representation theory

Representation Theory

A First Course

© 2004
William Fulton 0 ,
Joe Harris 1

Department of Mathematics, University of Michigan, Ann Arbor, USA

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Department of Mathematics, Harvard University, Cambridge, USA

Part of the book series: Graduate Texts in Mathematics (GTM, volume 129)

Part of the book sub series: Readings in Mathematics (READMATH)

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About this book

Hermann Weyl and representation theory

Representations of Complex Semi-simple Lie Groups and Lie Algebras

Unisingular representations in arithmetic and Lie theory

Abelian group
cohomology group
finite group
group action
Lie algebra
representation theory
Vector space

Table of contents (26 chapters)

Front matter, finite groups, representations of finite groups.

William Fulton, Joe Harris

Examples; Induced Representations; Group Algebras; Real Representations

Representations of \({\mathfrak{s}_{_d}}\) : young diagrams and frobenius’s character formula, representations of \({\mathfrak{u}_d}\) and \(g{l_2}\left( {{\mathbb{f}_q}} \right)\), weyl’s construction, lie groups and lie algebras, lie algebras and lie groups, initial classification of lie algebras, lie algebras in dimensions one, two, and three, representations of sl2ℂ, representations of sl3ℂ, part i, representations ofsl3ℂ, part ii: mainly lots of examples, the classical lie algebras and their representations, the general setup: analyzing the structure and representations of an arbitrary semisimple lie algebra, sl4ℂ and slnℂ, symplectic lie algebras, authors and affiliations.

William Fulton

Bibliographic Information

Book Title : Representation Theory

Book Subtitle : A First Course

Authors : William Fulton, Joe Harris

Series Title : Graduate Texts in Mathematics

DOI : https://doi.org/10.1007/978-1-4612-0979-9

Publisher : Springer New York, NY

eBook Packages : Springer Book Archive

Hardcover ISBN : 978-0-387-97527-6 Published: 22 October 1991

Softcover ISBN : 978-0-387-97495-8 Published: 22 October 1991

eBook ISBN : 978-1-4612-0979-9 Published: 01 December 2013

Series ISSN : 0072-5285

Series E-ISSN : 2197-5612

Edition Number : 1

Number of Pages : XV, 551

Topics : Topological Groups, Lie Groups

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Representation Theory

Representation theory is fundamental in the study of objects with symmetry.

It arises in contexts as diverse as card shuffling and quantum mechanics. An early success was the work of Schur and Weyl, who computed the representation theory of the symmetric and unitary groups; the answer is closely related to the classical theory of symmetric functions and deeper study leads to intricate questions in combinatorics.

More recently, methods from geometry and topology have greatly enhanced our understanding of these questions (“geometric representation theory”). The study of affine Lie algebras and quantum groups has brought many new ideas and viewpoints, and representation theory now furnishes a basic language for other fields, including the modern theory of automorphic forms.

All of these aspects are studied by Stanford faculty. Topics of recent seminars include combinatorial representation theory as well as quantum groups.

Daniel Bump

Persi Diaconis

Browse Course Material

Course info.

Prof. Pavel Etingof

Departments

Mathematics

As Taught In

Algebra and Number Theory
Linear Algebra

Learning Resource Types

Introduction to representation theory, lecture notes.

Students are assigned readings in these lecture notes each week. The present lecture notes arose from a representation theory course given by Prof. Etingof in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students. The students in that course — Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Elena Yudovina, and Dmitry Vaintrob — co-authored the lecture notes which are published here with their permission.

The book Introduction to Representation Theory based on these notes was published by the American Mathematical Society in 2016. A complete file of the book (PDF - 1.1MB) is on Prof. Etingof’s webpage. [Please note: This file cannot be posted on any website not belonging to the authors.]

Complete Lecture Notes ( PDF - 1.3MB )

Introduction ( PDF )

Chapter 1: basic notions of representation theory ( pdf ).

1.1 What is representation theory? 1.2 Algebras 1.3 Representations 1.4 Ideals 1.5 Quotients 1.6 Algebras defined by generators and relations 1.7 Examples of algebras 1.8 Quivers 1.9 Lie algebras 1.10 Tensor products 1.11 The tensor algebra 1.12 Hilbert’s third problem 1.13 Tensor products and duals of representations of Lie algebras 1.14 Representations of sl(2) 1.15 Problems on Lie algebras

Chapter 2: General Results of Representation Theory ( PDF )

2.1 Subrepresentations in semisimple representations 2.2 The density theorem 2.3 Representations of direct sums of matrix algebras 2.4 Filtrations 2.5 Finite dimensional algebras 2.6 Characters of representations 2.7 The Jordan-Hölder theorem 2.8 The Krull-Schmidt theorem 2.9 Problems 2.10 Representations of tensor products

Chapter 3: Representations of Finite Groups: Basic Results ( PDF )

3.1 Maschke’s Theorem 3.2 Characters 3.3 Examples 3.4 Duals and tensor products of representations 3.5 Orthogonality of characters 3.6 Unitary representations. Another proof of Maschke’s theorem for complex representations 3.7 Orthogonality of matrix elements 3.8 Character tables, examples 3.9 Computing tensor product multiplicities using character tables 3.10 Problems

Chapter 4: Representations of Finite Groups: Further Results ( PDF )

4.1 Frobenius-Schur indicator 4.2 Frobenius determinant 4.3 Algebraic numbers and algebraic integers 4.4 Frobenius divisibility 4.5 Burnside’s theorem 4.6 Representations of products 4.7 Virtual representations 4.8 Induced representations 4.9 The Mackey formula 4.10 Frobenius reciprocity 4.11 Examples 4.12 Representations of _S_ n 4.13 Proof of theorem 4.36 4.14 Induced representations for _S_ n 4.15 The Frobenius character formula 4.16 Problems 4.17 The hook length formula 4.18 Schur-Weyl duality 4.19 Schur-Weyl duality for GL (V) 4.20 Schur polynomials 4.21 The characters of L λ 4.22 Polynomial representations of GL (V) 4.23 Problems 4.24 Representations of _GL_ 2 (F q )

4.24.1 Conjugacy classes in _GL_ 2 (F q ) 4.24.2 1-dimensional representations 4.24.3 Principal series representations 4.24.4 Complementary series representations

4.25 Artin’s theorem 4.26 Representations of semidirect products

Chapter 5: Quiver Representations ( PDF )

5.1 Problems 5.2 Indecomposable representations of the quivers A1,A2,A3 5.3 Indecomposable representations of the quiver D 4 5.4 Roots 5.5 Gabriel’s theorem 5.6 Reflection functors 5.7 Coxeter elements 5.8 Proof of Gabriel’s theorem 5.9 Problems

Chapter 6: Introduction to Categories ( PDF )

6.1 The definition of a category 6.2 Functors 6.3 Morphisms of functors 6.4 Equivalence of categories 6.5 Representable functors 6.6 Adjoint functors 6.7 Abelian categories 6.8 Exact functors

Chapter 7: Structure of Finite Dimensional Algebras ( PDF )

7.1 Projective modules 7.2 Lifting of idempotents 7.3 Projective covers

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Representation Theory

Symmetries occur throughout mathematics and science. Representation theory seeks to understand all the possible ways that an abstract collection of symmetries can arise. Nineteenth-century representation theory helped to explain the structure of electron orbitals, and 1920s representation theory is at the heart of quantum chromodynamics. In number theory, p -adic representation theory is central the Langlands program, a family of conjectures that have guided a large part of number theory for the past forty years.

One fundamental problem involves describing all the irreducible unitary representations of each Lie group, the continuous symmetries of a finite-dimensional geometry. Doing so corresponds to identifying all finite-dimensional symmetries of a quantum-mechanical system. We've made great progress on this important problem, including work by MIT's strong faculty in this area. The representation theory of infinite-dimensional groups and supergroups is vital to string theory, statistical mechanics, integrable systems, tomography, and many other areas of mathematics and its applications.

Research interests of this group include vertex algebras, quantum groups, infinite-dimensional Lie algebras, representations of real and p -adic groups, Hecke algebras and symmetric spaces.

Department Members in This Field

Roman Bezrukavnikov Representation Theory, Algebraic Geometry
Alexei Borodin Integrable Probability
Pavel Etingof Representation Theory, Quantum Groups, Noncommutative Algebra
Victor Kac Algebra, Mathematical Physics
Ju-Lee Kim Representation Theory of p-adic groups
George Lusztig Group Representations, Algebraic Groups
David Vogan Group Representations, Lie Theory
Zhiwei Yun Representation Theory, Number Theory, Algebraic Geometry
Wei Zhang Number Theory, Automorphic forms, Arithmetic Geometry

Instructors & Postdocs

Joseph Berleant Representation theory, Geometric algebra
Elijah Bodish Representation theory, Link homology
Thomas Rüd Number theory, representation theory of p-adic groups, algebraic geometry
Robin Zhang Number Theory, Automorphic Forms, Arithmetic Geometry

Researchers & Visitors

David Roe Computational number theory, Arithmetic geometry, local Langlands correspondence

Graduate Students*

Anlong Chua Geometric representation theory
Ilya Dumanski Geometric and combinatorial methods in representation theory
Marisa Gaetz Group Representations, Lie Theory
Svetlana Gavrilova
Mikayel Mkrtchyan
Ivan Motorin Cluster Algebras, Resolution of Singularities, Representation Theory, Integrable Systems
Michael Panner
Andrew Riesen
Hamilton Wan
Frank Wang Geometric Representation Theory

*Only a partial list of graduate students