Representation Theory of Finite Monoids

Authors: Benjamin Steinberg



An entire part of the text is devoted to applications to Markov chains, combinatorics, and automata theory

Accessible to a wide readership of graduate students and researchers, including non-experts in semigroups

Contains exercises, chapter notes, and thoroughly worked examples  

This first text on the subject provides a comprehensive introduction to the representation theory of finite monoids. Carefully worked examples and exercises provide the bells and whistles for graduate accessibility, bringing a broad range of advanced readers to the forefront of research in the area. Highlights of the text include applications to probability theory, symbolic dynamics, and automata theory. Comfort with module theory, a familiarity with ordinary group representation theory,  and the basics of Wedderburn theory, are prerequisites for advanced graduate level study. Researchers in algebra, algebraic combinatorics, automata theory, and probability theory, will find this text enriching with its thorough presentation of applications of the theory to these fields.  

Prior knowledge of semigroup theory is not expected for the diverse readership that may benefit from this exposition. The approach taken in this book is highly module-theoretic and follows the modern flavor of the theory of finite dimensional algebras. The content is divided into 7 parts. Part I consists of 3 preliminary chapters with no prior knowledge beyond group theory assumed. Part II forms the core of the material giving a modern module-theoretic treatment of the Clifford –Munn–Ponizovskii theory of irreducible representations. Part III concerns character theory and the character table of a monoid. Part IV is devoted to the representation theory of inverse monoids and categories and Part V presents the theory of the Rhodes radical with applications to triangularizability. Part VI features 3 chapters devoted to applications to diverse areas of mathematics and forms a high point of the text. The last part, Part VII, is concerned with advanced topics. There are also 3 appendices reviewing finite dimensional algebras, group representation theory, and Möbius inversion.

Table of contents (21 chapters)

Front Matter

Elements of Monoid Theory
Front Matter
1 The Structure Theory of Finite Monoids
2 RR -trivial Monoids
3 Inverse Monoids

Irreducible Representations
Front Matter
4 Recollement: The Theory of an Idempotent
5 Irreducible Representations

Character Theory
Front Matter
6 The Grothendieck Ring
7 Characters and Class Functions

The Representation Theory of Inverse Monoids
Front Matter
8 Categories and Groupoids
9 The Representation Theory of Inverse Monoids

The Rhodes Radical
Front Matter
10 Bi-ideals and R. Steinberg’s Theorem
11 The Rhodes Radical and Triangularizability

Applications
Front Matter
12 Zeta Functions of Languages and Dynamical Systems
13 Transformation Monoids
14 Markov Chains

Advanced Topics
Front Matter
15 Self-injective, Frobenius, and Symmetric Algebras
16 Global Dimension
17 Quivers of Monoid Algebras
18 Further Developments
Appendix A Finite Dimensional Algebras
Appendix B Group Representation Theory
Appendix C Incidence Algebras and Möbius Inversion

Back Matter

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