Student Algebra Seminar, 2018 Semester 2

Week 2, Speaker: Joel Gibson

The goal of this talk will be to introduce a toolbox of results and techniques in the representation theory of finite groups, through the lens of the symmetric groups. I'll cover the classification of irreducible representations by partitions, and their basis of standard Young tableaux. Some examples will be given throughout the talk of how to leverage knowledge about representation theory to solve "real" problems, and how to think of representations of finite groups in general.

References:

• Chapter 4 of Introduction to Representation Theory by Etingof et. al. contains a very compact account of the general theory of representations of finite groups, and chapter 5 has a few pages devoted to Young symmetrisers, from an idempotents-in-algebras perspective.
• Chapter 7 of Young Tableaux by Fulton has an explicit construction of the Specht modules via tableaux, and is quite easy to follow.
• Chapter 2 of Linear and projective representations of symmetric groups by Kleshchev follows the approach of Okounkov and Vershik, and develops the representation theory of the symmetric group in a completely different way, by considering branching of representations, Gelfand-Zetlin bases, and Jucys-Murphy elements. This is a must read, if you work with symmetric group representations and you've never seen it before.

Week 3, Speaker: Victor Demcsak

In this seminar we will take a tour through complex semisimple Lie algebras. The root systems of the Classical Lie algebras will be reviewed in detail. The first landmark to see is the reduced root system, which follows by picking a Cartan subalgebra. Fixing a choice of ordering in the reduced root system leads to the second landmark, the Cartan matrix. The majority of the seminar will focus on the Lie algebra \(A_{n}=sl(n+1,\mathbb{C})\) and in a step by step calculation we produce the reduced root system and compute the Cartan matrix from which we can produce Dynkin diagrams.

References: An introduction to structure theory can be found in Chapter 2 from [SAM]. A detailed discussion of root systems can be found in Chapter 4 of [JAC] and Chapters 10-13 from [EW]. For an introduction to drawing Dynkin Diagrams see Chapters 7-8 in [CAH]. A more terse overview of root systems and Cartan matrices can be found in Chapter 5 from [SER] and Chapter 4 of [HUM].

• [CAH] Semi-Simple Lie Algebras and Their Representations, R. Cahn
• [EW] Introduction to Lie Algebras, K. Erdmann & M. Wildon
• [HUM] Introduction to Lie Algebras and Representation Theory, J. Humphreys
• [JAC] Lie Algebras, N. Jacobson
• [SAM] Notes on Lie Algebras, H. Samelson
• [SER] Complex Semisimple Lie Algebras, J.P. Serre

Week 4, Speaker: Yee Yau

The goal of this talk is to give an introduction to Coxeter groups and an overview of the classification of the finite Coxeter groups; closely following the classical exposition of Humphreys. The key result from which this classification follows is that the finiteness of a Coxeter group depends only on the associated bilinear form of its Tits representation. Hence, the talk will be very accessible to anyone with undergraduate linear algebra.

References:

• Reflection Groups and Coxeter Groups - James E. Humphreys (Chapter 2 for the classification).
• Combinatorics of Coxeter Groups - A. Bjorner, F. Brenti

Week 5, Speaker: Giulian Wiggins

We recount the definitions of projective and injective modules, and of projective covers and injective envelopes. We then derive the equivalence between projective indecomposable modules and simple modules of a finite dimensional algebra. Examples included.

References:

• A note by Henning Krause: https://arxiv.org/abs/1410.2822
• A note by Tom Leinster: https://arxiv.org/abs/1410.3671

Week 7, Speaker: Joel Gibson

We define quivers, the category of representations of quivers, and show some basic structure of simple and indecomposable representations. We cover Gabriel's theorem, which states that those quivers having finitely many indecomposable isomorphism classes of representations are precisely the quivers whose underlying graph is a Dynkin diagram of type A/D/E. We introduce the path algebra of a quiver, and point out some other structure on quiver representations which make them useful in geometric representation theory.

References:

• Chapter 6 of Introduction to Representation Theory by Etingof et. al. contains an account of Gabriel's theorem, although many of the theorems you will have to complete for yourself.
• An overview article on Finite-dimensional algebras and quivers by Alistair Savage.

Week 8, Speaker: Kane Townsend

We will introduce the basic definitions of reflections and unitary reflection groups (URG) noting some examples, the Shephard-Todd classification of irreducible URG and constructing the infinite family G(m,p,m) in detail. We will then see how unitary reflection groups are related to G-invariant polynomials on a complex vector space. In particular we will introduce the symmetric algebra and state the Shephard-Todd-Chevalley Theorem and associated facts that characterise URG in terms of of polynomial invariants. We will accompany this theorem and facts with the example of G(m,p,n).

References:

• Main source: Unitary Reflection Groups by Lehrer and Taylor.
• Supplementary: Reflection Groups and Coxeter Groups by Humphreys, Reflection Groups and Invariant Theory by Kane.

Week 9, Speaker: Gaston Burrull

In this talk, we will present the quantized universal enveloping algebra \(U_q(\mathfrak{sl}_2)\) of the Lie algebra \(\mathfrak{sl}_2\). Let \(V\) be its natural representation. We will define the Jones-Wenzl idempotent as the idempotent endomorphism of the $n$-fold tensor product \(V^{\otimes n}\) into its maximal irreducible module \(V_n\) when \(q\) is not a root of the unity. We will give at least three equivalent definitions of the Jones-Wenzl projector and prove two of them. We will give a diagrammatic description of the Jones-Wenzl projectors in the diagrammatic Temperley-Lieb algebra.

References:

• Appendix A of the paper of Ben Elias and Nicolas Libedinsky "Indecomposable Soergel bimodules for universal Coxeter groups” (2017) for a proof of the existence of Jones-Wenzl projectors and some properties.
• Chapter 2 of Gisa Schäfer thesis "Categorified Uq(sl2)-theory using Bar-Natan’s approach" for general background about the quantized universal enveloping algebra of \(\mathfrak{sl}_2\).

Speaker: Joel Gibson

Worksheet: pdf, tex

In this talk, I cover the basics of category theory, going into some examples such as products, coproducts (and their universal properties), functors, and natural transformations. There is a double-sided worksheet I gave out during the lecture, the first side has a brief listing of definitions (including a couple not covered in the lecture), and the second has some exercises, some of which you should try before the exercise sessions.

For learning stuff about basic category theory, I recommend the following sources:

• Ravi Vakil has a hugenormous set of notes about algebraic geometry. The first chapter of these notes are about category theory and abelian categories, and are written extremely well, with many (often do-it-yourself) proofs and examples. You can find the latest version of his notes at the bottom of this page.
• Categories for the working mathematician by Mac Lane is a great read.

Speaker: Giulian Wiggins

Worksheet: pdf

Speaker: Alex Kerschl

Worksheet: pdf, tex

Speaker: Giulian Wiggins

We will be covering Sections 1-2 of Alistair Savage's notes on categorification.

Speaker: Joseph Baine

We will be covering Section 3 of Alistair Savage's notes on categorification.