##
**Combinatorics of Coxeter groups.**
*(English)*
Zbl 1110.05001

Graduate Texts in Mathematics 231. New York, NY: Springer (ISBN 3-540-44238-3/hbk). xiv, 363 p. (2005).

A Coxeter group is a group defined by the finite presentation
\[
G=\langle a_1,\dots,a_n\mid a_i^2,(a_ia_j)^{m_{ij}},i\neq j\rangle.
\]
The general theory of Coxeter groups naturally involves combinatorics, geometry and algebra. The aim of the book under review is to present the core combinatorial aspects of the theory of Coxeter groups.

Chapter 1 gives basic definitions and examples. Elementary combinatorial facts underlying the rest of the book are derived. In Chapter 2 Bruhat order is introduced and its basic combinatorial properties are derived. Chapers 3 and 4 deal with problems concerning the combinatorics of reduced words. As a by-product, a computational device for working with reduced decompositions, the so-called “numbers game” is presented. The remaining four chapters are dealing with more advanced topics and are independent of each other. Some external references are necessary for Chapters 5–7. Chapter 8 is elementary and discusses permutation representations of the most important finite and affine Coxeter groups.

Each chapter contains a set of exercises. Easy exercises to test understanding of the material are well formulated and comprehensive. For harder exercises, which represent results from research literature, references are provided. Several open problems appear among the exercises, they are marked by an asterisk.

Endnotes to each chapter provide historical connections and references for related materials, including the algebraic and geometric aspects of the topics. Four appendices and a comprehensive bibliography containing 559 entries complete this book to a useful research monograph. Part I of the book is also well suited to serve as a graduate textbook.

Chapter 1 gives basic definitions and examples. Elementary combinatorial facts underlying the rest of the book are derived. In Chapter 2 Bruhat order is introduced and its basic combinatorial properties are derived. Chapers 3 and 4 deal with problems concerning the combinatorics of reduced words. As a by-product, a computational device for working with reduced decompositions, the so-called “numbers game” is presented. The remaining four chapters are dealing with more advanced topics and are independent of each other. Some external references are necessary for Chapters 5–7. Chapter 8 is elementary and discusses permutation representations of the most important finite and affine Coxeter groups.

Each chapter contains a set of exercises. Easy exercises to test understanding of the material are well formulated and comprehensive. For harder exercises, which represent results from research literature, references are provided. Several open problems appear among the exercises, they are marked by an asterisk.

Endnotes to each chapter provide historical connections and references for related materials, including the algebraic and geometric aspects of the topics. Four appendices and a comprehensive bibliography containing 559 entries complete this book to a useful research monograph. Part I of the book is also well suited to serve as a graduate textbook.

Reviewer: Herman J. Servatius (Worcester)

### MSC:

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

68R15 | Combinatorics on words |

### Online Encyclopedia of Integer Sequences:

Number of trees on n labeled nodes: n^(n-2) with a(0)=1.a(n) = (2n+1)^n.

a(n) = (2*n+1)^(n+1).

Triangle read by rows: T(n,k) = number of elements in the Coxeter group D_n with descent set contained in {s_k}, for 0<=k<=n-1.

The number of maximal paths in the Bruhat graph for S_n.

Number of isomorphism classes of Coxeter systems with n reflections