Representation Theory of Finite Monoids

  • Benjamin Steinberg

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Elements of Monoid Theory

    1. Front Matter
      Pages 1-1
    2. Benjamin Steinberg
      Pages 3-16
    3. Benjamin Steinberg
      Pages 17-24
    4. Benjamin Steinberg
      Pages 25-37
  3. Irreducible Representations

    1. Front Matter
      Pages 39-39
    2. Benjamin Steinberg
      Pages 41-52
    3. Benjamin Steinberg
      Pages 53-91
  4. Character Theory

    1. Front Matter
      Pages 93-93
    2. Benjamin Steinberg
      Pages 95-102
    3. Benjamin Steinberg
      Pages 103-122
  5. The Representation Theory of Inverse Monoids

    1. Front Matter
      Pages 123-123
    2. Benjamin Steinberg
      Pages 125-135
    3. Benjamin Steinberg
      Pages 137-152
  6. The Rhodes Radical

    1. Front Matter
      Pages 153-153
    2. Benjamin Steinberg
      Pages 155-162
    3. Benjamin Steinberg
      Pages 163-173
  7. Applications

    1. Front Matter
      Pages 175-175
    2. Benjamin Steinberg
      Pages 177-189
    3. Benjamin Steinberg
      Pages 191-204
    4. Benjamin Steinberg
      Pages 205-225
  8. Advanced Topics

    1. Front Matter
      Pages 227-227
    2. Benjamin Steinberg
      Pages 229-234
    3. Benjamin Steinberg
      Pages 235-244
    4. Benjamin Steinberg
      Pages 245-262
    5. Benjamin Steinberg
      Pages 263-268
    6. Benjamin Steinberg
      Pages 269-278
    7. Benjamin Steinberg
      Pages 279-289
    8. Benjamin Steinberg
      Pages 291-293
  9. Back Matter
    Pages 295-317

About this book


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.


automata theory finite monoids monoid applications Markov chains monoid applications automata theory monoid applications combinatorics monoid applications probability representation theory finite monoids representation theory inverse monoids zeta function languages monoid algebras

Authors and affiliations

  • Benjamin Steinberg
    • 1
  1. 1.Department of MathematicsThe City College of New York (CCNY)New YorkUSA

Bibliographic information

  • DOI
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-43930-3
  • Online ISBN 978-3-319-43932-7
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site