A Transducer Approach to Coxeter Groups

Let (W; S) be a Coxeter system ((1] ch. 4), where we assume S to be nite, with n elements , n 1. The theory of these groups exhibits a deep interplay between geometry and combinatorics; in fact, many basic combinatorial facts about them are most conveniently proved using an explicit geometric realization, and are best understood in that setting. This is also the approach taken by most computer programs dealing with these groups (say in the nite case, which is the principal case of interest in this paper.) In contrast, we would like to show here that once the exchange condition is known, all the computations in these groups can be explicitly handled at the combinatorial level, and point out in particular how parabolic decompositions appear naturally in these questions. This is particularly eecient in the case of nite groups, where we obtain a cascade of very small transducers (cf. sect. 3) handling the main \word processing" problems that one would like to deal with. In addition, we show how parabolic decompositions lead to a very eecient determination of the Bruhat order | a further indication of their relevance in computational questions. On a theoretical level, the results that we use are due to Deodhar ((3], 4]); our contribution has been the realization of their practical value in terms of computer implementations. We also present a general algorithm for nding the normal form (cf. sect. 1) of an arbitrary word in a general Coxeter group. We did not attempt to analyze the complexity of this algorithm in general, but it certainly becomes very eecient when used for the construction of the transducer tables by induction on the length; once this is done, the general algorithm is not needed any more. Of course the transducer tables could also have been constructed using, for instance, a geometric realization of the group, but our algorithm has the advantages of simplicity, of handling the non-cristallographic cases with equal ease, and may also yield some additional information on the structure of normal forms. It is natural in this setting to ask about the Knuth-Bendix relations for the presentation (cf. sect. 4). It turns out that for the nite Coxeter groups they can be readily read oo from our transducer tables. We have indicated them in order to give some examples of complete presentations that appear to be inaccessible to the general-purpose methods used for …