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

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 $\le 8$ (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.

26 pages; the second version contains minor corrections

References in corpus (1)

• Hecke algebras and involutions in Weyl groups

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• The development version of the CHEVIE package of GAP3
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• The low-dimensional homology of finite-rank Coxeter groups
• On the Complexity of Multiplication in the Iwahori--Hecke Algebra of the Symmetric Group
• On canonical bases and induction of $W$-graphs
• On the cells and associated varieties of highest weight Harish-Chandra modules