Author(s): Wushi Goldring
Series: version 2020-05-26
Year: 2020
Language: English
Commentary: Downloaded from https://kurser.math.su.se/pluginfile.php/87309/mod_resource/content/12/Course%20Notes.pdf
1. March 19, 2020: Comparing bases, the first orthogonality relation and the character table
1.1. Set-up
1.2. Recall our two bases of Z(C[G])
1.3. Expressing one basis in the other
1.4. The generalized and 1st orthogonality relations
1.5. Application I: The inner product on class functions
1.6. Proof of the generalized orthogonality relation
1.7. The commutator subgroup and abelianization
1.8. Application of the commutator subgroup to degree 1 characters
1.9. The character Table
2. March 26, 2020: The second orthogonality relation and applications. Pullback and restriction. Symmetric and exterior powers, and their characters. Application of symmetric and exterior powers to nonabelian groups of order 8 and the character table of S5
2.1. Restriction
2.2. Application of the change of basis 1.3.2
2.3. Application of the 1st orthogonality relation II: The 2nd orthogonality relation
2.4. Applications of the 2nd orthogonality relation I: Another proof of 2.2.1, following Gorenstein
2.5. Interlude: Pulling back representations
2.6. Applications of the 2nd orthogonality relation II: Comparing centralizers in G and its quotients
2.7. Tensor, symmetric and exterior powers: Characters
2.8. Using symmetric and exterior powers to distinguish between groups with the same character table
2.9. Using symmetric and exterior powers to complete character tables
2.10. Question about the kernel of chi 2 for the quaternion group
3. April 2, 2020:
3.1. The kernel and centralizer of a character
3.2. Locating normal subgroups via the character table
3.3. Twisting representations by outer automorphisms
4. April 9, 2020
4.1. Twisting representations by automorphism (continued)
4.2. The character table of A5
4.3. More about character centralizers
4.4. Existence of a faithful irreducible character
4.5. Bounding character degrees
4.6. More about permutation representations
4.7. Other problems from Chapter 2 of
4.8. Applications of the 2nd orthogonality relation III : Real elements
5. April 16, 2020
5.1. Explicit tensor-generation: The Brauer-Burnside theorem
5.2. Counting solutions to equations in G as class functions: The Frobenius-Schur Theorem
5.3. Involutions: The Alperin-Feit-Thompson Theorem
5.4. Algebraic integers
6. April 23, 2020
6.1. The central character again
6.2. Some more preliminaries about algebraic and integral elements
6.3. Some of Burnside's applications of integrality
6.4. More integrality: Irreducible character degrees divide the order of the group
6.5. Applications of the 2nd orthogonality relation IV: Real-valued characters and odd-order groups
7. April 30, 2020
7.1. Reduction of chi(1) divides [G:Z(chi)] to chi(1) divides |G| via tensor products (d'après Tate)
7.2. Induced representations and induced Characters
7.3. Monomial characters and M-groups
8. May 7, 2020. Part I: Frobenius groups and other applications of induction
8.1. More on induced Characters
8.2. Frobenius groups
8.3. Deeper results about Frobenius groups
8.4. Using characters to produce `large' subgroups
9. May 7, 2020. Part II: Every simple group of order 360 is isomorphic to A6
9.1. Review & Motivation
9.2. The local group theory of G simple of order 360
9.3. Simple groups of order 360 I: Via Characters
9.4. Simple groups of order 360 II: A geometric approach
9.5. Connection to Artin L-functions and Artin's conjecture (not required for the course but very cool)
10. Summary
References
