An Algorithmic Approach To Crystallographic Coxeter Groups

Abstract

Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. It turns out that the finite Coxeter groups are precisely the finite Euclidean reflection groups. Coxeter studied these groups and classified all finite ones in 1935, however they were known as reflection groups until J. Tits coined the term Coxeter groups for them in the sixties. Finite crystallographic Coxeter groups, also known as finite Weyl groups, play a prominent role in many branches of mathematics like combinatorics, Lie theory, number theory, and geometry. The computational aspects of these groups are of great interests and play a very important role in representation theory. Since it’s enough to study only the irreducible class of groups in order to understand any Coxeter group, we discuss irreducible crystallographic Coxeter groups here. Our goal is to try to deal with some of the fundamental computational problems that arise in working with the structures such as Weyl groups, root system, Weyl characters. For the classical cases, especially type A, many of these problems are not very subtle and have been solved completely. However, these solutions often do not generalize. In this report, our emphasis is on algorithms which do not really depend on the classifications of root systems. The canonical example, we always keep in mind is E8. In chapter 1, we fix the notations and give some basic results which have been used in this report. In chapter 2, we explain algorithms to various Weyl group problems like membership problem; how to find the length of an element; how to check if two words in a Weyl group represent the same element or not; finding the coset representative for an element for a given parabolic subgroup; and list all the expressions possible for an element. In chapter 3, the main goal is to write an algorithm to compute the weight multiplicities of the irreducible representations using Freudenthal’s formula. For this, we first compute the positive roots and dominant weights for a given root system and then finally find the weight multiplicities. We argue this mathematically using the results given in chapter 1. The crystallographic hypothesis is unnecessary for much of what is discussed in chapter 2. In the last chapter, we give codes of the computer programs written in C++ which implement the algorithms described in the previous chapters in this report.