Combinatorics of Coxeter Groups

Coxeter groups are groups which are generated by involutions. More formally, we define a Coxeter group by a presentation with generators r₁ ...r_n with the relation that r_ir_j has order m_ij where m_ii = 1 for all i (forcing the generators to all be involutions) and m_ij to be greater than 1 (and is allowed to be infinite, meaning there will be no relation between r_i and r_j) if i is not equal to j. Coxeter groups can be encoded as symmetric matrices (whose entries are the m_ij from above) or as graphs where the vertices represent the generators and the edges are labeled depending on the value of m_ij. Examples of Coxeter groups include the symmetric groups, the dihedral groups, and all Weyl groups, and the theory of Coxeter groups has many applications in algebra, geometry, and combinatorics.

It is this last group of applications which the new book by Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups is concerned with. They are particularly interested in the partial order structure that these groups possess, which is known as Bruhat order, and the ability to describe the lengths of elements using this ordering. They devote several chapters of the book to developing the idea of this partial ordering, and to finding reduced expressions of words and enumerating the structures of these groups. The first half of the book consists of four chapters which are written in a very expository manner, and serve as a very nice introduction to the subject which a graduate student would be able to follow. These chapters are self-contained, and the book has several appendices to help the reader who does not have the full background.

Darren Glass teaches at Gettysburg College.