Intersecting principal Bruhat ideals and grades of simple modules

TL;DR

This paper establishes a combinatorial link between grades of simple modules indexed by boolean permutations and Lusztig's a-function, using intersection properties of Bruhat order ideals and the Robinson-Schensted correspondence.

Contribution

It provides a new combinatorial characterization of module grades and identifies conditions for modules to be perfect over the incidence algebra of the symmetric group.

Findings

Grades of simple modules match Lusztig's a-function for boolean permutations.

Intersection of principal order ideals described combinatorially.

Characterization of perfect modules as those indexed by longest elements of parabolic subgroups.

Abstract

We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's a-function. Our arguments are combinatorial, and include a description of the intersection of two principal order ideals when at least one permutation is boolean. An important object in our work is a reduced word written as minimally many runs of consecutive integers, and one step of our argument shows that this minimal quantity is equal to the length of the second row in the permutation's shape under the Robinson-Schensted correspondence. We also prove that a simple module over the above-mentioned incidence algebra is perfect if and only if its index is the longest element of a parabolic subgroup.