Stratifying KLR algebras of affine ADE types

Abstract

We generalize imaginary Howe duality for KLR algebras of affine ADE types, developed in our previous paper, from balanced to arbitrary convex preorders. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that these KLR algebras are properly stratified.Dedicated to the memory of Professor J.A. Green.

Affine zigzag algebras and imaginary strata for KLR algebras

“…Suppose for a moment that is a field. It is shown in [20,22] that the R nδ -module ∆ •n δ = Ind δ,...,δ (∆ ⊠n δ ) factors through to a projective C nδ -module, and ∆ •n δ is a projective generator for C nδ if char = 0 or char > n. We will build on the previous section to explicitly describe for all n the algebra End C nδ (∆ •n δ ) as the rank n affine zigzag algebra Z aff n (Γ), defined in §4.2, where Γ is the finite type Dynkin diagram of type C ′ . This gives a Morita equivalence between C nδ and Z aff n when char = 0 or char > n. In fact, our proof that End C nδ (∆ •n δ ) ∼ = Z aff n (Γ) works over any commutative until ground ring .…”

“…From now on until the end of this subsection we assume that is a field. The irreducible C nδmodules are parametrized canonically by the l-multipartitions λ ∈ P n , see [19,20,22,26]. The irreducible corresponding to λ is denoted by L(λ), and its projective cover in C nδ -mod is denoted ∆(λ).…”

“…Proof. By the Mackey Theorem [15,Proposition 2.18] (see also [20,Theorem 4.3]), the restriction Res δ,...,δ Ind δ,...,δ (∆ ⊠n δ ) has filtration with n! subquotients all of which are isomorphic to ∆ ⊠n δ .…”

“…Proof. In this situation the module ∆ •n δ is a projective generator for B nδ , see [20,Lemma 6.22], so B nδ is Morita equivalent to End C nδ (∆ •n δ ) op ∼ = (Z aff n ) op . Then the result follows by Lemma 4.11.…”

“…McNamara [22] shows that these algebras are explicitly properly stratified if p = 0. McNamara's result is generalized in [20] to the case where p > min{n i | i ∈ I}.…”