Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

TL;DR

This paper presents an efficient algorithm for computing Kazhdan-Lusztig polynomials and explores their applications in understanding characters of reductive groups and properties of finite groups, including counterexamples to longstanding conjectures.

Contribution

It introduces a practical algorithm for parabolic Kazhdan-Lusztig polynomial computation and applies it to derive new insights into finite group representations and subgroup structures.

Findings

Computed Kazhdan-Lusztig polynomials relevant to character formulas for reductive groups.

Discovered finite groups with unusually high first cohomology groups.

Constructed finite groups with many maximal subgroups, countering G.E. Wall's conjecture.

Abstract

We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig. Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases. Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conjecture by G.~E.~Wall.