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

@article{Geck2012PyCoxCW, title={PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras}, author={Meinolf Geck}, journal={Lms Journal of Computation and Mathematics}, year={2012}, volume={15}, pages={231-256} }

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 E 8 ). We also discuss the computation of Lusztig’s leading coefficients of character values and… Expand

#### 21 Citations

On the Kazhdan-Lusztig cells in type E8

- Computer Science, Mathematics
- Math. Comput.
- 2015

Algorithmic aspects of this theory for finite W which are important in applications are addressed, e.g., run explicitly through all left cells, determine the values of Lusztig’s a-function, identify the characters of left cell representations. Expand

On the complexity of multiplication in the Iwahori-Hecke algebra of the symmetric group

- Mathematics, Computer Science
- J. Symb. Comput.
- 2017

This work presents new efficient data structures for elements of Coxeter groups of type Am and their associated IwahoriHecke algebras H(Am), which leads to a huge improvement in the cost of multiplication in H( Am). Expand

Title On the complexity of multiplication in the Iwahori-Heckealgebra of the symmetric group

- 2018

We present new efficient data structures for elements of Coxeter groups of type Am and their associated Iwahori–Hecke algebras H(Am). Usually, elements of H(Am) are represented as simple coefficient… Expand

Conjugacy classes of involutions and Kazhdan-Lusztig cells

- Mathematics
- 2012

According to an old result of Sch\"utzenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this… Expand

W-graphs for Hecke algebras with unequal parameters

- Mathematics
- 2015

In (Howlett and Nguyen in J Algebra 361:188–212, 2012), the concept of W-graph ideal in a Coxeter group was introduced, and it was showed that a W-graph can be constructed from a given W-graph ideal… Expand

On Kottwitz' conjecture for twisted involutions

- Mathematics
- 2012

Kottwitz' conjecture is concerned with the intersections of Kazhdan--Lusztig cells with conjugacy classes of involutions in finite Coxeter groups. In joint work with Bonnaf\'e, we have recently found… Expand

Hecke algebras with unequal parameters and Vogan's left cell invariants

- Mathematics
- 2015

In 1979, Vogan introduced a generalised τ-invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of… Expand

Vogan classes in type $B_n$.

- Mathematics
- 2017

Consider a weighted Coxeter system $(W,S,\mathscr{L})$. Via its associated Iwahori-Hecke algebra, we may determine the partition of $W$ into Kazhdan-Lusztig cells. In this paper, we use the theory of… Expand

ON CANONICAL BASES AND INDUCTION OF $W$-GRAPHS

- Mathematics
- Nagoya Mathematical Journal
- 2018

A canonical basis in the sense of Lusztig is a basis of a free module over a ring of Laurent polynomials that is invariant under a certain semilinear involution and is obtained from a fixed “standard… Expand

Finding characters satisfying a maximal condition for their unipotent support

- Mathematics
- 2014

In this article we extend independent results of Lusztig and Hezard concerning the existence of irreducible characters of finite reductive groups (defined in good characteristic and arising from… Expand

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Leading coefficients and cellular bases of Hecke algebras

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Abstract Let H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer,… Expand

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CHEVIE is a computer algebra package which collects data and programs for the representation theory of finite groups of Lie type and associated structures. We explain the theoretical and conceptual… Expand

Positivity results for the Hecke algebras of non-crystallographic finite Coxeter groups

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Inducing W-graphs

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Kazhdan--Lusztig cells and the Frobenius--Schur indicator

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Let $W$ be a finite Coxeter group. It is well-known that the number of involutions in $W$ is equal to the sum of the degrees of the irreducible characters of $W$. Following a suggestion of Lusztig,… Expand