Summer 2021
Organizers: Cecily Santiago and Eric Nathan Stucky
The seminar will be held virtually this year. We will meet on Tuesdays at 3:30pm CDT
Please email Cecily if you're interested in giving a talk or to request more information
Schedule
June 22
Speaker: Andy Hardt
Title: Basics of Representation Theory
Abstract: We will define and introduce representations in the nicest case: finite groups and complex vector spaces. We'll keep a running example involving the symmetric group \mathfrak{S}_3, which will help us explore basic concepts such as subrepresentations, irreducibility, and Maschke's Theorem. One important tool is character theory, and this will help us learn about the representations of a group and even the group itself.
June 29
Speaker: Esther Banaian
Title: Introduction to Quiver Representations
Abstract: We will introduce quiver representations from scratch. We will discuss the category of quiver representations, compare it to some module categories, and study some special indecomposable representations. There will be examples and exercises to keep things engaging and hands-on!
July 6
Speaker: Kayla Wright
Title: Quiver Representations Part II
Abstract: Last week, Esther introduced quiver representations. In this talk, we will be talking about a classification of quiver representations. We will also discuss why quiver representations are way more useful than they have any right to be by introducing the notion of Morita equivalence.
July 13
Speaker: Eric Nathan Stucky
Title: Group Algebras & Induced Representations
Abstract: In the first part of this talk, we take a meandering path to connect the two "flavors" of representation theory that have been discussed so far in the seminar. The second part concerns induction, a method for constructing new representations from simpler ones that is rather more subtle than other tools that we have seen thus far. The two parts are largely independent from each other and also from previous weeks: relevant results will be reviewed as they are discussed.
July 20
Speaker: Trevor Karn
Title: Equivariant Kazhdan-Lusztig polynomials and their combinatorics
Abstract: Tableaux combinatorics and representation theory (in particular the representation theory of the symmetric group) are deeply intertwined. In the past five years, there has been deep study into a polynomial known as the matroid Kazhdan-Lusztig polynomial. Among the results which have been proven over this time are a description of the coefficients of the polynomial in terms of standard skew Young tableaux due to Lee, Nasr, and Radcliffe which was proven purely combinatorially, and a generalization of the polynomial via representation theory due to Gedeon, Proudfoot, and Young. In May of 2021, Gao, Xie, and Yang, showed that the representation theoretic approach provides an alternate proof of the description in terms of tableaux. I plan to outline this general thread of research and if time permits state questions raised by the work of Gao, Xie, and Yang which are currently being examined by Nasr and myself.
July 27
Speaker: Robbie Angarone
Title: Characters of the infinite symmetric group and total positivity
Abstract: Character theory for finite groups is a particularly neat and beautiful subtopic in representation theory, with the characters of the symmetric group being its crown jewel. Total positivity, on the other hand, is a fascinating topic from linear algebra—originally studied in physics, it eventually gave way to beautiful combinatorics, including the discovery of cluster algebras. In this talk, we explore a natural generalization of characters to an infinite version of the symmetric group. We will see that these characters are in bijection with certain infinite totally positive matrices. En route to this surprising connection, we will utilize symmetric functions and discuss their basic properties, partially revealing why they are so important to both representation theory and total positivity.
August 3
Speaker: Esther Banaian
Title: Auslander-Reiten theory
Abstract: We dig deeper into the category of representations of a quiver by learning how to draw the Auslander-Reiten quiver of a quiver. Hands-on examples will accompany almost every concept.
August 10
Speaker: Libby Farrell
Title: Cluster Categories and Dissected Polygons
Abstract: We will explore the interplay between combinatorics, algebra, and representation theory by using the quiver representations to categorify cluster algebras from triangulated polygons. Along the way, we will discover how the representation theoretic data of the cluster category is captured by triangulations of polygons.
August 17
Speaker: Patty Commins
Title: Representations of Type B Groups
Abstract: The representation theory of symmetric groups (groups of type A) is a celebrated and well-understood subject, highlighting the value of combinatorics in understanding algebraic structures. It is natural to wonder which ideas carry over to related groups. In this talk, we will consider the representation theory of groups of type B (the hyperoctahedral groups) and how it compares to that of the symmetric groups. Any necessary ideas from symmetric group representation theory will be reviewed.
August 24
Speaker: William Dudarov
Title: Quantum Groups and Crystal Bases
Abstract: Quantum groups, introduced by Drinfeld and Jimbo in the study of the solvability of lattice models in mathematical physics, are, of course: (1) not quantum, and (2) not groups. The "quantum" here refers to a "deformation" of the universal enveloping algebra of a semisimple Lie algebra, or a Kac-Moody algebra, with a Hopf algebra structure with a parameter q, where the original universal enveloping algebra is recovered at q=1. The combinatorial theory of crystal bases developed by Kashiwara helps us think about the representation theory of these objects.