Explicit matrix representations for type D Coxeter groups.

Abstract

This paper presents some specific matrix representations of Coxeter groups of type <italic>D</italic>. To accomplish this, we describe how to construct an orthonormal basis for each irreducible representation, and produce a set of orthogonal matrices relative to that basis. In a second construction, we find a diagonal change of basis that produces matrices with rational entries. By using hereditary bases, we are guaranteed that the matrices constructed will be sparse, and also that the matrix entries in the orthogonal case are canonical up to sign. Similar matrix representations have been previously produced for the Coxeter groups of type <italic>A</italic>, type <italic>B/C</italic>, and the exceptional Weyl groups. This work completes the construction of explicit matrices for the irreducible representations of all finite Weyl groups.