A note on the fusion product decomposition of Demazure modules
aa r X i v : . [ m a t h . R T ] F e b A NOTE ON THE FUSION PRODUCTDECOMPOSITION OF DEMAZURE MODULES
R. VENKATESH AND SANKARAN VISWANATH
Abstract.
We settle the fusion product decomposition theorem for higher level affineDemazure modules for the cases E (1)6 , , , F (1)4 and E (2)6 , thus completing the main theo-rems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). Weobtain a new combinatorial proof for the key fact, that was used in Chari et al. (opcit.), to prove this decomposition theorem. We give a case free uniform proof for thiskey fact. Introduction
Affine Demazure modules of higher level have been the subject of intensive study dueto their connection with the representation theory of quantum affine algebras [5], crystalbases [13], the X = M conjecture [11], Macdonald polynomials [1] and more.One of the key structural results concerning these modules is their decomposition intoa fusion product of smaller Demazure modules of the same level. This Steinberg typedecomposition theorem , proved in [4, 7, 14] (see also Theorem 1 below) plays a vitalrole in understanding their graded structure and has many applications. For instance,this is used in [2] to prove that certain level two affine Demazure modules in type A coincide with the prime representations of the quantum affine algebra introduced byHernandez-Leclerc [9] in the context of monoidal categorification of cluster algebras.The decomposition theorem also serves as a base case for the study of fusion productsof Demazure modules of unequal levels, and can be used to obtain defining relations forspecial modules of this kind [12].However, the decomposition theorem was only proved for the untwisted affine algebrasother than E (1)6 , , and F (1)4 in [4] and for twisted affine algebras other than E (2)6 in [7].The obstruction lay in a technical result concerning the action of the affine Weyl groupon weights, which was established in [4, Proposition 3.5] first for the untwisted affinesof type C and then by separate root sytem arguments in each of the types A, B, D and G ; additionally an appendix tabulated computational evidence in types E and F .In this note, we formulate a stronger version of this technical result (Proposition 1)and give a short, uniform proof for all affine types. We also point out in section 4 the RV is partially funded by the grants DST/INSPIRE/04/2016/000848, MTR/2017/000347, and anInfosys Young Investigator Award. SV is partially funded by the grant MTR/2019/000071. new cases of the decomposition theorem and other key results of [4] which now followas consequences. 2.
Preliminaries
We assume that the base field is complex numbers throughout the paper. Werefer to [10] for the general theory of affine Lie algebras. We denote by A an indecom-posable affine Cartan matrix, and by S the corresponding Dynkin diagram with thelabeling of vertices as in Table Aff 1 − S be the Dynkin dia-gram obtained from S by dropping the 0 th node and let ˚ A be the Cartan matrix, whoseDynkin diagram is ˚ S . Let g and ˚ g be the affine Lie algebra and the finite–dimensionalsimple Lie algebra associated to A and ˚ A , respectively. We shall realize ˚ g as a subalge-bra of g . We let h and ˚ h denote the Cartan subalgebras of g and ˚ g respectively, and let h ∗ R and ˚ h ∗ R denote their real forms.We let Φ and ˚Φ denote the sets of roots of g and ˚ g respectively. We fix Π = { α , . . . , α n } a basis for Φ such that ˚Π = { α , . . . , α n } is a basis for ˚Φ. The weightlattice (resp. coweight lattice) of Φ is denoted by P (resp. P ∨ ) and the set of dominantintegral weights is denoted by P + . Similarly, the weight lattice (resp. coweight lattice)of ˚Φ is denoted by ˚ P (resp. ˚ P ∨ ) and the set of dominant integral weights by ˚ P + . We recall the key facts about affine Weyl groups following[10] and [3]. Let W and ˚ W be the Weyl groups of g and ˚ g respectively. We denoteby s i the reflection associated to the simple root α i for 0 ≤ i ≤ n . Then we have W = h s , s , · · · , s n i and ˚ W = h s , · · · , s n i . Let h , i be the standard non-degeneratesymmetric invariant form on g . This determines a bijection h → h ∗ . There is an actionof W on h and h ∗ , compatible with this bijection.For each α ∈ ˚ h ∗ , we define t α : h ∗ → h ∗ by t α (Λ) = Λ + Λ( c ) α − ( h Λ , α i + h α, α i c )) δ for Λ ∈ h ∗ . Here c denotes the canonical central element of g and δ the unique indivisiblepositive imaginary root of Φ.Let θ denote the highest root of ˚Φ and define a = 2 if g is of type A ( ) ( n ≥ ) and a = 1 otherwise. Then W = ˚ W ⋉ t M , where t M = { t µ : µ ∈ M } and M is the sublatticeof ˚ h ∗ R generated by the elements w (cid:16) a θ (cid:17) for all w ∈ ˚ W. The explicit description of thelattice M can be found in [3, Page 414]. The affine Weyl group W acts on the set h ∗ R , = { Λ ∈ h ∗ R : Λ( c ) = 1 } and this induces an action on the orbit space h ∗ R , / R δ ∼ = ˚ h ∗ R .The element t µ ∈ M acts on ˚ h ∗ R as the translation x x + µ . Thus, W acts as affinetransformations on ˚ h ∗ R .Let H α ∨ ,k = { λ ∈ ˚ h ∗ : λ ( α ∨ ) = k } ; this defines an affine hyperplane in ˚ h ∗ for α ∈ ˚Φ , k ∈ Z , where α ∨ is the coroot corresponding to the root α . NOTE ON THE FUSION PRODUCT DECOMPOSITION OF DEMAZURE MODULES 3
Let ˚ H be the set of hyperplanes n H α ∨ , : α ∈ ˚Φ o . The connected components of ˚ h ∗ R − S H ∈ ˚ H H are the Weyl chambers of ˚ h ∗ R . The elements of ˚ W permute the hyperplanes in ˚ H and hence act on the set of Weyl chambers. The set C = n λ ∈ ˚ h ∗ R : λ ( α ∨ i ) > ≤ i ≤ n o is called the fundamental Weyl chamber. The map w w ( C ) gives a bijection from ˚ W to the set of Weyl chambers and the closure ¯ C is a fundamental region for the ˚ W actionon ˚ h ∗ R .Let H be a set of affine hyperplanes n H α ∨ ,k : α ∈ ˚Φ , k ∈ Z α o in ˚ h ∗ R , where the sets Z α are defined as follows (see [3, Page 414]): Z α = Z if g is of type B (1) n ( n ≥ , C (1) n ( n ≥ , F (1)4 and α is short3 Z if g is of type G (1)2 and α is short Z , if g is of type A (2)2 n ( n ≥
1) and α is long Z otherwiseThe connected components of ˚ h ∗ R − S H ∈H H are called the alcoves of ˚ h ∗ R . The elementsof W permute the affine hyperplanes in H and hence act on the set A of alcoves. Theset A = n λ ∈ ˚ h ∗ R : λ ( α ∨ i ) > ≤ i ≤ n, λ ( θ ∨ ) < /a o is called the fundamental alcove. The map w w ( A ) gives a bijection from W ontothe set of alcoves A . Moreover, the closure¯ A = n λ ∈ ˚ h ∗ R : λ ( α ∨ i ) ≥ ≤ i ≤ n, λ ( θ ∨ ) ≤ /a o is a fundamental region for the W action on ˚ h ∗ R . Finally, let Λ ∈ P + be the fundamentalweight corresponding to the 0 th vertex of the Dynkin diagram of g and let w denotethe unique longest element in ˚ W . We will need the following well-known elementary lemma.
Lemma 1.
Let F denote the set of alcoves contained in the fundamental Weyl chamber.Then ¯ C = S B ∈F ¯ B . The technical result
Set M + = M ∩ ¯ C. The following is our main technical result which is crucial in provingTheorem 1. We refer to [4, Section 4] and [7, Section 6] for more details.
Proposition 1.
Let g be an affine Lie algebra. Given ℓ ∈ N and λ ∈ ˚ P + , there exists µ ∈ M + and w ∈ ˚ W such that wt µ ( ℓ Λ − λ ) ∈ P + . R. VENKATESH AND SANKARAN VISWANATH
Proof.
Let λ ′ = − w ( λ ). Since λ ∈ ˚ P + and − w ( ˚ P + ) = ˚ P + , we have λ ′ ∈ ˚ P + . Thus,the element λ ′ /ℓ ∈ ¯ C. There exists B ∈ F such that λ ′ /ℓ ∈ ¯ B by Lemma 1. Since W acts simply transitively on the set of alcoves A , we have ¯ B = t µ ′ u ¯ A for some µ ′ ∈ M and u ∈ ˚ W . Since 0 ∈ ¯ A , we have µ ′ = t µ ′ u (0) ∈ ¯ B . This implies that µ ′ ∈ ¯ B ∩ M ⊆ ¯ C ∩ M = M + . Set µ = − w µ ′ ∈ M + and w = u − w ∈ ˚ W and consider wt µ ( ℓ Λ − λ ) ≡ ℓ Λ + w ( ℓµ − λ ) mod Z δ. We claim that wt µ ( ℓ Λ − λ ) ∈ P + . This will follow if we prove that (i) w ( ℓµ − λ ) ∈ ˚ P + and (ii) ℓ Λ ( α ∨ ) + w ( ℓµ − λ )( α ∨ ) ≥
0. Both these facts follow from the followingobservation: w ( ℓµ − λ ) = u − ( λ ′ − ℓµ ′ ) = ℓu − (cid:18) λ ′ ℓ − µ ′ (cid:19) = ℓu − t − µ ′ (cid:18) λ ′ ℓ (cid:19) ∈ ℓ ¯ A ∩ ˚ P. Since this belongs to ℓ ¯ A , we have w ( ℓµ − λ )( θ ∨ ) ≤ ℓ/a . Recalling that α ∨ = c − a θ ∨ [10, Chapter 6], we conclude ℓ + w ( ℓµ − λ )( α ∨ ) = ℓ − a w ( ℓµ − λ )( θ ∨ ) ≥
0, proving (ii).Since ℓ ¯ A ∩ ˚ P ⊆ ˚ P + we have w ( ℓµ − λ ) ∈ ˚ P + , proving (i). (cid:3) Remarks
In the interest of completeness, we briefly point out the new cases of the main resultsof [4] which hold, now that Proposition 1 has been established.
As before, let ˚ g be a finite-dimensional simple Lie algebra and g the corre-sponding untwisted affine Lie algebra. We consider the Demazure modules of g thatare stable under the current algebra ˚ g ⊗ C [ t ]. These modules are parametrized by pairs( ℓ, λ ) with ℓ a positive integer and λ a dominant integral weight of ˚ g . Let D ( ℓ, λ ) denotethe Demazure module corresponding to the pair ( ℓ, λ ). Theorem 1.
Let ˚ g be one of E , E , E or F . Let λ ∈ ˚ P + , ℓ ∈ N and suppose that λ = ℓ ( P ki =1 λ i ) + λ with λ ∈ ˚ P + and λ i ∈ ( ˚ P ∨ ) + for ≤ i ≤ k . Then there is anisomorphism of ˚ g [ t ] –modules D ( ℓ, λ ) ∼ = D ( ℓ, ℓλ ) ∗ D ( ℓ, ℓλ ) ∗ · · · ∗ D ( ℓ, ℓλ k ) ∗ D ( ℓ, λ )Here ∗ denotes the fusion product, and we refer to [4, Section 2.7] for the definitionof fusion products of ˚ g ⊗ C [ t ]–modules. As mentioned earlier, this was established in[4, Theorem 1] for all untwisted affines other than those of types E, F and in [7] forthe twisted cases (modulo an extra hypothesis for E (2)6 ). It now also follows that [7,Theorem 7] holds without this extra condition. NOTE ON THE FUSION PRODUCT DECOMPOSITION OF DEMAZURE MODULES 5
Now let ˚ g be simply-laced, so that ( ˚ P ∨ ) + = ˚ P + . Let α i and ω i for 1 ≤ i ≤ n denote the simple roots and fundamental weights respectively of ˚ g . Given ℓ ≥ λ ∈ ˚ P + , there is a unique decomposition λ = ℓ ( P ni =1 m i ω i ) + λ where λ = P ni =1 r i ω i with 0 ≤ r i < ℓ and m i ≥ i . Theorem 1 implies that for ˚ g of type E , one has D ( ℓ, λ ) ∼ = ˚ g [ t ] D ( ℓ, ℓω ) ∗ m ∗ D ( ℓ, ℓω ) ∗ m ∗ · · · ∗ D ( ℓ, ℓω n ) ∗ m n ∗ D ( ℓ, λ ) (4.1)Recall that a ˚ g [ t ]-module is said to be prime if it is not isomorphic to a fusion productof non-trivial ˚ g [ t ] modules. In conjunction with [4, Proposition 3.9], Theorem 1 yieldsthe following corollary: Corollary 1.
Let ˚ g be one of E , E or E . The decomposition (4.1) gives a primefactorization of the Demazure module D ( ℓ, λ ) , i.e., an expression as a fusion product ofprime ˚ g [ t ] -modules. This was established in [4] for types A and D . A second corollary concerns the notion of a Q -system [8]. Roughly speaking a Q -system is a short exact sequence of ˚ g -modules:0 → O j ∼ i V ( ℓω j ) → V ( ℓω i ) ⊗ V ( ℓω i ) → V (( ℓ + 1) ω i ) ⊗ V (( ℓ − ω i ) → ω i is a miniscule weight of ˚ g and j ∼ i means that nodes i and j are connected byan edge in the Dynkin diagram. Generalizations of Q -systems considered in [4, 5, 6, 7]involve replacing the tensor products above by fusion products of certain ˚ g [ t ]-modules.The following result was established in [4, Section 5]: Proposition 2. ( [4] ) Let ˚ g be simply-laced. Let ℓ ≥ , λ ∈ ˚ P + with ℓ ≥ max { λ ( α ∨ ) : α ∈ ˚Φ + } and suppose ω i is a miniscule weight such that λ ( α ∨ i ) > . Let µ = ℓω i + λ − λ ( α ∨ i ) α i .Then there exists a short exact sequence of ˚ g [ t ] -modules: → τ ∗ λ ( α ∨ i ) D ( ℓ, µ ) → D ( ℓ, λ + ℓω i ) → D ( ℓ + 1 , λ + ℓω i ) → V , τ ∗ d V denotes V with its grading shifted by d ; we refer to [4]for a fuller explanation of the notations. Proposition 2 and Theorem 1 together implythe following generalized Q -system in type E : Corollary 2.
Assume that ˚ g is of type E or E , and retain the other notations ofProposition 2. Write µ = ℓµ + µ for some µ , µ ∈ ˚ P + . Then there exists a naturalshort exact sequence of ˚ g [ t ] -modules: → τ ∗ λ ( α ∨ i ) ( D ( ℓ, ℓµ ) ∗ D ( ℓ, µ )) → D ( ℓ, ℓω i ) ∗ D ( ℓ, λ ) → D ( ℓ + 1 , ( ℓ + 1) ω i ) ∗ D ( ℓ + 1 , λ − ω i ) → . We note that ˚ g = E is excluded since it does not have miniscule weights. This resultwas established in types A, D in [4, Theorem 2].
R. VENKATESH AND SANKARAN VISWANATH
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Department of Mathematics, Indian Institute of Science, Bangalore 5600112
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