For summer 2020, this seminar was organized by myself and Cecily Santiago. It met Tuesdays/Thursdays at 12.20pm for the first two weeks as a rep. theory bootcamp, then Tuesdays at 12.20pm for the remainder of the summer. All seminar meetings were on Zoom, due to COVID.
8/25/20: Suki Dasher
Title: Galois Representations
Abstract: A Galois representation is a representation of an absolute Galois group. In contrast with the simplicity of this definition, there is mounting evidence that Galois representations are intimately connected to other areas of mathematics in deep and mysterious ways. This talk will introduce the concepts of infinite Galois theory and Galois representations and will describe the absolute Galois group of the rational numbers and its representations. We will also touch on some results and conjectures about the relationship between Galois representations and other areas of mathematics. The content will be mostly non-technical and little background beyond basic graduate algebra will be assumed, but it sure would help.
8/18/20: Neelima Borade
Title: Reflection Functors and Gabriel's Theorem
Abstract: Several problems in linear algebra can be studied using ideas such as the Coxeter Weyl group and Dynkin diagrams. In this talk we will illustrate this technique by discussing Gabriel's Theorem and the correspondence between classes of indecomposable quiver representations and positive roots. We will also define reflection functors, which are used to prove Gabriel's theorem. No prior background on root systems will be assumed.
8/11/20: Sunita Chepuri
Title: Combinatorics of Kazhdan--Lusztig Polynomials
Abstract: In this talk we introduce Kazhdan--Lusztig polynomials and how they arise in the representation theory of Hecke algebras. Although these polynomials have many uses in representation theory and algebraic geometry, they are notoriously difficult to compute. We will discuss combinatorial formulas to compute these polynomials in several cases.
8/4/20: Andy Hardt
Title: Crystal Bases
Abstract: Crystal bases were defined by Kashiwara as certain nice bases of quantum group modules. These bases respect (in a technical sense) the raising and lowering operators of the quantum group, and allow us to perform certain functorial operations, such as tensor products and restriction to Levi subgroups, on the level of bases instead of modules. Since Kashiwara, the study of "crystals", the axiomatic definition of a crystal basis, has taken on a life of its own, with connections to symmetric functions, tableaux combinatorics, reduced words in Coxeter groups, cluster algebras, alcove walks, Demazure characters, and more.
In this talk, we're going to stick close to the representation theory. We'll define a crystal basis, give a concrete type A definition in terms of tableaux, and finish with a short description of Kashiwara's "universal highest weight crystal" that allows one to obtain Lusztig's canonical basis for the quantum group itself.
This talk will assume a basic familiarity with root systems and highest weight theory. Any standard Lie theory course or textbook should be sufficient. Alternatively, the following lectures by Ben Brubaker give a concise exposition of the relevant material: semisimple Lie algebras, Cartan matrices, highest weight theory.
7/28/20: Elizabeth Kelley
Title: Introduction to AR quivers
Abstract: In this talk, we'll use an orientation of A5 as the central example for exploring some basic notions of AR quivers. Along the way, we'll define irreducible morphisms, almost split sequences, the Nakayama functor, and the AR translation. We'll explicitly use the "knitting algorithm" to draw the AR quiver and will briefly mention other techniques. Finally, we'll show how the AR quiver can be used to compute dim Hom(M,N) and dim Ext1 (M,N), and find short exact sequences of the form 0 → N → E → M → 0 for any indecomposable representations M, N of Q.
7/21/20: Libby Farrell
Title: Is it Cake? Is It a Quiver Rep?
Abstract: This week, a popular internet trend has taught us that yes, everything is cake. For most of us, this is an exciting realization because cake is a sweet and delicious treat.
In this talk, we will learn that everything is a quiver rep. More precisely, every representation of a finite dimensional algebra is a quiver rep in disguise. This is even more exciting than everything being cake, as we will see that quiver reps are easy to work with and well understood.
(For more info about Everything is Cake this article explains the recent phenomenon: https://www.google.com/amp/s/amp.theguardian.com/travel/2020/jul/16/cake-slicing-meme-videos-explanation)
7/14/20: Cecily Santiago
Title: Representation Stability
Abstract: Representation stability is a tool used to study sequences {Vn} of representations for sequences of groups {Gn}. If such a sequence stabilizes, it tells us about the representations for large n. This talk gives an overview of representation stability, including its history, some examples, a few major tools.
7/7/20: Esther Banaian
Title: Cluster Category from Marked Surfaces
Abstract: We will discuss the categorification of cluster algebras from marked surfaces. This will allow us to understand some technical but important concepts with pictures. All definitions will be provided and there will be many examples - no familiarity with cluster algebras or categories required!
6/30/20: John O'Brien
Title: Spherical varieties and Hecke modules
Abstract: Of increasing importance to the Langlands program are spherical varieties--varieties with a reductive group action with an open, dense Borel orbit. Examples include such ubiquitous spaces as the projective line, flag varieties, and toric varieties. In this talk, we will discuss a result of Mars and Springer relating the algebraic geometry of spherical varieties to modules over Hecke algebras. Loosely, the geometry of a spherical variety relates to the geometry of the reductive group in the same way an R-module relates to R. Using categories of sheaves on spherical varieties and the notions of Grothendieck groups, we will make this statement precise.
6/23/20: Claire Frechette
Title: New Yang-Baxter Equations for Metaplectic Ice
Abstract: Last week, Emily talked about representations of the general linear group GL2 over a finite field. Much of the spirit behind that case extends up into the representations of GLr, for all r over a local field, and then even further to representations of covering groups of GLr over local fields. Recently I've been working with a special set of functions called Whittaker functions, which are defined on principal series representations of these covering groups, extending a result of Brubaker, Buciumas, and Bump that shows that they can in fact be given by the partition functions of a particular set of lattice models. In this talk, we will construct these covering groups explicitly and talk about when we can distinguish them from one another, then construct the principal series representations on them, and finally, if we have time, talk about Whittaker functions and their relationship to lattice models
6/18/20: Emily Tibor
Title: Representations of GL(2) over a finite field
Abstract: In this last talk of the boot camp, we will apply our knowledge from previous talks to study complex representation theory of GL2(q), the group of 2x2 invertible matrices with entries in a finite field of order q>2. I'll briefly describe some motivation for the study of the representation theory of this group, from the perspective of automorphic representations. There will be a brief aside, where we add a couple more tools to our toolbox, namely induction and restriction. Then, we will be ready to describe the irreducible representations of GL2(q) and, given time, observe the analogue of the Local Langlands Correspondence for this group.
6/16/20: Cecily Santiago
Title: Representations of the Symmetric Group
Abstract: The symmetric group, Sn, is a classic example in representation theory. It is particularly nice, because in characteristic 0 we can construct its irreducible representations. These are called Specht Modules. This talk will walk us through the construction of Specht modules, from Sn through partitions, Young Diagrams, Young Tableaux, and tabloids. Lots of examples will be given.
6/11/20: Andy Hardt
Title: Representations of Finite Dimensional Algebras, Schur's Lemma, and Semisimplicity
Abstract: In this second bootcamp talk, we're going to start looking at representations from a slightly more sophisticated perspective. We'll introduce representations of finite dimensional algebras, which generalize those of finite groups. One thing we lose when we step away from finite groups (or from characteristic zero) is the property that every representation decomposes into a direct sum of irreducibles. This property is called semisimplicity, and it is highly desirable! We will explain what it means for an algebra to be semisimple, and use a powerful object called an endomorphism ring to decompose the regular representation of a semisimple algebra.
6/9/20: Claire Frechette
Title: Introduction To Representation Theory (Bootcamp Talk 1)
Abstract: This talk will cover the basic definitions and tools of representation theory for finite groups, assuming no prior background in the subject. Topics include: irreducible representations are (and how to find them), Maschke's theorem, and an introduction to character theory. If you've ever heard the word "representation" and never really known what it meant or why you should care, this is a good talk for you!
For summer 2019, this seminar met Mondays and Wednesdays at 1.30 pm in Vincent Hall 364. These are the talk announcements and notes in reverse chronological order.
8/12/19: Dev Hegde
Title: Lie groups representations, some history and a story.
Abstract: The talk will give some historical details and some results in the representation theory of Lie groups surrounding the Plancherel formula.
7/31/19: Sarah Brauner
Title: Eulerian representations for reflection groups of coincidental type
Abstract: The Eulerian representations of the symmetric group are a topic of longstanding interest to combinatorialists, representation theorists and topologists. Less understood are the surprising connections between this family of representations and coincidental type reflection groups---the complex reflection groups whose degree sequence forms an arithmetic progression, and which can be generated by as many reflections as the dimension of the vector space on which they act. I will summarize many of the features that make the Eulerian representations so compelling, and discuss efforts to generalize these properties to the context of complex reflection groups of coincidental type.
7/29/19: Andy Hardt
Title: Representations of Rook and Renner Monoids
Abstract: We talk about Solomon’s reconception of Munn’s rook monoid representation theory. The rook monoid is the set of 0,1 matrices with at most one 1 in each row and column. Rook monoids also fit into a larger picture: they are “Renner monoids” of type A, the monoid equivalent of the Weyl group of a reductive group. Li, Li, and Cao extended Solomon’s results to all Renner monoids, and other work has been done on the associated Hecke algebras.
7/24/19: Claire Frechette
Title: Metaplectic Ice (i.e. Almost REU Problem 2 meets What REU Problem 8 Was Supposed to Be)
Abstract: We explore the principal series representations of central extensions of reductive groups in the p-adic case (i.e. REU Problem 8 in the non-finite case). On each of these representations, we can define Whittaker functions, which are important functions for number theory. Amazingly, we can construct a six-vertex ice model that gives these Whittaker functions as its partition function.
7/22/19: Jorin Schug
Title: Representation Rings and theorems in Equivariant Geometry
Abstract: While representation theory of groups pops up throughout algebraic geometry, one of the applications that gained traction in the mid-20th century was to equivariant geometry, the study of algebraic groups and their actions on varieties. In this talk, we will begin by examining in detail a remarkable association between varieties associated to representation rings and the original group. Then we will take a quick tour of how this association informs research in equivariant geometry and intersection theory.
7/17/19: Esther Banaian
Title: A link between quiver representations and cluster algebras
Abstract: We will discuss the Caldero-Chapton (or cluster character) function, which, in nice cases, assigns a cluster variable to each indecomposable representation of a quiver. Nearly all statements will be supplemented with examples, and no background in cluster algebras is assumed for this talk.
Notes and Main Reference (“Solutions” are available in Examples 3.1 and 3.3. The left side of top of table is P1, the right is I2 in 3.1. Note that I2 should have numerator 1+ 2u2 + u2^2 + u1u3.)
7/15/19: Emily Tibor
Title: Representations of GL2 over a finite field
Abstract: In this talk we will study complex representation theory of GL2(q), the group of 2x2 invertible matrices with entries in a field of order q>2. Many constructions and phenomena for representation theory of GL2(q) generalize in some way to p-adic representation theory for reductive algebraic groups, which are of particular interest to number theorists. We will begin with an overview of the representation theory for GL2(q), and then observe the analogue of the Local Langlands Correspondence for this group.
7/10/19: Lilly Webster
Title: An Introduction to Quiver Representations
Abstract: In this talk, we will cover the basics of the representation theory of quivers. We will start with the definition of quiver and a representation, then proceed to discuss indecomposable representations and ways to get new representations from old ones. We will also discuss the category Rep(Q) and its relationship to the path algebra of a quiver. No background in quivers is assumed for this talk.
7/3/19: Craig Corsi
Title: Cyclic sieving on semistandard Young tableaux via representation theory
Abstract: This talk explores the cyclic sieving phenomenon (CSP) as it applies to the set of semistandard Young tableaux of a fixed shape. We present two representation-theoretic approaches to exhibiting CSP, one arising from jeu-de-taquin promotion, and the other coming from a type An crystal structure.
Notes
7/1/19: Peter Webb
Title: A panorama of representation theory
Abstract: What is representation theory and why should we study it? Often it seems that we study a piece of theory for the reason that it is there to be studied, or some other person (a teacher, the person giving a seminar, some famous mathematician) appears to think it is worth studying. Character theory of finite groups is wonderfully elegant and surprising, but why should we bother to study it in the first place? I will show how representation theory applies in diverse interesting situations, outside the subject of representation theory and even outside of mathematics. My examples will all be representations of categories, which include representations of groups and representations of quivers.
Notes: official and unofficial
6/26/19: Andy Hardt
Title: Highest Weight Representations of sl2(ℂ)
Abstract: A Lie group is “a group that is also a manifold”, and its Lie algebra can be seen as the tangent space at the identity. This important object is both amenable to study (it is a vector space, for instance) and closely related to the original Lie group (in many cases, there is a correspondence between their representations). We will look at the case of sl2(ℂ), the Lie algebra of SL2(ℂ). This most basic example is both a building block for, and gives the flavor of, the beautiful general picture. Each finite-dimensional irreducible representation of sl2(ℂ) is the direct sum of “weight spaces”, and the “highest weight” classifies the representation. This classification leads to the result that every finite dimensional representation of sl2(ℂ) is completely reducible. We won’t have time to go into detail with any of this, but we’ll do our best to paint the rough picture.
6/24/19: Claire Frechette
Title: Representations of Sn
Abstract: The symmetric group is one of the most fundamental groups in all of group theory, so it would be almost criminal to end our crash course in representation theory without discussing its representations. In this talk, we'll go through the construction of these representations and discuss some of their wonderful properties, including the Branching Rule and the Murnaghan-Nakayama Rule, which give us nice ways to construct the character table for Sn.
6/19/19: Imposter Syndrome Panel
Abstract: In place of the usual mathematical Student Summer Representation Theory Seminar, we're going to talk about a more psychological kind of (self-)representation theory today: we're hosting a panel on Impostor Syndrome with grad student panelists. Whether you haven't heard about it yet, think you've maybe already had it, or just want to hear about our experiences, come and ask any questions you might have. Older grad students are also free to come sit in and share your experiences.
6/17/19: Katy Weber
Title: Schur-Weyl Duality
Abstract: In this talk, we give an overview of the classical version of Schur-Weyl duality, which describes a precise relationship between certain representations of the general linear group and of the symmetric group. We motivate the discussion with the simplest interesting example, namely that of GL2(ℂ) and S2, and then detail the key features that underlie the general duality.
6/12/19: Eric Stucky
Title: Decomposing Symmetric Algebras
Abstract: Given a finite-dimensional G-representation, we can produce an infinite-dimensional (!) G-representation called its symmetric algebra. We will start by writing down some examples of these algebras, and then state Molien's formula, which describes how this representation decomposes into irreducibles. We will give a detailed sketch of the proof, since it offers several opportunities to point out generally useful techniques. Time permitting, we may compare and contrast with the exterior algebra, which has a similar construction.
6/10/19: Andy Hardt
Title: Basic Operations on Representations
Abstract: We continue our crash course in finite group representation theory by looking at some important operations on representations. We start by defining the group algebra of a finite group; group representations naturally biject with modules over the group algebra. After that, we'll talk through a variety of ways to construct new representations from old, such as restriction, induction, inflation, tensor product, and we may even squeeze in symmetric and exterior powers.
6/6/19: Andy Hardt and Sarah Brauner
Title: A crash course in representation theory for finite groups
Abstract: In this talk, we will cover the basics of representation theory for finite groups, assuming no prior background in the subject. This will include a discussion of what irreducible representations are (and how to find them), Maschke's theorem, and an introduction to character theory. If you've ever heard the word "representation" and never really known what it meant or why you should care, this is a good talk for you!