Bigrassmannian permutations and Verma modules

TL;DR

This paper explores the relationship between bigrassmannian permutations and the structure of Verma modules in type A, revealing combinatorial and algebraic insights into their socular constituents and extensions.

Contribution

It establishes a connection between bigrassmannian permutations and socle structures of cokernels of Verma module inclusions, providing combinatorial and degree computations.

Findings

Socular constituents are indexed by Weyl group elements from the penultimate two-sided cell.

The socular constituents are determined by the essential set of the permutation.

Degrees of socular constituents are computable via the associated rank function.

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 $w\in S_n$ 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.