Algebraic and Combinatorial approaches to representation theory

Abstract. The Weyl algebra arises in many different contexts in mathematics and physics and plays an essential role in the representation theory of nilpotent Lie algebras. This talk will focus on a family of Weyl-like algebras that includes the Weyl algebra itself, various algebras that arise in noncommutative geometry, and many others. The aim is to give a uniform approach to the structure and representations of these algebras.

Abstract. We will define and study deformations along several parameters of the enveloping algebra of a semisimple finite dimensional Lie algebra, called the Generalized Quantum Enveloping (GQE) algebras. We will see how GQE algebras can be used to define Langlands Interpolating Quantum (LIQ) groups and solve conjectures motivated by the geometric Langlands program. These conjectures have been suggested by E. Frenkel and D. Hernandez in an original work on different LIQ groups.

Abstract. This talk is an overview of the known results and open problems about vertex-algebraic structure of the principal subspaces of standard modules for certain untwisted and twisted affine Lie algebras. We discuss presentations of these subspaces and their fermionic characters. The talk is based on joint work with James Lepowsky and Antun Milas.

Abstract. In this talk, we will use Hall Algebras to count rational points of moduli spaces and flag varieties of quiver representations. One of our main tools is the counting character (first used by M. Reineke) from Hall algebras to various quantum tori. This character can play with the full Hopf structure of Hall algebras of quivers to generate interesting results. When playing with comultiplication, the counting character can be specialized to Qin and Rupel's quantum cluster character, which leads to several easy proofs of hard theorems in the cluster algebra.

Abstract. We shall present the explicit forms of certain elementary characters for the classical groups and describe some relations between their explicit forms and the monomial realizations of crystal bases. We also mention the refined polyhedral realizations of crystal bases.

Abstract. Kazhdan-Lusztig cells of Weyl groups are related to primitive ideals of complex semi-simple Lie algebras. We will explain a conjectural description of left cells in terms of Galois theory for Calogero-Moser spaces, which are deformations of symplectic singularities constructed by Etingof and Ginzburg from rational Cherednik algebras (joint work with Cedric Bonnafé).

Abstract. A tight fusion frame is a sequence of orthogonal projection matrices which sum to a scalar multiple of the identity. To any such sequence, we can associate a weakly decreasing sequence of positive integers given by the ranks of these projections.

The question we address is the following: For which sequences of positive integers do tight fusion frames exist?

In this talk, I will discuss joint work with K. Luoto and M. Bownik where we explore this problem. In particular, we give a combinatorial characterization in terms of nonvanishing Littlewood-Richardson coefficients. Classically, Littlewood-Richardson coefficients appear in the representation theory of $GL_n$ as the structure constants when decomposing a tensor product of irreducible representations. This connection between frame theory and algebraic combinatorics yields several interesting results in both fields of mathematics.

Abstract. In this talk I will describe the non-initial cluster variables in a quantum cluster algebra in terms of the representation theory of a corresponding quiver. Time permitting, I will explain why quantum cluster characters should have ever been discovered by realizing them as the image under a certain algebra homomorphism from the Ringel-Hall algebra of the quiver to the initial quantum torus.

Abstract. In my talk, we would like to discuss the following questions: Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, and fix a dominant integral weight $\lambda$. Assume that $\mu$ runs over all dominant integral weights such that $\lambda-\mu$ are also dominant.

Question 1. Determine the condition for $\mu$ so that the dimension of $V(\lambda) \otimes V(\lambda-\mu)$ is maximal. Here, $V(\lambda)$ and $V(\lambda-\mu)$ are irreducible $\mathfrak{g}$-modules of highest weight $\lambda$ and $\lambda-\mu$, respectively.

Question 2. Determine the condition for $\mu$ so that the character of $V(\lambda) \otimes V(\lambda-\mu)$ is "maximal" in the sense that the multiplicity of $V(\nu)$ in $V(\lambda) \otimes V(\lambda-\mu)$ is maximal for every dominant integral weight $\nu$.

Abstract. Suppose a finite group acts on a scheme (or algebraic variety) $X$ and a "target'' Lie superalgebra $\mathfrak g$. Then the space of equivariant algebraic maps from $X$ to $\mathfrak g$ is a Lie superalgebra under pointwise multiplication. We call this an "equivariant map superalgebra". An important class of examples are the (twisted) loop superalgebras, where the variety $X$ is the one-dimensional torus.

In this talk we will present a classification of the irreducible finite-dimensional representations of an equivariant map superalgebra where the target is a basic classical Lie superalgebra and the group in question acts freely on $X$. It turns out that all irreducible finite-dimensional representations are generalized evaluation representations. In the case that the even part of $\mathfrak g$ is semisimple, they are in fact all evaluation representations. As a corollary of our general result, we obtain the first classification of the twisted loop superalgebras.

Abstract. Consider a symmetrizable Kac-Moody algebra. In finite type, Mirkovic-Vilonen (MV) polytopes give nice combinatorial realizations of Kashiwara's crystals. These polytopes were originally defined using the geometry of the affine grassmannian, but they also arise naturally in several other contexts. For instance, they can be understood in terms of quiver varieties, or in terms of Khovanov-Lauda-Rouquier (KLR) algebras. Both of these points of view have the advantage that they make sense beyond finite type. I will explain as much of this story as I can, focusing on recent work with Ben Webster developing the KLR algebra construction. In particular, we obtain MV polytopes in all affine types which can be described combinatorially and which have many properties analogous to the finite-type situation.

Abstract. Suppose $V$ is a representation of a complex simple Lie algebra that can be written as a tensor product of irreducible representations. A theorem of C.S. Rajan states that the irreducible factors that occur are uniquely determined, up to reordering, by the isomorphism class of $V$. I will present an elementary proof of Rajan's theorem, which generalizes with no extra effort to the infinite dimensional (Kac-Moody) Lie algebras. This is joint work with S. Viswanath.