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An Invitation to Representation Theory

Polynomial Representations of the Symmetric Group

  • Textbook
  • © 2022

Overview

Authors:
  1. R. Michael Howe

    1. Department of Mathematics, University of Wisconsin–Eau Claire, Eau Claire, USA

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  • First text accessible to undergraduates having taken a course in linear algebra
  • Contains over 250 exercises with complete solutions
  • Introduces many important notions of algebra

Part of the book series: Springer Undergraduate Mathematics Series (SUMS)

Part of the book sub series: SUMS Readings (SUMSR)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. First Steps

    • R. Michael Howe
    Pages 1-24

  3. Polynomials, Subspaces and Subrepresentations

    • R. Michael Howe
    Pages 25-37

  4. Intertwining Maps, Complete Reducibility, and Invariant Inner Products

    • R. Michael Howe
    Pages 39-59

  5. The Structure of the Symmetric Group

    • R. Michael Howe
    Pages 61-66

  6. S n-Decomposition of Polynomial Spaces for n = 1,  2,  3

    • R. Michael Howe
    Pages 67-75

  7. The Group Algebra

    • R. Michael Howe
    Pages 77-83

  8. The Irreducible Representations of S n: Characters

    • R. Michael Howe
    Pages 85-101

  9. The Irreducible Representations of S n: Young Symmetrizers

    • R. Michael Howe
    Pages 103-123

  10. Cosets, Restricted and Induced Representations

    • R. Michael Howe
    Pages 125-152

  11. Direct Products of Groups, Young Subgroups and Permutation Modules

    • R. Michael Howe
    Pages 153-164

  12. Specht Modules

    • R. Michael Howe
    Pages 165-186

  13. Decomposition of Young Permutation Modules

    • R. Michael Howe
    Pages 187-208

  14. Branching Relations

    • R. Michael Howe
    Pages 209-223

  15. Back Matter

    Pages 225-229
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Keywords

About this book

An Invitation to Representation Theory offers an introduction to groups and their representations, suitable for undergraduates. In this book, the ubiquitous symmetric group and its natural action on polynomials are used as a gateway to representation theory.

The subject of representation theory is one of the most connected in mathematics, with applications to group theory, geometry, number theory and combinatorics, as well as physics and chemistry. It can however be daunting for beginners and inaccessible to undergraduates. The symmetric group and its natural action on polynomial spaces provide a rich yet accessible model to study, serving as a prototype for other groups and their representations. This book uses this key example to motivate the subject, developing the notions of groups and group representations concurrently.


With prerequisites limited to a solid grounding in linear algebra, this book can serve as a first introduction to representation theory at the undergraduate level, for instance in a topics class or a reading course. A substantial amount of content is presented in over 250 exercises with complete solutions, making it well-suited for guided study.

Reviews

“The book under review is a nice introduction to the representation theory of the symmetric group. … The book is well structured and enriched with numerous exercises, many of which are solved or with hints for the solution.” (Enrico Jabara, zbMATH 1514.20002, 2023)

Authors and Affiliations

  • Department of Mathematics, University of Wisconsin–Eau Claire, Eau Claire, USA

    R. Michael Howe

About the author

R. Michael Howe spent 20 years in various roles in the music industry and earned a PhD in mathematics at the University of Iowa, becoming a professor at the University of Wisconsin-Eau Claire, where he is now Emeritus Professor. As a mathematics professor he has supervised research and independent study projects of scores of undergraduate students, at least a dozen of whom have gone on to earn a PhD in mathematics. He still enjoys playing music and his other hobbies include hiking, mountaineering, kayaking, biking and skiing.

Bibliographic Information

  • Book Title: An Invitation to Representation Theory

  • Book Subtitle: Polynomial Representations of the Symmetric Group

  • Authors: R. Michael Howe

  • Series Title: Springer Undergraduate Mathematics Series

  • DOI: https://doi.org/10.1007/978-3-030-98025-2

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Softcover ISBN: 978-3-030-98024-5Published: 29 May 2022

  • eBook ISBN: 978-3-030-98025-2Published: 28 May 2022

  • Series ISSN: 1615-2085

  • Series E-ISSN: 2197-4144

  • Edition Number: 1

  • Number of Pages: XV, 229

  • Topics: Group Theory and Generalizations

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