Lectures in algebraic combinatorics : Young's construction, seminormal representations, sl(2) representations, heaps, basics on finite fields / Adriano M. Garsia, Ömer Eğecioğlu.

Format:

Book

Author/Creator:

Garsia, Adriano M., 1928- author.

Eğecioğlu, Ömer Nuri, 1954- author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 2277.

Lecture Notes in Mathematics, 0075-8434 ; volume 2277

Language:

English

Subjects (All):

Combinatorial analysis.

Algebra.

Field theory (Physics).

Group theory.

Commutative algebra.

Commutative rings.

Field Theory and Polynomials.

Group Theory and Generalizations.

Commutative Rings and Algebras.

Local Subjects:

Field Theory and Polynomials.

Group Theory and Generalizations.

Commutative Rings and Algebras.

Physical Description:

xiv, 230 pages : illustrations ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2020]

Summary:

Capturing Adriano Garsia's unique perspective on essential topics in algebraic combinatorics, this book consists of selected, classic notes on a number of topics based on lectures held at the University of California, San Diego over the past few decades. The topics presented share a common theme of describing interesting interplays between algebraic topics such as representation theory and elegant structures which are sometimes thought of as being outside the purview of classical combinatorics. The lectures reflect Garsia's inimitable narrative style and his exceptional expository ability. The preface presents the historical viewpoint as well as Garsia's personal insights into the subject matter. The lectures then start with a clear treatment of Alfred Young's construction of the irreducible representations of the symmetric group, seminormal representations and Morphy elements. This is followed by an elegant application of SL(2) representations to algebraic combinatorics. The last two lectures are on heaps, continued fractions and orthogonal polynomials with applications, and finally there is an exposition on the theory of finite fields. The book is aimed at graduate students and researchers in the field.

Contents:

Intro

Preface

Lecture 1: Alfred Young's Construction of the Irreducible Representations of Sn

Lecture 2: Young's Seminormal Representation, Murphy Elements, and Content Evaluations

Lecture 3: On Finite Dimensional sl(2) Representations and an Application to Algebraic Combinatorics

Lecture 4: Heaps, Continued Fractions, and Orthogonal Polynomials

Lecture 5: Basics on Finite Fields

Acknowledgements

Contents

1 Alfred Young's Construction of the Irreducible Representations of Sn

1.1 Introduction

1.2 The Natural Representation Matrices

Problems

3 On Finite Dimensional sl(2) Representations and an Application to Algebraic Combinatorics

3.1 Basic Identities

3.2 Diagonalizability of H

3.3 Spernerity of L[m, n]

4 Heaps, Continued Fractions, and Orthogonal Polynomials

4.1 Introduction

4.2 Heaps of Monomers and Dimers

4.3 The Cartier-Foata Languages

4.4 Orthogonal Polynomials and Continued Fractions

4.5 Moments and Motzkin Paths

4.6 Chebyshev Polynomials

4.7 The Rogers-Ramanujan Continued Fraction

4.8 Partitions and Hermite Polynomials

4.9 The Legendre Polynomials

4.10 The Laguerre Polynomials

5 Finite Fields

5.1 Introduction

5.2 The Euclidean Algorithm

5.3 Polynomial Factorization

5.4 Cyclotomic Polynomials

Problems

5.5 The Frobenius Map

5.6 A Factorization Algorithm

Glossary

References

Notes:

Current copyright fee: GBP19.00 42\0.

Includes bibliographical references.

ISBN:

9783030583729

3030583724

OCLC:

1203133141

Publisher Number:

10.1007/978-3-030-58373-6