Algebraic and Combinatorial approaches to representation theory

Department of Mathematics, University of California Riverside

Last modified on October 4, 2012


Friday, May 18 Saturday, May 19 Sunday, May 20
9:30-10:30Breakfast 9:30-10:30P. Tingley
10:00-11:00Coffee10:30-11:30 R. Rouquier10:45-11:15A. Bouayad
11:00-12:00G. Benkart 11:45-12:45 E. Richmond11:30-12:30A. Savage
Lunch Lunch
2:00-3:00C. Calinescu 2:15-3:15J. Fei
3:15-3:45R. Venkatesh 3:30-4:00D. Rupel
4:00-5:00D. Sagaki 4:15-5:15T. Nakashima


Georgia Benkart (University of Wisconsin, Madison, USA)
A Talk on the Weyl Side

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.

Alexandre Bouayad (Université Paris VII, France)
Generalized quantum enveloping algebras and Langlands interpolating quantum groups

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.

Corina Calinescu (Yale University, USA)
Vertex-algebraic structure of representations of affine Lie algebras

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.

Jiarui Fei (UC Riverside, USA)
Hall algebra and counting

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.

Toshiki Nakashima (Sophia University, Tokyo, Japan)
Elementary characters and monomial realizations of crystal bases

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.

Raphaël Rouquier (UCLA, USA and University of Oxford, United Kingdom)
Kazhdan-Lusztig cells and Calogero-Moser spaces

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).

Edward Richmond (University of British Columbia, Canada)
Littelwood-Richardson coefficients

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.

Dylan Rupel (University of Oregon, Eugene, USA)
Quantum Cluster Characters

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.

Daisuke Sagaki (University of Tsukuba, Japan)
Maximal dimension of tensor products and Schur positivity for classical Lie algebras

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 $\nu$.

Alistair Savage (University of Ottawa, Canada)
Equivariant map superalgebras

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.

Peter Tingley (MIT, USA)
Various constructions of Mirkovic-Vilonen polytopes

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.

R. Venkatesh (Institute of Mathematical Sciences, Chennai, India)
Unique factorization of tensor products for Kac-Moody algebras

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.

List of participants

Alyssa ArmstrongNorth Carolina State University, USA
Irfan BagciUC Riverside, USA
Georgia Benkart*University of Wisconsin Madison, USA
Matthew BennettUC Riverside, USA
Alexandre Bouayad*Université Paris VII, France
Kathryn BrennemanNorth Carolina State University, USA
Corina Calinescu*Yale University, USA
Vyjayanthi ChariUC Riverside, USA
Konstantina ChristodoulopoulouUniversity of Connecticut, USA
John DuselUC Riverside, USA
Christina ErbacherNorth Carolina State University, USA
Jiarui Fei*UC Riverside, USA
Joel GeigerLouisiana State University, USA
Jacob GreensteinUC Riverside, USA
Jonas HartwigStanford University, USA
Ines HenriquesUC Riverside, USA
Mana IgarashiSophia University, Tokyo, Japan
Garrett JohnsonNorth Carolina State University, USA
Miroslav JerkovicUniversity of Zagreb, Croatia
Gizem KaraaliPomona College, USA
Apoorva KhareStanford University, USA
Deniz KuzUniversität zu Köln, Germany
Mathew LundeUC Riverside, USA
Chad MangumNorth Carolina State University, USA
Nathan ManningUC Riverside, USA
Kailash MisraNorth Carolina State University, USA
Peter McNamaraStanford University, USA
Adriano de Moura UNICAMP, Brazil
Swarnava MukhopadhyayUNC Chapel Hill, USA
Toshiki Nakashima*Sophia University, Tokyo, Japan
Daniel OrrUNC Chapel Hill, USA
Tu PhamUC Riverside, USA
Edward Richmond*University of British Columbia, Canada
Tim RidenourNorthwestern University, USA
Raphaël Rouquier*UCLA, USA and University of Oxford, United Kingdom
Dylan Rupel*University of Oregon, Eugene, USA
Daisuke Sagaki*University of Tsukuba, Japan
Ben SalisburyUniversity of Connecticut, USA
Alistair Savage*University of Ottawa, Canada
Prasad SenesiCatholic University of America, USA
Sachin SharmaInstitute of Mathematical Sciences, Chennai, India
Jie SunUC Berkeley, USA
Akaki TikaradzeUniversity of Toledo, USA
Peter Tingley*MIT, USA
R. Venkatesh*Institute of Mathematical Sciences, Chennai, India

* speaker

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