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El. knyga: Invitation to Representation Theory: Polynomial Representations of the Symmetric Group

  • Formatas: PDF+DRM
  • Serija: SUMS Readings
  • Išleidimo metai: 28-May-2022
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030980252
Kitos knygos pagal šią temą:
Invitation to Representation Theory: Polynomial Representations of the Symmetric GroupDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: SUMS Readings
  • Išleidimo metai: 28-May-2022
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030980252
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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.

Recenzijos

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)

1 First Steps
1(24)
1.1 Permutations and Groups
1(3)
1.2 Group Actions and Representations
4(2)
1.3 More About the Symmetric Group
6(2)
1.4 More Groups and Subgroups
8(3)
1.5 Group Homomorphisms and More About Representations
11(4)
1.6 Representations on Function Spaces
15(2)
1.7 Hints and Additional Comments
17(8)
2 Polynomials, Subspaces and Subrepresentations
25(14)
2.1 Polynomials
25(2)
2.2 Subspaces and Subrepresentations
27(1)
2.3 Partitions and More Subrepresentations
27(2)
2.4 Vector Space Direct Sums
29(2)
2.5 Projection Maps
31(1)
2.6 Irreducible Subspaces
32(2)
2.7 Hints and Additional Comments
34(5)
3 Intertwining Maps, Complete Reducibility, and Invariant Inner Products
39(22)
3.1 Intertwining Maps
39(3)
3.2 Complete Reducibility
42(2)
3.3 Invariant Inner Products and Another Proof of Complete Reducibility
44(3)
3.4 Dual Spaces and Contragredient Representations
47(2)
3.5 Hints and Additional Comments
49(12)
4 The Structure of the Symmetric Group
61(6)
4.1 Cycles and Cycle Structure
61(1)
4.2 Generators and Parity
62(3)
4.3 Conjugation and Conjugacy Classes
65(1)
4.4 Hints and Additional Comments
66(1)
5 Sn-Decomposition of Polynomial Spaces for n = 1, 2, 3
67(10)
5.1 S1
67(1)
5.2 S2
67(1)
5.3 53
68(3)
5.4 Isotypic Subspaces and Multiplicities
71(2)
5.5 Hints and Additional Comments
73(4)
6 The Group Algebra
77(8)
6.1 Version One
77(2)
6.2 Version Two
79(2)
6.3 Hints and Additional Comments
81(4)
7 The Irreducible Representations of Sn: Characters
85(18)
7.1 Characters and Class Functions
85(3)
7.2 Characters of S3
88(1)
7.3 Orthogonality of Characters, Bases
89(7)
7.4 Another Look
96(3)
7.5 Hints and Additional Comments
99(4)
8 The Irreducible Representations of Sn: Young Symmetrizers
103(22)
8.1 Partitions Again: Young Tableaux
103(1)
8.2 Orderings on Partitions
104(1)
8.3 Young Symmetrizers
105(4)
8.4 Construction of Irreducible Representations in C[ Sn]
109(7)
8.5 More Representations
116(2)
8.6 Hints and Additional Comments
118(7)
9 Cosets, Restricted and Induced Representations
125(28)
9.1 Restriction
125(1)
9.2 Quotient Spaces
126(1)
9.3 Cosets
127(3)
9.4 Coset Representations of a Group
130(1)
9.5 Induced Representations: Version One
131(3)
9.6 Matrix Realizations and Characters of Induced Representations
134(1)
9.7 Construction of Induced Representations
135(1)
9.8 Frobenius Reciprocity
136(1)
9.9 Induced Representations: Version Two
137(3)
9.10 Hints and Additional Comments
140(13)
10 Direct Products of Groups, Young Subgroups and Permutation Modules
153(12)
10.1 Direct Products of Groups
153(2)
10.2 Young Subgroups and Permutation Modules
155(2)
10.3 Decomposition of Polynomial Spaces into Permutation Modules
157(1)
10.4 More Permutation Modules: Tabloids and Polytabloids
158(4)
10.5 Hints and Additional Comments
162(3)
11 Specht Modules
165(22)
11.1 Construction of Specht Modules
165(1)
11.2 Irreducibility of Specht Modules
166(2)
11.3 Inequivalence of Specht Modules
168(1)
11.4 The Standard Basis for Specht Modules
169(11)
11.4.1 Linear Independence
170(2)
11.4.2 Span
172(1)
11.4.3 A Straightening Algorithm
173(7)
11.5 Application to Polynomial Spaces
180(1)
11.6 Hints and Additional Comments
181(6)
12 Decomposition of Young Permutation Modules
187(22)
12.1 Generalized and Semistandard Young Tableaux
187(2)
12.2 The Space C[ Tλμ] and Its Equivalence to C[ Tμ]
189(2)
12.3 The Space HomC|Sn|(Sλ, C[ Tλμ])
191(3)
12.4 Column Equivalence and Ordering
194(1)
12.5 The Semistandard Basis for HomC|Sn|(Snλ, Mμ)
195(5)
12.6 Young's Rule
200(1)
12.7 Hints and Additional Comments
201(8)
13 Branching Relations
209(16)
13.1 The Hook Length Formula
209(5)
13.2 Branching Relations
214(6)
13.3 Hints and Additional Comments
220(5)
Bibliography 225(2)
Index 227
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.
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