• June 20: Robbie Angarone

  • 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).

  • Recording of talk.

• June 22: Carolyn Stephen

  • 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.

  • Slides.

• June 27: Lilly Webster

  • 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!

  • Recording of talk.

  • Slides.

• June 29: Craig Corsi

  • 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.

  • Recording of talk.

  • Slides.

• July 4: No seminar

• July 6: Andy Hardt

  • 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).

  • Recording of talk.

Part 2: Special topics, July 8th - August 3rd

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.

• July 11: Anh Hoang

  • 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. 

  • Recording of talk

• July 13 Special Session: Son Nguyen

  • 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.

  • Recording of talk.

  • Slides.

• July 13: Patty Commins

  • 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!

  • Recording of talk.

  • Slides.

• July 18: Sasha Pevzner

  • 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.

• July 20: Eric Nathan Stucky

  • 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.

  • Recording of talk.

  • Slides.

• July 25: No seminar

• July 27: Sai Sivakumar

  • 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.

  • Recording of talk.

• August 1: No seminar

• August 3: Kaelyn Willingham

  • Title: Quiver representations and neural networks