**Fields Institute Monographs**

Volume: 23;
2006;
181 pp;
Softcover

MSC: Primary 05; 06; 16; 20; 51;

Print ISBN: 978-0-8218-5354-2

Product Code: FIM/23.S

List Price: $62.00

Individual Member Price: $49.60

**Electronic ISBN: 978-1-4704-1787-1
Product Code: FIM/23.E**

List Price: $62.00

Individual Member Price: $49.60

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#### Supplemental Materials

# Coxeter Groups and Hopf Algebras

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*Marcelo Aguiar; Swapneel Mahajan*

A co-publication of the AMS and Fields Institute

An important idea in the work of G.-C. Rota is
that certain combinatorial objects give rise to Hopf algebras that
reflect the manner in which these objects compose and
decompose. Recent work has seen the emergence of several interesting
Hopf algebras of this kind, which connect diverse subjects such as
combinatorics, algebra, geometry, and theoretical physics. This
monograph presents a novel geometric approach using Coxeter complexes
and the projection maps of Tits for constructing and studying many of
these objects as well as new ones. The first three chapters introduce
the necessary background ideas making this work accessible to advanced
graduate students. The later chapters culminate in a unified and
conceptual construction of several Hopf algebras based on
combinatorial objects which emerge naturally from the geometric
viewpoint. This work lays a foundation and provides new insights for
further development of the subject.

To read more about Coxeter groups
see The Coxeter Legacy:
Reflections and Projections.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Readership

Graduate students and research mathematicians interested in algebraic combinatorics, Coxeter groups, and Hopf algebras.

#### Reviews & Endorsements

This monograph is the first to present structural interactions between the theory of Coexter groups and the theory of Hopf algebras. ...It is not just the results but also a unified conceptual treatment of the theory that make this monograph a very valuable one.

-- EMS Newsletter

Despite the formidable notational complexity, the book is well-organized and quite readable. In particular, there is a useful notation index.

-- Earl J. Taft for Zentralblatt MATH

This is a fascinating research monograph with many new and interesting ideas for the combinatorial study of algebras.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Coxeter Groups and Hopf Algebras

- Cover Cover11
- Title page i2
- Foreword iii4
- Contents v6
- List of tables xi12
- List of figures xiii14
- Preface xv16
- Coxeter groups 118
- Left regular bands 1330
- Hopf algebras 3148
- A brief overview 3956
- The descent theory for Coxeter groups 4360
- The construction of Hopf algebras 8198
- The Hopf algebra of pairs of permutations 133150
- The Hopf algebra of pointed faces 155172
- Bibliography 165182
- Author index 171188
- Notation index 173190
- Subject index 177194
- Back Cover Back Cover1201