Bigrassmannian permutations and Verma modules

Abstract

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $$w\in S_n$$ w ∈ S n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

Some Homological Properties of Category for Lie Superalgebras

“…where x , y are the shortest representatives in the cosets xW μ and yW μ , respectively; see [KMM2,Proposition 15]. For ν ∈ h * , we recall the weight ν + defined in (3-4).…”

Some homological properties of category $\mathcal O$ for Lie superalgebras

“…where x ′ , y ′ are the shortest representatives in the cosets xW µ and yW µ , respectively, see [KMM2,Proposition 15].…”

“…The recent paper [KMM2] discovered an interesting connection between bigrassmannian permutations and cokernels of inclusions of Verma modules for the Lie algebra sl n . The second motivation of this paper is to investigate to which extent this result generalizes to the super-setting.…”

“…In Section 4, we describe the socle of the cokernel of a nonzero homomorphism between Verma supermodules for pe(n). We manage to reduce the problem for pe(n) to the corresponding problem for gl(n), where the main result of [KMM2] applies.…”

“…Let λ be a dominant and regular weight. By Theorem 7, for any y ∈ W , the quotient ∆(λ)/∆(y • λ) has simple socle if and only if y is a bigrassmannian permutation, see[KMM2, Section 1.1, Theorem 1.3].…”