Representation Theory: A Combinatorial ViewpointThis book discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. The first chapter provides a detailed account of necessary representation-theoretic background. An important highlight of this book is an innovative treatment of the Robinson–Schensted–Knuth correspondence and its dual by extending Viennot's geometric ideas. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. The treatment of Schur–Weyl duality reveals the directness and simplicity of Schur's original treatment of the subject. In addition, each exercise is assigned a difficulty level to test readers' learning. Solutions and hints to most of the exercises are provided at the end. |
Contents
1 Basic Concepts of Representation Theory | 1 |
2 Permutation Representations | 32 |
3 The RSK Correspondence | 51 |
4 Character Twists | 70 |
5 Symmetric Functions | 96 |
6 Representations of General Linear Groups | 141 |
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Common terms and phrases
algebraically closed field algorithm basis bijection boxes character table character values characteristic class functions coefficient column combinatorial complete symmetric functions completely reducible Compute conjugacy classes conjugate Corollary cycle type decreasing monomials defined Definition denote the number diagonal dimension distinct odd divide G dual shadow element EndKV example Exercise finite group finite-dimensional follows G-set GLn(K identity intertwiner invariant subspace inverse irreducible characters ith row K-algebra lexicographic order maximal entries Mn(K monomial non-negative integer odd permutation one-dimensional permutation representation polynomial representation positive integer Proof R-module regular representation representation of G representation theory representations of GLm(K reverse lexicographic order S K(m Schur algebras Schur functions Section semisimple sequence set of partitions shadow path shadow points Show simple modules simple representations Solution subgroup Suppose symmetric groups tensor product Theorem theory of symmetric variables vector space weak composition Young diagram Young tableau