An Invitation to Representation Theory: Polynomial Representations of the Symmetric Group (Springer Undergraduate Mathematics Series)

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. 1 First Steps 1.1 Permutations and Groups 1.2 Group Actions and Representations 1.3 More About the Symmetric Group 1.4 More Groups and Subgroups 1.5 Group Homomorphisms and More About Representations 1.6 Representations on Function Spaces 1.7 Hints and Additional Comments 2 Polynomials, Subspaces and Subrepresentations 2.1 Polynomials 2.2 Subspaces and Subrepresentations 2.3 Partitions and More Subrepresentations 2.4 Vector Space Direct Sums 2.5 Projection Maps 2.6 Irreducible Subspaces 2.7 Hints and Additional Comments 3 Intertwining Maps, Complete Reducibility, and Invariant Inner Products 3.1 Intertwining Maps 3.2 Complete Reducibility 3.3 Invariant Inner Products and Another Proof of Complete Reducibility 3.4 Dual Spaces and Contragredient Representations 3.5 Hints and Additional Comments 4 The Structure of the Symmetric Group 4.1 Cycles and Cycle Structure 4.2 Generators and Parity 4.3 Conjugation and Conjugacy Classes 4.4 Hints and Additional Comments 5 Sn-Decomposition of Polynomial Spaces for n=1,2,3 5.1 S1 5.2 S2 5.3 S3 5.4 Isotypic Subspaces and Multiplicities 5.5 Hints and Additional Comments 6 The Group Algebra 6.1 Version One 6.2 Version Two 6.3 Hints and Additional Comments 7 The Irreducible Representations of Sn: Characters 7.1 Characters and Class Functions 7.2 Characters of S3 7.3 Orthogonality of Characters, Bases 7.4 Another Look 7.5 Hints and Additional Comments 8 The Irreducible Representations of Sn: Young Symmetrizers 8.1 Partitions Again: Young Tableaux 8.2 Orderings on Partitions 8.3 Young Symmetrizers 8.4 Construction of Irreducible Representations in C[Sn] 8.5 More Representations 8.6 Hints and Additional Comments 9 Cosets, Restricted and Induced Representations 9.1 Restriction 9.2 Quotient Spaces 9.3 Cosets 9.4 Coset Representations of a Group 9.5 Induced Representations: Version One 9.6 Matrix Realizations and Characters of Induced Representations 9.7 Construction of Induced Representations 9.8 Frobenius Reciprocity 9.9 Induced Representations: Version Two 9.10 Hints and Additional Comments 10 Direct Products of Groups, Young Subgroups and Permutation Modules 10.1 Direct Products of Groups 10.2 Young Subgroups and Permutation Modules 10.3 Decomposition of Polynomial Spaces into Permutation Modules 10.4 More Permutation Modules: Tabloids and Polytabloids 10.5 Hints and Additional Comments 11 Specht Modules 11.1 Construction of Specht Modules 11.2 Irreducibility of Specht Modules 11.3 Inequivalence of Specht Modules 11.4 The Standard Basis for Specht Modules 11.4.1 Linear Independence 11.4.2 Span 11.4.3 A Straightening Algorithm 11.5 Application to Polynomial Spaces 11.6 Hints and Additional Comments 12 Decomposition of Young Permutation Modules 12.1 Generalized and Semistandard Young Tableaux 12.2 The Space C[Tλμ] and Its Equivalence to C[Tμ] 12.3 The Space HomC[Sn](Sλ, C[Tλμ]) 12.4 Column Equivalence and Ordering 12.5 The Semistandard Basis for Hom C[Sn](Sλ, Mμ) 12.6 Young's Rule 12.7 Hints and Additional Comments 13 Branching Relations 13.1 The Hook Length Formula 13.2 Branching Relations 13.3 Hints and Additional Comments Bibliography Over the years I have had the opportunity to work with undergraduate mathematics students on various research and independent study projects, and I’ve found “representation theory of the symmetric group” to be an excellent vehicle to introduce them to more advanced mathematical concepts relatively early in their undergraduate careers. The basic idea is accessible to anyone with a solid background in linear algebra, and new content can be introduced as needed. Of course, it’s also a beautiful subject in its own right that has many important applications.