Summer Representation Theory Seminar

Welcome 🥰

June 20 Craig Corsi [UMN Alum]

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. 

• Slides

June 27 Sarbartha Bhattacharya [UMN]

Title: Deligne-Lustig Theory : Finite groups of Lie type and beyond. 

Abstract: The representations of finite groups of Lie type (for example Gln(F_q)) has a long history. Perhaps the most significant contribution in this field was made by Deligne and Lustig in the 70s, who gave a complete and uniform description of the representations of these groups. Ideas used in their construction have since been used in several other contexts, and has been an integral part of geometric representation theory. We will explore the main ideas in this theory, often keeping the specific example of Sl2(F_q) in mind. Time permitting, we will see how this idea has been used in the representation theory of several other objects, maybe most important of them being p-adic groups. 

July 11 Eric Nathan Stucky [UMN Alum] website

Title: To Lie or Not to Lie: Associative Analogues of the PBW Theorem 

Abstract: The Poincaré-Birkhoff-Witt Theorem states that the universal enveloping algebra of a Lie algebra has a basis of standard monomials even though it is usually non-commutative. Similar results exist in the associative setting, notably for the Weyl algebra. In this talk we will first define the Weyl algebra, stating its PBW theorem, and discussing the analogy to the classical Lie-theoretic result. We then describe a "universal enveloping algebra" analogue for group representations, and precisely describe when such an algebra satisfies a PBW theorem 

• Slides

July 18 Andrew Douglas Frohmader [UWM]

Title: Quantum Groups, Crystal Bases, and Tableaux 

Abstract:  Quantum groups were introduced independently by Drinfeld and Jimbo in the 80s in their study of the quantum Yang-Baxter equation. Since their introduction, quantum groups have turned up in many branches of math. We will introduce the quantum groups appearing as deformations of universal enveloping algebras of Kac-Moody algebras and discuss their crystal bases. Specializing to enveloping algebras of Lie algebras, the crystal bases can be realized as tableaux and provide combinatorial models of representations of Lie groups. 

July 25 Laura Colmenarejo [NCSU] website

Title: An insertion algorithm on multiset partitions with applications to diagram algebras. 

Abstract: In algebraic combinatorics, the Robinson-Schensted-Knuth algorithm is a fundamental correspondence between words and pairs of semistandard tableaux illustrating identities of dimensions of irreducible representations of several groups. 

In this talk, I will present a generalization of the Robinson-Schensted-Knuth algorithm to the insertion of two-row arrays of multisets. This generalization leads to new enumerative results that have representation-theoretic interpretation as decomposition of centralizer algebras and the spaces they act on. I will also present a variant of this algorithm for diagram algebras that has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. 

August 1 Alexander Wilson [Oberlin] website

Title: Crystal Structures on Bitableaux 

Abstract: The Kronecker problem is a long-standing problem in combinatorial representation theory. The problem asks for a combinatorial interpretation of the Kronecker coefficients which appear in the decomposition of the tensor products of irreducible symmetric group representations. In this talk we will see how objects called lexicographic bitableaux arise naturally in the study of these coefficients and how the problem can be rephrased as putting something called a crystal structure on these tableaux.