Computation in Coxeter Groups-I. Multiplication

An efficient and purely combinatorial algorithm for calculating products in arbitrary Coxeter groups is presented, which combines ideas of Fokko du Cloux and myself. Proofs are largely based on geometry. The algorithm has been implemented in practical Java programs, and runs surprisingly quickly. It seems to be good enough in many interesting cases to build the minimal root reflection table of Brink and Howlett, which can be used for a more efficient multiplication routine. MR subject classifications: 20H15, 20-04 Submitted March 28, 2001; accepted August 25, 2001. A Coxeter group is a pair (W, S) where W is a group generated by elements from its subset S, subject to relations (st)s,t = 1 for all s and t in S, where (a) the exponent ms,s = 1 for each s in S and (b) for all s 6= t the exponent ms,t is either a non-negative integer or ∞ (indicating no relation). Although there some interesting cases where S is infinite, in this paper no harm will be done by assuming S to be finite. Since ms,s = 1, each s in S is an involution: s = 1 for all s ∈ S . If we apply this to the other relations we deduce the braid relations: st . . . = ts . . . (ms,t terms on each side) . The array ms,t indexed by pairs of elements of S is called a Coxeter matrix. A pair of distinct elements s and t will commute if and only if ms,t = 2. The labeled graph whose nodes are elements of S, with an edge linking non-commuting s and t, labeled by ms,t, is called the associated Coxeter graph. (For ms,t = 3 the labels are often omitted.) Coxeter groups are ubiquitous. The symmetry group of a regular geometric figure (for example, any of the five Platonic solids) is a Coxeter group, and so is the Weyl group of any Kac-Moody Lie algebra (and in particular any finite-dimensional semi-simple Lie algebra). The Weyl groups of finite-dimensional semi-simple Lie algebras are those associated to the finite root systems An (n ≥ 1), Bn (n ≥ 2), Cn (n ≥ 2), Dn (n ≥ 4), En (n = 6, 7, 8), F4, and G2. The Coxeter groups determined by the affine root systems the electronic journal of combinatorics 9 (2002), #R25 1 associated to these are also the Weyl groups of affine Kac-Moody Lie algebras. The other finite Coxeter groups are the remaining dihedral groups Ip (p 6= 2, 3, 4, 6), as well as the symmetry group H3 of the icosahedron and the group H4, which is the symmetry group of a regular polyhedron in four dimensions called the 120-cell. In spite of their great importance and the great amount of effort spent on them, there are many puzzles involving Coxeter groups. Some of these puzzles are among the most intriguing in all of mathematics—suggesting, like the Riemann hypothesis, that there are whole categories of structures we haven’t imagined yet. This is especially true in regard to the polynomials Px,y associated to pairs of elements of a Coxeter group by Kazhdan and Lusztig in 1981, and the W -graphs determined by these polynomials. In another direction, the structure of Kac-Moody algebras other than the finite-dimensional or affine Lie algebras is still largely uncharted territory. There are, for example, many unanswered questions about the nature of the roots of a non-symmetrizable Kac-Moody Lie algebra which probably reduce to understanding better the geometry of their Weyl groups. The puzzles encountered in studying arbitrary Coxeter groups suggests that it would undoubtedly be useful to be able to use computers to work effectively with them. This is all the more true since many computational problems, such as computing Kazhdan-Lusztig polynomials, overwhelm conventional symbolic algebra packages. Extreme efficiency is a necessity for many explorations, and demands sophisticated programming. In addition to the practical interest in exploring Coxeter groups computationally, there are mathematical problems interesting in their own right involved with such computation. In this paper, I shall combine ideas of Fokko du Cloux and myself to explain how to program the very simplest of operations in an arbitrary Coxeter group—multiplication of an element by a single generator. As will be seen, this is by no means a trivial problem. The key idea is due to du Cloux, who has used it to design programs for finite Coxeter groups, and the principal accomplishment of this paper is a practical implementation of his idea without the restriction of finiteness. I have not been able to determine the efficiency of the algorithms in a theoretical way, but experience justifies my claims of practicality. It would seem at first sight that the techniques available for Coxeter groups are rather special. Nonetheless, it would be interesting to know if similar methods can be applied to other groups as well. Multiplication in groups is one place where one might expect to be able to use some of the extremely sophisticated algorithms to be found in language parsing (for example, those devised by Knuth to deal with LR languages), but I have seen little sign of this (in spite of otherwise interesting work done with, for example, automatic groups). For this reason, the results of this paper might conceivably be of interest to those who don’t care much about Coxeter groups per se. the electronic journal of combinatorics 9 (2002), #R25 2