Algorithms for Garside calculus

Garside calculus is the common mechanism that underlies a certain type of normal form for the elements of a monoid, a group, or a category. Originating from Garside’s approach to Artin’s braid groups, it has been extended to more and more general contexts, the latest one being that of categories and what are called Garside families. One of the benefits of this theory is to lead to algorithms solving effectively the naturally occurring problems, typically the Word Problem. The aim of this paper is to present and solve these algorithmic questions in the new extended framework. In 1969, F.A.Garside [20] solved the Word and Conjugacy Problems in Artin’s braid group Bn [2] by describing the latter as a group of fractions and analyzing the involved monoid in terms of its divisibility relation. This approach was continued and extended in several steps, first to Artin-Tits groups of spherical type [3, 16, 1, 24, 19, 18, 4], then to a larger family of groups now known as Garside groups [10, 8]. More recently, it was realized that going to a categorical context allows for capturing further examples [17], and a coherent theory has recently emerged with a central unifying notion called Garside families [14, 15]: the central notion is a certain way of decomposing the elements of the reference category or its groupoid of fractions and a Garside family is what makes the construction possible. What we do in this paper is to present and analyse the main algorithms arising in this new, extended context of Garside families, with two main directions, namely recognizing that a candidate family is a Garside family and using a Garside family to compute in the category, typically finding distinguished decompositions and solving the Word Problem along the lines of [11]. This results in a corpus of about twenty algorithms that are proved to be correct, analysed, and given examples. We do not address the Conjugacy Problem here, as extending the methods of [21, 22] will require further developments that we keep for a subsequent work. The paper consists of five sections. Section 1 is a review of Garside families and the derived notions involved in the approach, together with some basic results that appear in other sources. Next, we address the question of effectively recognizing Garside families and we describe and analyse algorithms doing it: in Section 2, we consider the case when the ambient category is specified using a presentation (of a certain type), whereas, in Section 3, we consider the alternative approach when the category is specified using what is called a germ. Finally, the last two sections are devoted to those computations that can be developed once a Garside family is given. In Section 4, we consider computations taking place in the reference category or monoid (“positive case”), whereas, in Section 5, we address similar questions in the groupoid or group of fractions of the reference category (“signed case”). 1. The general context In this introductory section, we present the background of categories and Garside families, together with some general existence and uniqueness results that will be used and, often, refined in the sequel of the paper. Proofs appear in other sources and they will be omitted in general. 1991 Mathematics Subject Classification. 20F10, 18B40, 20F36, 68Q17.