Bigrassmannian permutations and Verma modules

2020/08/20 by Hankyung Ko, Volodymyr Mazorchuk, Rafael Mrđen
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Extension (predicate logic) #Rank (graph theory) #Set (abstract data type) #Simple (philosophy) #Type (biology) #Verma module #Weyl group #math.CO #math.RT

paper · pdf · doi:10.1007/s00029-021-00672-z

published as Selecta Mathematica volume 27:55 (2021) · 18 pages

arxiv created 2020/08/20 · openalex created_date 2020/08/24 · openalex publication_date 2021/06/21 · arxiv updated 2021/06/22 · openalex updated_date 2026/08/05

Abstract

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∈ S_n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> 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.

Citations

• Koszul Duality Patterns in Representation Theory
• Flags, Schubert polynomials, degeneracy loci, and determinantal formulas
• Enumeration of Bigrassmannian Permutations Below a Permutation in Bruhat Order
• On the Representations of the Symmetric Group
• Longest Increasing and Decreasing Subsequences