PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras

Abstract

We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older 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 8 (except E8). We also discuss the computation of Lusztig's leading coefficients of character values and distinguished involutions (which works for E8 as well). Our experiments suggest a re-definition of Lusztig's 'special' representations which, conjecturally, should also apply to the unequal parameter case.

Lifting of elements of Weyl groups

“…If n = 2, so G is of type D 4 , the three elements x, y, z of Spin(8) are related by automorphisms of Spin (8). Since Spin(8)/ z ≃ SO(8), we conclude Semispin(8) ≃ SO(8), and W lifts by the previous discussion.…”

“…The only other exceptional case is D 4 . It follows from a tedious and not very enlightening argument that W lifts to SO (8), uniquely up to T -conjugacy and multiplication by −I, and the lifting to P SO(8) is unique up to T -conjugacy. We leave the details to the reader.…”

“…See [8,Section 3.4] or [7,Section 3] for the untwisted cases, and [6,Section 3 & 4] or [9,Section 7] for the twisted ones.…”

Determination of the Representations and a Basis for the Yokonuma–Temperley–Lieb Algebra

“…The last result can be also independently obtained with the use of Tits's deformation theorem (see, for example, [GePf,Theorem 7.4.6…”

Imprimitive irreducible modules for finite quasisimple groups. II

“…is of type A, and f ψ is a power of 2, otherwise (see [14,Corollary 9.3.6] and [20, (4.1.1),(4.14.2)]). Here, a ψ is the value of Lusztig's a-function associated to ψ.…”