\[ \DeclareMathOperator{\End}{End} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ker}{ker} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\span}{span} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\ad}{ad} \newcommand\d{\mathrm{d}} \newcommand\pref[1]{(\ref{#1})} \]

I'm working through the textbook Introduction to Representation Theory by Pavel Etingof et al.. These pages document my solutions to problems as well as notes I took on sections that I found hard to follow.

I got interested in representation theory because I wanted to understand spinors. I think I understand spinors representations now, but I still don't understand the role played by the Clifford algebra. Oh well. One day.

Section and problem numbers are based on the text published by the American Mathematical Society. This differs from the lecture notes found online. I recommend buying the book; it contains a number of amusing anecdotes about mathematicians.

A reminder that in the text, fields are assumed to be algebraically closed unless it is stated otherwise, and all algebras are unital.

I use the convention that 0 is a natural number.

• Chapter 2: Basic notions of representation theory
  • Section 2.3: Representations
  • Section 2.4: Ideals
  • Section 2.5: Quotients
  • Section 2.7: Examples of algebras
  • Section 2.8: Quivers
  • Section 2.9: Lie algebras
  • Section 2.11: Tensor products
  • Section 2.13: Hilbert's third problem
  • Section 2.14: Tensor products and duals of representations of Lie algebras
  • Section 2.15: Representations of sl(2)
  • Section 2.16: Problems on Lie algebras
• Chapter 3: General results of representation theory
  • Section 3.3: Representations of direct sums of matrix algebras
  • Section 3.6: Characters of representations
  • Section 3.8: The Krull–Schmidt theorem
  • Section 3.9: Problems
  • Section 3.10: Representations of tensor products
• Chapter 4: Representations of finite groups: Basic results
  • Section 4.1: Maschke's theorem
  • Section 4.2: Characters
  • Section 4.3: Examples
  • Section 4.5: Orthogonality of characters
  • Section 4.6: Unitary representations. Another proof of Maschke's theorem for complex representations
  • Section 4.7: Orthogonality of matrix elements
  • Section 4.12: Problems
• Chapter 5: Representations of finite groups: Further results
  • Section 5.1: Frobenius–Schur indicator
  • Section 5.2: Algebraic numbers and algebraic integers
  • Section 5.3: Frobenius divisibility
  • Section 5.4: Burnside's theorem
  • Section 5.8: Induced representations
  • Section 5.10: Frobenius reciprocity
  • Section 5.11: Examples
  • Section 5.12: Representations of \(S_n\)
  • Section 5.13: Proof of the classification theorem for representations of \(S_n\)
  • Section 5.14: Induced representations for \(S_n\)
  • Section 5.15: The Frobenius character formula