PyCox: Computing with (finite) Coxeter groups and Iwahori-Hecke algebras

TL;DR

PyCox is a Python-based computer algebra system for Coxeter groups and Hecke algebras, featuring efficient algorithms for Kazhdan--Lusztig cells, W-graphs, and Lusztig's character coefficients, extending to complex cases like E8.

Contribution

Introduces PyCox, a Python package with new algorithms for Coxeter groups and Hecke algebras, including efficient Kazhdan--Lusztig computations for all finite groups up to rank 8.

Findings

Efficient computation of Kazhdan--Lusztig cells and W-graphs for finite Coxeter groups.

Successful calculation of Lusztig's leading coefficients and involutions, including E8.

Proposes a re-definition of Lusztig's 'special' representations, applicable to unequal parameters.

Abstract

We introduce the computer algebra package {\sf PyCox}, written entirely in the {\sf Python} language. It implements a set of algorithms - in a spirit similar to the older {\sf CHEVIE} system - for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan--Lusztig cells and $W$-graphs, which works efficiently for all finite groups of rank $\leq 8$ (except $E_8$). We also discuss the computation of Lusztig's leading coefficients of character values and distinguished involutions (which works for $E_8$ as well). Our experiments suggest a re-definition of Lusztig's "special" representations which, conjecturally, should also apply to the unequal parameter case.