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 win 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.
Highlights
In Corollary 19 we show that the socular constituents in the cokernel of the inclusion of a Verma module indexed by w ∈ Sn into the dominant Verma module are determined by the essential set of w
In Proposition 22 we show how the associated rank function can be used to compute the degrees of these simple socular constituents in the graded picture
Y ∈ Sn, the coefficient of v in pw,y + py,w is denoted by μ(w, y) = μ(y, w), defining the (Kazhdan–Lusztig), μ-function
Summary
The associated KL combinatorics divides Sn into subsets, called two-sided cells, ordered with respect to the two-sided order ≤J. This coincides with the division of Sn given by the RobinsonSchensted correspondence: to each w ∈ Sn, we associate a pair (Pw, Qw) of standard Young tableaux of the same shape λ which is a partition of n, see [32,33,34]. The element wi, j is just the product of simple reflections along the unique path in the Dynkin diagram from the vertex corresponding to si to the vertex corresponding to w0s j w0, written from left to right
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