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\(\mathsf{PyCox}\): computing with (finite) Coxeter groups and Iwahori-Hecke algebras. (English) Zbl 1296.20009
Summary: We introduce the computer algebra package \mathsfPyCox, written entirely in the \mathsfPython language. It implements a set of algorithms, in a spirit similar to the older \mathsfCHEVIE 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 \(\leqslant 8\) (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.

MSC:
20C40 Computational methods (representations of groups) (MSC2010)
20C08 Hecke algebras and their representations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
68W30 Symbolic computation and algebraic computation
Software:
CHEVIE; GAP; PyCox
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