Welcome to the homepage of the UMN Student Summer Representation Theory Seminar, summer 2023! This year, I am organizing the seminar with fellow graduate student Carolyn Stephen.
The seminar meets at 3:30pm Central on Tuesdays and Thursdays in Vincent Hall 570.
You can also join virtually via Zoom; a meeting link will be shared with those on the mailing list. Feel free to reach out if you would like the Zoom link.
The goal of this seminar is to get the UMN community excited about representation theory! It's a beautiful subject with applications in essentially every other area of math. The seminar will be broken into two parts: first, a three-week boot camp to introduce the main ideas of representation theory; and second, a variety of talks about special topics in representation theory.
All members of the UMN math community are encouraged to attend; this includes (but is not limited to) incoming first-year graduate students, undergraduates, and participants in the UMN mathematics REU programs. Of course, other current graduate students, postdocs, professors, staff, and alumni are encouraged to attend as well!
During the boot camp, we will meet Tuesdays and Thursdays. Our goal is to provide a broad overview of classical and well-studied aspects of representation theory. We intend to focus on examples, explicit computation, and do exercises together. All members of the UMN community are welcome to attend! We will assume a familiarity with group theory and linear algebra, but no previous exposure to representation theory specifically.
The seminar meets at 3:30pm Central on Tuesdays and Thursdays in Vincent Hall 570.
Title: The language of representation theory
Abstract: In this introductory talk and exercise session, we'll discuss what a representation is and compute examples. We'll also unpack some the many equivalent ways people talk about representations. Finally, we'll introduce two of the most important theorems in representation theory: Maschke's theorem and Schur's lemma. The goal of this session is to build intuition and vocabulary about representations, so that when they appear later on, they seem less mysterious.
Slides and exercises (scroll to the end for exercises).
Title: Characters of finite groups
Abstract: Characters of group representations are quite straightforward to compute, but encode essential information about a group and its representations. In this talk, we will explore character theory of finite groups over fields of characteristic 0, paying special attention to character tables and orthogonality relations. We will also use guiding examples and exercises to illustrate some of the most remarkable results in character theory.
Title: A Crash Course in Representations of Sn
Abstract: Our first deep dive into a particular family of representations will be focused on the symmetric group. In this talk, we will give an overview of the classical theory of symmetric group representations, focusing on concrete examples. We will explicitly construct the irreducible representations of the symmetric group and give an overview of their remarkable properties. If you’ve ever heard “the irreducible representations of Sn are indexed by the partitions of n” and wondered what was going on, this is the talk for you!
Title: Quiver algebras and quiver representations
Abstract: Quiver algebras are rings with the combinatorial structure of a finite directed graph, and their module theory can be extensively described in the language of this combinatorial structure. We define quiver algebras and quiver representations and discuss several examples. We identify the simple, semisimple, projective, and injective modules of these algebras. We then discuss the extent to which quiver algebras can be a general framework for the representation theory of arbitrary finite-dimensional algebras.
Title: An introduction to the representations of Lie groups and Lie algebras
Abstract: Lie groups are groups that are also manifolds, and these encompass some of the most useful infinite groups such as the general linear group, the special linear group, the orthogonal group, and the symplectic group. Every Lie group has a Lie algebra, which is its tangent space at the identity. Their representation theory is related, and so we often study the Lie algebra since it has the advantage of being a vector space. In this talk, we will primarily consider the simplest case of sl_2(C), the Lie algebra of SL_2(C). Each finite-dimensional irreducible representation of sl_2(C) is the direct sum of “weight spaces”, and the “highest weight” classifies the representation. We will work through concrete examples, and hence understand the classification of representations of sl_2(C).
For the second part of the seminar, we will meet on Tuesdays and Thursdays. Our goal is to provide a space for the UMN community to learn about the diverse applications of representation theory. We hope to place special emphasis on the role of representation theory in cutting-edge & ongoing research.
The seminar meets at 3:30pm Central on Tuesdays and Thursdays in Vincent Hall 570.
Title: Introduction to representation stability
Abstract: We will motivate and survey a few examples of the representation stability phenomenon, with an emphasis on those in topology.
Title: Growth Diagrams for Bumpless Pipe Dream Insertion
Abstract: The well-known RSK correspondence is a bijection between matrices with non-negative integer entries and pairs of column-strict tableaux. The growth diagram, developed by Fomin, is a tool to break down this complicated correspondence into a simple set of local rules. Recently, Huang and Pylyavskyy introduced RSK insertion on bumpless pipe dreams that recovers the classical RSK as a special case. In this talk, we will introduce the growth diagram and local rules for this bumpless pipe dream insertion.
Title: The representation theory of finite monoids
Abstract: The representation theory of finite groups is a widely appreciated and well-understood subject. What happens if we relax our algebraic object of study to a monoid? In this talk, we'll outline the main ideas of the surprisingly beautiful subject of representation theory of finite monoids, focusing our examples on monoids with combinatorial connections. Although the basic theory has been known to semigroup specialists for quite some time, it has gained popularity with more mathematicians in recent years due to emerging connections with Markov chains and algebraic combinatorics. If you were sad to miss Peter Webb's reading course on this topic, this talk is for you!
Title: Representation stability part II: FI modules
Abstract: FI modules encode the information of symmetric group representations and equivariant maps between them, all in one compact package. Church, Ellenberg, and Farb showed that finiteness properties of FI modules imply representation stability for a sequence of representations. In this talk, we will provide an introduction to the topic of FI modules and discuss which conditions must hold in order to deduce stability. Time permitting, we will discuss some applications of FI modules in commutative algebra and (characteristic p) group cohomology. This talk will include plenty of examples.
Title: The Many Functions of Parking Spaces
Abstract: Parking functions arose independently in statistics and computer science. However, mathematicians quickly noted that these objects have a natural symmetric group action, yielding a representation that today is called the "parking space." At the turn of the century this representation gained significantly more research interest as the central object in a network of algebro-geometric conjectures. In this talk we will provide several constructions of the parking space, including abstractly as a sum of irreducibles. Time permitting, we will describe further structure on the parking space needed to state the Shuffle Conjecture.
Title: Fourier analysis on LCA groups and Pontryagin Duality
Abstract: We will lay the foundations for recovering the classical Fourier transforms on the circle group and on the real line, and time permitting, briefly talk about Pontryagin duality. Topics to be discussed are irreducible unitary representations of locally compact Abelian groups, their characters, the dual group, the Fourier transform, Pontryagin duality, and Fourier inversion.
Title: Quiver representations and neural networks
Information about past iterations of the SSRTS can be found here: 2021, 2019-2020, 2016-2018.
Here are some of the speakers' favorite resources for self-studying these topics. (The list is still being updated!)
Representation theory in general:
Fulton and Harris, Representation Theory
Representations of the symmetric group and symmetric functions:
Sagan, The Symmetric Group
Stanley, Enumerative Combinatorics, vol. 2