i , Λ ) = a − δ i , (Λ , Λ ) = 0. It is generated by e i , f i , i ∈ I , and q h , h ∈ P ∗ := Hom Z ( P, Z ). Let U ′ q ( g ) ⊂ U q ( g ) be the subalgebra generated by e i , f i , i ∈ I , and q h , h ∈ P ∗ cl = L i ∈ I Z α ∨ i . A U q ( g ) -crystal (resp. U ′ q ( g )-crystal) is a set B equipped with a weight function wt : B → P (resp. wt : B → P cl ) and crystal operators ˜ e i , ˜ f i : B ⊔ { } → B ⊔ { } ( i ∈ I ) such that for all i ∈ I and b ∈ B , there holds ˜ e i (0) = ˜ f i (0) = 0 andwt(˜ e i b ) = wt( b ) + α i whenever ˜ e i b = 0 , and wt( ˜ f i b ) = wt( b ) − α i whenever ˜ f i b = 0; ε i ( b ) := max { k ≥ | ˜ e ki b = 0 } < ∞ , φ i ( b ) := max { k ≥ | ˜ f ki b = 0 } < ∞ ; h α ∨ i , wt( b ) i = φ i ( b ) − ε i ( b ); HE DARK SIDE OF GENERALIZED DEMAZURE CRYSTALS 3 ˜ f i (˜ e i b ) = b whenever ˜ e i b = 0, and ˜ e i ( ˜ f i b ) = b whenever ˜ f i b = 0 . This agrees with the notion of a seminormal crystal in [12, § crystal to mean either a U q ( g )-crystal or U ′ q ( g )-crystal.A strict embedding of a crystal B into a crystal B ′ is an injective map Ψ : B ⊔ { } → B ′ ⊔ { } such that Ψ(0) = 0 and Ψ commutes with wt, ε i , φ i , ˜ e i , and ˜ f i for all i ∈ I . It isnecessarily an isomorphism from B onto a disjoint union of connected components of B ′ .For a U q ( g )-crystal B with weight function wt : B → P , its U ′ q ( g ) -restriction is the U ′ q ( g )-crystal with the same edges as B and weight function cl ◦ wt : B → P cl .For two crystals B and B , their tensor product B ⊗ B = { b ⊗ b | b ∈ B , b ∈ B } is the crystal with weight function wt( b ⊗ b ) = wt( b ) + wt( b ) and crystal operators˜ e i ( b ⊗ b ) = ( ˜ e i b ⊗ b if φ i ( b ) ≥ ε i ( b ) ,b ⊗ ˜ e i b if φ i ( b ) < ε i ( b ) . (2.1)˜ f i ( b ⊗ b ) = ( ˜ f i b ⊗ b if φ i ( b ) > ε i ( b ) ,b ⊗ ˜ f i b if φ i ( b ) ≤ ε i ( b ) . (2.2)Kirillov-Reshetikhin modules W r,s are finite-dimensional U ′ q ( g )-modules parameterizedby ( r, s ) ∈ I × Z ≥ . For nonexceptional g , the W r,s have crystal pseudobases [9, 21, 22],and these yield U ′ q ( g )-crystals B r,s known as KR crystals. We are interested in the subclassof perfect KR crystals (see [8]); we will not work with the definition directly, but onlyneed the following from [3]: a KR crystal B r,s is perfect if and only if c r = max { , a r /a ∨ r } divides s .2.3. Dynkin diagram automorphisms and crystals.
For τ ∈ Σ and U q ( g )-crystals(resp. U ′ q ( g )-crystals) B, B ′ , a bijection of sets z : B → B ′ is a τ -twist if τ (wt( b )) = wt( z ( b )) , and z (˜ e i b ) = ˜ e τ ( i ) z ( b ) , z ( ˜ f i b ) = ˜ f τ ( i ) z ( b ) for all i ∈ I , where z (0) := 0 . Since τ ( P ) = P and τ ( δ ) = δ , τ yields automorphisms of P and P cl , and thus τ (wt( b ))belongs to P (resp. P cl ). Proposition 2.1 ([23, Lemma 6.5], [20, Proposition 5.5]) . For any KR crystal B and τ ∈ Σ , there exists a unique τ -twist of U ′ q ( g ) -crystals F Bτ : B → B . There is also a unique τ -twist F Λ τ : B (Λ) → B ( τ (Λ)) for any Λ ∈ P + , where B (Λ) isthe U q ( g )-crystal of the irreducible U q ( g )-module of highest weight Λ [10, 12].It is easily verified that if z : B → B ′ and z : B → B ′ are τ -twists, then so is z ⊗ z : B ⊗ B → B ′ ⊗ B ′ . Thus the tensor product of maps F Λ τ ⊗ F Λ τ is the naturalchoice of τ -twist from any tensor product B (Λ ) ⊗ B (Λ ) of highest weight U q ( g )-crystals,Λ , Λ ∈ P + . Using in addition Proposition 2.1, a similar τ -twist exists from any tensorproduct of KR crystals and highest weight crystals to another such product, and we denoteit F τ (these are the only crystals we will consider in this paper); for example, for a KRcrystal B , we denote by F τ the map F Λ τ ⊗ F Bτ : B (Λ ) ⊗ B → B (Λ τ (0) ) ⊗ B , where B (Λ )and B (Λ τ (0) ) are regarded as U ′ q ( g )-crystals by restriction. JONAH BLASIAK
Generalized Demazure crystals.
For a crystal B , S ⊂ B , and i ∈ I , define F i S := { ˜ f ki b | b ∈ S, k ≥ } \ { } ⊂ B. For a reduced expression w = s i · · · s i m ∈ W , we write F w S for F i · · · F i m S when this iswell defined, i.e., does not depend on the choice of reduced expression. A Demazure crystal is a subset of some highest weight U q ( g )-crystal B (Λ) of the form B w (Λ) := F w { u Λ } forsome w ∈ W , where u Λ is the highest weight element of B (Λ); it is well defined by [11].A generalized Demazure crystal is a subset of a tensor product of highest weight crystalsof the form F w F τ (cid:0) u Λ ⊗F w F τ (cid:0) u Λ ⊗· · · F w p F τ p ( { u Λ p } ) · · · (cid:1)(cid:1) for some Λ , . . . , Λ p ∈ P + , w , . . . , w p ∈ W , and τ , . . . , τ p ∈ Σ. The combinatorial excellent filtration theorem [14],[5] states that u Λ ⊗ F w { u Λ } ⊂ B (Λ ) ⊗ B (Λ ) is a disjoint union of Demazure crystals.It follows that (see [20, Lemma 4.3]) Theorem 2.2.
Any generalized Demazure crystal is a disjoint union of Demazure crystals(and the above expression is well defined). Matching generalized Demazure and DARK crystals
Lemma 3.1 ([10]) . For any Λ , Λ ′ ∈ P + , there is a strict embedding of U q ( g ) -crystals B (Λ + Λ ′ ) ֒ → B (Λ) ⊗ B (Λ ′ ) determined by u Λ+Λ ′ u Λ ⊗ u Λ ′ ; it maps B (Λ + Λ ′ ) isomor-phically onto a connected component of B (Λ) ⊗ B (Λ ′ ) .Proof. It is well known [10] that B (Λ) ⊗ B (Λ ′ ) is isomorphic to a disjoint union of highestweight crystals. Since ˜ e i ( u Λ ⊗ u Λ ′ ) = 0 for all i ∈ I , it is the highest weight element of aconnected component isomorphic to B (Λ + Λ ′ ). (cid:3) The following result gives a beautiful connection between Demazure and KR crystals;part (i) is due to [8] and (ii) to [23, Theorem 6.1]. Let w be the longest element of W . Theorem 3.2.
Let B = B r,c r s be a perfect KR crystal. There is a unique element b r,s ∈ B satisfying ε ( b r,s ) = s and ε i ( b r,s ) = 0 for i ∈ I . Put µ = c r w ( ̟ r ) and write t µ = yτ with y ∈ W , τ ∈ Σ . (i) There is a U ′ q ( g ) -crystal isomorphism θ : B ( s Λ ) ⊗ B ∼ = −→ B ( s Λ τ (0) ) which maps u s Λ ⊗ b r,s u s Λ τ (0) . Here, B ( s Λ ) and B ( s Λ τ (0) ) are regarded as U ′ q ( g ) -crystals by restriction—see § (ii) θ maps the subset u s Λ ⊗ B onto the Demazure crystal B y ( s Λ τ (0) ) ⊂ B ( s Λ τ (0) ) . Remark 3.3.
It is convenient to allow s = 0 in Theorem 3.2, which holds (trivially)with B r, = { b r, } defined to be the trivial U ′ q ( g )-crystal, i.e., wt( b r, ) = 0 and ˜ e i ( b r, ) =˜ f i ( b r, ) = 0 for all i ∈ I . Note that B (0Λ i ) = B (0) = { u } is the trivial U q ( g )-crystal. Lemma 3.4.
Let A and Z be U ′ q ( g ) -crystals. Let u ∈ A , z ∈ Z , and j , . . . , j m ∈ I , andsuppose that F j · · · F j m ( u ⊗ z ) ⊂ u ⊗ Z in A ⊗ Z . Then for any G = ˜ f d j · · · ˜ f d m j m , d i ∈ Z ≥ , G ( u ⊗ z ) = u ⊗ G ( z ) . HE DARK SIDE OF GENERALIZED DEMAZURE CRYSTALS 5
Proof.
The containment tells us that each application of ˜ f i in computing G ( u ⊗ z ) canonly be applied to u if ˜ f i ( u ) = 0, but this would mean φ i ( u ) = 0 and ˜ f i is applied on theleft tensor factor, which is not allowed by the tensor product rule (2.2). (cid:3) Lemma 3.5.
Maintain the notation of Theorem 3.2 and in addition let w ≤ y in Bruhatorder and let w = s j · · · s j m be a reduced expression for w . Let C be any U ′ q ( g ) -crystal.Then for any G = ˜ f d j · · · ˜ f d m j m , d i ∈ Z ≥ , and c ∈ C , there holds G ( u s Λ ⊗ b r,s ⊗ c ) = u s Λ ⊗ G ( b r,s ⊗ c ) in the U ′ q ( g ) -crystal B ( s Λ ) ⊗ B ⊗ C .Proof. By Theorem 3.2, u s Λ ⊗ B = θ − ( F y ( u s Λ τ (0) )) = F y ( θ − ( u s Λ τ (0) )) = F y ( u s Λ ⊗ b r,s ).Hence F j · · · F j m ( u s Λ ⊗ b r,s ) ⊂ F y ( u s Λ ⊗ b r,s ) = u s Λ ⊗ B in B ( s Λ ) ⊗ B . This implies F j · · · F j m ( u s Λ ⊗ b r,s ⊗ c ) ⊂ F j · · · F j m ( u s Λ ⊗ b r,s ) ⊗ F j · · · F j m ( c ) ⊂ u s Λ ⊗ B ⊗ C . Theresult then follows from Lemma 3.4 with A = B ( s Λ ), Z = B ⊗ C . (cid:3) Theorem 3.6.
Let B j = B r j ,c rj λ j for j ∈ [ p ] be perfect KR crystals with r = ( r , . . . , r p ) ∈ ( I ) p and λ = ( λ ≥ λ ≥ · · · ≥ λ p ≥ , and set λ j = λ j − λ j +1 with λ p +1 = 0 . Put µ j = c r j ω ( ̟ r j ) and write t µ j = y j τ j with y j ∈ W and τ j ∈ Σ . There is a strict embedding(see § U ′ q ( g ) -crystals Θ r ,λ : B ( λ Λ ) ⊗ B ⊗ · · · ⊗ B p → B ( λ Λ τ (0) ) ⊗ · · · ⊗ B ( λ p Λ τ τ ··· τ p (0) ) . (3.1) Proof.
Apply the isomorphism θ of Theorem 3.2 to the left two factors, then the strictembedding of Lemma 3.1, then apply F τ θ F − τ to the second and third factors, and so on: B ( λ Λ ) ⊗ B ⊗ B ⊗ · · · ⊗ B p ∼ = −→ B ( λ Λ τ (0) ) ⊗ B ⊗ · · · ⊗ B p ֒ → B ( λ Λ τ (0) ) ⊗ B ( λ Λ τ (0) ) ⊗ B ⊗ B ⊗ · · · ⊗ B p ∼ = −→ B ( λ Λ τ (0) ) ⊗ B ( λ Λ τ τ (0) ) ⊗ B ⊗ · · · ⊗ B p ֒ → B ( λ Λ τ (0) ) ⊗ B ( λ Λ τ τ (0) ) ⊗ B ( λ Λ τ τ (0) ) ⊗ B ⊗ · · · ⊗ B p · · ·→ B ( λ Λ τ (0) ) ⊗ B ( λ Λ τ τ (0) ) ⊗ · · · ⊗ B ( λ p Λ τ τ ··· τ p (0) ) . (cid:3) Theorem 3.7.
Maintain the notation of Theorem 3.6 and in addition let w , . . . , w p ∈ W with w i ≤ y i for all i . Then the subset B := F w (cid:0) b r ,λ ⊗ F τ F w (cid:0) b r ,λ ⊗ · · · F τ p − F w p ( { b r p ,λ p } ) · · · (cid:1)(cid:1) ⊂ B ⊗ · · · ⊗ B p (3.2) is well defined—we call it a DARK crystal—and the image of u λ Λ ⊗ B under the strictembedding Θ r ,λ is a generalized Demazure crystal: Θ r ,λ ( u λ Λ ⊗ B ) = F w F τ (cid:0) u λ Λ ⊗ F w F τ (cid:0) u λ Λ ⊗ · · · F w p F τ p ( { u λ p Λ } ) · · · (cid:1)(cid:1) . (3.3) Proof.
Choose reduced expressions w i = s j i, · · · s j i,mi for all i ∈ [ p ]. For now, interpret F w i in (3.2) as F j i, · · · F j i,mi . Then we can specify an arbitrary element of u λ Λ ⊗ B asin (3.4) by choosing arbitrary G i = ˜ f d i, j i, · · · ˜ f d i,mi j i,mi with d i, , . . . , d i,m i ∈ Z ≥ for i ∈ [ p ]. JONAH BLASIAK
Tracing through the maps making up Θ r ,λ , we obtain u λ Λ ⊗ G (cid:16) b r ,λ ⊗ F τ G (cid:0) b r ,λ ⊗ · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17) (3.4)= G (cid:16) u λ Λ ⊗ b r ,λ ⊗ F τ G (cid:0) b r ,λ ⊗ · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17) by Lemma 3.5 G (cid:16) u λ Λ τ ⊗ F τ G (cid:0) b r ,λ ⊗ · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17) by Theorem 3.2 (i)= G F τ (cid:16) u λ Λ ⊗ G (cid:0) b r ,λ ⊗ · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17) by § G F τ (cid:16) u λ Λ ⊗ u λ Λ ⊗ G (cid:0) b r ,λ ⊗ · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17) by Lemma 3.1= G F τ (cid:16) u λ Λ ⊗ G (cid:0) u λ Λ ⊗ b r ,λ ⊗ · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17) by Lemma 3.5 G F τ (cid:16) u λ Λ ⊗ G (cid:16) u λ Λ τ ⊗ F τ G (cid:0) · · · F τ p − G p ( b r p ,λ p ) (cid:1)(cid:17)(cid:17) by Theorem 3.2 (i) · · ·7→ G F τ (cid:16) u λ Λ ⊗ G F τ (cid:16) u λ Λ ⊗ G τ (cid:0) · · · G p F τ p ( u λ p Λ ) (cid:1)(cid:17)(cid:17) , which is an arbitrary element of the right side of (3.3). Moreover, by Theorem 2.2, theright side of (3.3) does not depend on the chosen reduced expressions for w i , so the samegoes for B since Θ r ,λ is injective. (cid:3) Let Z [ P ] denote the group ring of P with Z -basis { e µ } µ ∈ P . The Demazure operators arelinear operators D i on Z [ P ] defined for each i ∈ I by D i ( f ) = f − e − αi · s i ( f )1 − e − αi , where s i actson Z [ P ] by s i ( e µ ) = e s i ( µ ) . We also have an action of Σ on Z [ P ] given by τ ( e µ ) = e τ ( µ ) .For a reduced expression w = s i · · · s i m ∈ W , define the operator D w = D i · · · D i m on Z [ P ]; it is independent of the choice of reduced expression [13, Corollary 8.2.10].Naoi [20, Theorem 7.1] showed that Θ r ,λ matches the statistic h d, wt( b ) i on U q ( g )-crystals to energy. Combining this with Theorem 2.2 and [20, Corollary 4.6] we obtain Corollary 3.8.
Maintain the notation of Theorem 3.7. The energy adjusted characterof the DARK crystal B agrees with the character of the generalized Demazure crystal in (3.3) (call it B ′ ), and both have the following Demazure operator formula: e λ Λ + δC X b ∈B e aff(wt( b )) − δ D ( b ) a = X b ∈B ′ e wt( b ) = D w τ (cid:0) e λ Λ · D w τ (cid:0) e λ Λ · · · D w p τ p ( e λ p Λ ) (cid:1)(cid:1) , where D ( b ) is the energy of b and C ∈ Q is a constant which depends only on λ and r . Remark 3.9.
When w i = y i for all i ∈ [ p ], the DARK crystal B in (3.2) is equal to B ⊗ · · · ⊗ B p (this follows from F y i { b r i ,λ i } = B i and [20, Lemma 5.15]). Thus Proposi-tion 5.16/Corollary 7.2 of [20] are encompassed by Theorem 3.7/Corollary 3.8. Note thatthe a appearing in Corollary 3.8 corrects a typo in [20, Corollary 7.2]. Remark 3.10.
In Theorem 3.7 and Corollary 3.8, we can more generally allow w i of theform w i = v i w ′ i where v i ∈ W and w ′ i ≤ y i . Indeed, in the setting of Lemma 3.5, for any j ∈ I , ˜ f j G ( u s Λ ⊗ b r,s ⊗ c ) = ˜ f j ( u s Λ ⊗ G ( b r,s ⊗ c )) = u s Λ ⊗ ˜ f j G ( b r,s ⊗ c ) as ˜ f j ( u s Λ )= ˜ e j ( u s Λ ) = 0; hence the lemma holds more generally with w = vw ′ with v ∈ W , w ′ ≤ y . HE DARK SIDE OF GENERALIZED DEMAZURE CRYSTALS 7
For g of type A (1) n and r = = (1 , . . . , Corollary 3.11.
Let λ = ( λ ≥ · · · ≥ λ p ≥ and set λ j = λ j − λ j +1 with λ p +1 = 0 . Let τ be the Dynkin diagram automorphism given by j j + 1 mod n + 1 . Then there is astrict embedding of U ′ q ( b sl n +1 ) -crystals Θ ,λ : B ( λ Λ ) ⊗ B ,λ ⊗ · · · ⊗ B ,λ p → B ( λ Λ ) ⊗ · · · ⊗ B ( λ p Λ p ) . (3.5) Moreover, for any w , . . . , w p ∈ W , u λ Λ ⊗ F w (cid:0) b ,λ ⊗ F τ F w (cid:0) b ,λ ⊗ · · · F τ F w p ( { b ,λ p } ) · · · (cid:1)(cid:1) Θ ,λ w (cid:0) u λ Λ ⊗ F τ F w (cid:0) u λ Λ ⊗ · · · F τ F w p ( { u λ p Λ } ) · · · (cid:1)(cid:1) . This result is used in [1] to connect the katabolism operations of Lascoux [15] andShimozono-Weyman [25] to generalized Demazure crystals. The combinatorial significanceof Theorem 3.7 for more general r and in other types remains to be explored. Acknowledgments.
We thank Katsuyuki Naoi and Jennifer Morse for helpful discus-sions and Elaine So for help typing.
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