REPRESENTATION THEORY

### COURSE DESCRIPTION

Lecturer: Roland van der Veen, Assistant: Maurizio Moreschi. Contact us at: representationtheory18@gmail.com

Lectures: Mondays 11-12:45 Leiden, Snellius 402. Maurizio's office hours: Fridays 11:00-13:00 in Snellius 207 or the library.

REPRESENTATION THEORY is about using linear algebra to understand and exploit symmetry to the fullest. As such it plays a major role in many subjects in mathematics and physics. For example it provides a framework for understanding special functions and generalizes Fourier analysis to a non-commutative setting. Also modular forms in number theory are intimately related to representations of the Galois group. In physics one describes particles scattering into smaller elementary particles in terms of the corresponding representation decomposing into irreducible representations.

CONCRETELY, representation theory is about how a given subgroup of GL(V) decomposes vector space V into invariant subspaces. Even more concretely this often amounts to block-diagonalizing matrices. Working with representations of algebras we are able to introduce the theory for both Lie algebras and finite groups. Along the way we will also meet tensor products and categories.

FINAL GRADE: Written exam. Bi-weekly homework counts as bonus (max one point on the final grade, it can only count positively). Homework is preferrably typed in latex and handed in as pdf to representationtheory18@gmail.com. Alternatively it can be put in Maurizio's pigeon hole on the second floor of the math department. Please send an email that you have done so and do not email scans as they are difficult to read. English language is preferred.

PREREQUISITES: Linear algebra 1,2, Algebra 1.

LITERATURE: Etingof, Introduction to Representation theory, freely available on the author's website.

MATERIAL TREATED: The following topics in the book were treated and may be expected on the exam:

• Chapter 2: 2.2 algebras, 2.3 representations, Schur lemma, 2.4 Ideals 2.5 Quotients, 2.6 generators and relations 2.7 Examples, 2.9 Lie algebras 2.11 Tensor products, 2.12 tensor algebra, 2.14 tensor products of reps of Lie algebras.
• Chapter 3: 3.1 Subreps in semi-simple reps. 3.2 Density theorem, 3.3 Reps of direct sums of matrix algs, 3.4 filtrations, 3.5 finite dimensional algebras. 3.6 Characters of reps. 3.7 Jordan-Holder.
• Chapter 4: 4.1 Maschke's theorem, 4.2 Characters, 4.3 Examples, 4.4 Duals and tensor products of reps, 4.5 Orthogonality of characters, 4.6 Unitary representations, 4.8 Character tables, 4.9 Tensor product multiplicities using character tables, 4.10 Frobenius determinant.
• Chapter 5: 5.7 Virtual representations, 5.8 Induced representations, 5.10 Frobenius reciprocity. 5.11 examples, 5.12 representations of Sn, 5.13 proof of classification theorem of Sn reps.
• Chapter 7: 7.1 Categories, 7.2 functors, 7.3 natural transformations.

$\hat{f}(\chi) = \sum_{g\in G}f(g)\bar{\chi}(g)$
$\mathbb{C}[G] = \bigoplus_\rho \rho\ \mathrm{dim}(\rho)$
$\det X_G = \prod_{j=1}^r P_j(x)^{\mathrm{deg} P_j}$
$V_i\otimes V_j = \bigoplus_{k = |i-j|}^{i+j}V_k$
SCHEDULE
• Feb. 5: NO CLASS
• Feb. 12: Introduction, algebras and representations, Section 2.2, 2.3 up to Schur's lemma.
• Feb. 19: Ideals Quotients 2.4, 2.5 Generators and relations 2.6, Lie algebras 2.9 Tensor products 2.11
• Feb. 26: Finishing up Chapter 2: Tensor algebra and sl(2). Homework TBA
• Mar. 5: Semisimple algebras 3.1, Density theorems 3.2
• Mar. 12: NO CLASS
• Mar. 19: Matrix algebras 3.3, Filtrations, 3.4
• Mar. 26: Filtrations and Finite dimensional algebras Section 3.4, 3.5
• Apr. 2: NO CLASS
• Apr. 9: Characters, Section 3.6.3.7 Jordan-Holder, Finite groups 4.1 Homework Additional Exercises 7-9. Deadline Monday Apr 16 before class.
• Apr. 16: Examples, duals and tensor products 4.3,
• Apr. 23: NO CLASS
• Apr. 30: Orthogonality of the characters, matrix elements, Section 4.5, 4.6, 4.7, 4.8
• May. 7: Representations of the symmetric group, Chapter 5.12,5.13.
Homework, due Monday may 14 before class: Additional Exercises 10,11,12.
• May. 14: Induced representations 5.8, Frobenius reciprocity 5.10, Categories and functors, chapter 7.
• May. 21: NO CLASS
• May. 28: Recap, natural transformations, unitary representations.
• June 11: 11am: Questions and answers to the practice exam available here.
• June 20, 14:00-17:00: WRITTEN EXAM, location Snellius 408.