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E-raamat: Representation Theory: A Combinatorial Viewpoint

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Representation Theory: A Combinatorial ViewpointSuurem pilt
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This 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 RobinsonSchenstedKnuth 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 SchurWeyl 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.

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This book discusses representation theory of symmetric groups, theory of symmetric functions and polynomial representation theory of general linear groups.
List of Tables
vii
Preface ix
1 Basic Concepts of Representation Theory
1(31)
1.1 Representations and Modules
1(4)
1.2 Invariant Subspaces and Simplicity
5(2)
1.3 Complete Reducibility
7(4)
1.4 Maschke's Theorem
11(2)
1.5 Decomposing the Regular Module
13(6)
1.6 Tensor Products
19(3)
1.7 Characters
22(7)
1.8 Representations over Complex Numbers
29(3)
2 Permutation Representations
32(19)
2.1 Group Actions and Permutation Representations
32(2)
2.2 Permutations
34(5)
2.3 Partition Representations
39(2)
2.4 Intertwining Permutation Representations
41(3)
2.5 Subset Representations
44(2)
2.6 Intertwining Partition Representations
46(5)
3 The RSK Correspondence
51(19)
3.1 Semistandard Young Tableaux
51(5)
3.2 The RSK Correspondence
56(12)
3.3 Classification of Simple Representations of Sn
68(2)
4 Character Twists
70(26)
4.1 Inversions and the Sign Character
70(3)
4.2 Twisting by a Multiplicative Character
73(2)
4.3 Conjugate of a Partition
75(4)
4.4 Twisting by the Sign Character
79(1)
4.5 The Dual RSK Correspondence
80(3)
4.6 Representations of Alternating Groups
83(13)
5 Symmetric Functions
96(45)
5.1 The Ring of Symmetric Functions
96(2)
5.2 Other Bases for Homogeneous Symmetric Functions
98(9)
5.3 Specialization to m Variables
107(3)
5.4 Schur Functions and the Frobenius Character Formula
110(7)
5.5 Frobenius' Characteristic Function
117(2)
5.6 Branching Rules
119(1)
5.7 Littlewood--Richardson Coefficients
120(4)
5.8 The Hook--Length Formula
124(3)
5.9 The Involution Sλ Sλ'
127(2)
5.10 The Jacobi--Trudi Identities
129(3)
5.11 The Recursive Murnaghan--Nakayama Formula
132(4)
5.12 Character Values of Alternating Groups
136(5)
6 Representations of General Linear Groups
141(19)
6.1 Polynomial Representations
141(1)
6.2 Schur Algebras
142(6)
6.3 Schur Algebras and Symmetric Groups
148(2)
6.4 Modules of a Commutant
150(3)
6.5 Characters of the Simple Representations
153(2)
6.6 Polynomial Representations of the Torus
155(3)
6.7 Weight Space Decompositions
158(2)
Hints and Solutions to Selected Exercises 160(22)
Suggestions for Further Reading 182(3)
References 185(4)
Index 189
Amritanshu Prasad is a Professor of Mathematics at the Institute of Mathematical Sciences (IMSc), Chennai. Before joining IMSc in 2003, he was a CRM-CICMA fellow at the Centre de Recherche Mathématiques, a Canadian centre for research in the fundamental sciences located at the Université de Montréal. He has held visiting positions at the Max Planck Institute for Mathematics in Bonn and the Institut des Hautes Études Scientifiques in Bur-sur-Yvette, near Paris. He has been an Associate of the Indian Academy of Sciences (20052010) and is a winner of the Young Scientist Medal of the Indian National Science Academy (2010). He completed his PhD under the supervision of Robert E. Kottwitz at the University of Chicago (2001). He holds a Masters degree from the University of Chicago (1996) and a Bachelor's degree in Statistics from the Indian Statistical Institute, Kolkata (1995). He has taught undergraduate and graduate students in the United States, Canada, and India. His mathematical interests include representation theory, number theory, harmonic analysis and combinatorics.
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