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Department of Mathematics

University of North Carolina at Chapel Hill

Fridays 4pm, PH-367 or PH-385

The aim of this seminar is to bring speakers from this area and outside to speak on topics related to Representation Theory (specially geometric and topological methods employed in Representation Theory). The speakers are expected to give their talks at a level suitable for graduate students. The seminar is organized by Shrawan Kumar.

Zoom Meeting ID: 930 8567 8674, passcode: 2^10

2022 Fall

Date Speaker Affiliation Mode Title
Dec 2 Mark Ebert USC In-person TBA
Nov 18 Daping Weng UC Davis In-person TBA
Nov 11 Matt Hogancamp Northeastern In-person TBA
Nov 4 Peter Koroteev Berkeley In-person TBA
Oct 28 Junliang Shen Yale In-person The P=W conjecture for GL_n
October 7 Xuqiang Qin UNC Chapel Hill In-person Birational geometry of Beauville-Mukai systems on K3 surfaces
Sept 16 Till Wehrhan University of Bonn and MPIM In-person The rim hook rule in equivariant quantum Schubert calculus via Bethe and Clifford algebras

Junliang Shen: The P=W conjecture for GL_n
Abstract: In 2010, de Cataldo-Hausel-Migliorini proposed a conjecture connecting topology of the Hitchin system and Hodge theory of the corresponding character variety via the non-abelian Hodge theory. This conjecture is now referred to as the P=W conjecture. The purpose of this talk is to explain a recent proof of this conjecture (for GL_n) in joint work with Davesh Maulik for any rank and genus, where we combine tools from algebraic geometry and representation theory.

Xuqiang Qin: Birational geometry of Beauville-Mukai systems on K3 surfaces
Abstract: A Beauville-Mukai system on a K3 surface is a moduli space of stable torsion sheaves, which admits a Lagrangian fibration given by mapping each sheaf to its support. In this talk, we will focus on a class of Beauville-Mukai systems which are birational to Hilbert schemes of points on the surface. Using wall-crossing techniques from Bridgeland stability, we decompose the birational map into a sequence of flops, whose exceptional loci are Brill-Noether type subsets. As a result, we give full description of the birational geometry of such systems. This is based on joint work with Justin Sawon.

Till Wehrhan: The rim hook rule in equivariant quantum Schubert calculus via Bethe and Clifford algebras
Abstract: It was proven by Bertiger, Milicevic and Taipale that the the strucutre coefficients in the equivariant quantum cohomology of a Grassmannian can be determined by computing the usual cup product in a certain larger Grassmannian and then applying a combinatorial rim hook algorithm. In this talk, we discuss a generalization of this result using the realization of the equivariant quantum cohomology of Grassmannians as Bethe algebras of a specific integrable model established by Gorbounov, Korff and Stroppel. If time permits, we also discuss further possible generalizations to equivairant quantum K-theory.

2022 spring

Date Speaker Affiliation Mode Title
April 29 Elijah Bodish Oregon In-person Semisimpliflying type C tilting modules in medium-low quantum characteristic
April 29 2:45pm David Rose UNC In-person Type C Webs
April 22 Dinakar Muthiah Glasgow Zoom Fundamental monopole operators and affine Grassmannian slices
April 1 Tommaso Botta ETH Zurich In-person Shuffle products for stable envelopes of Nakajima varieties
March 25 Ivan Loseu Yale In-person Harish-Chandra modules over quantizations of nilpotent orbits
March 4 Joshua Kiers Ohio State In-person Demazure polytopes and saturation
Feb 18 Shiliang Gao UIUC In-person Newell-Littlewood numbers
Feb 11 Cristian Lenart SUNY Albany Zoom A combinatorial Chevalley formula for semi-infinite flag manifolds and related topics
Jan 21 Prakash Belkale UNC Hybrid Rigid local systems and the multiplicative eigenvalue problem

Elijah Bodish
Abstract: Given a semisimple Lie algebra and a positive integer L, one obtains the monoidal category of tilting modules for the associated quantum group at a primitive L-th root of unity. The category of tilting modules has a unique semisimple monoidal quotient. The images of indecomposable tilting modules with nonzero dimension exhaust the isomorphism classes of simple objects in the quotient category.

If L is larger than the Coxeter number of the Lie algebra, then the indecomposable tilting modules with non-zero dimension are exactly those with highest weight in the fundamental alcove (with respect to the L dilated affine Weyl group). When L is less than the Coxeter number much less is known. The state of the art is that in early 2020 Brundan-Entova-Etingof-Ostrik determined which tilting modules have nonzero dimension for gl_n when L < n+1 (note that the Coxeter number for gl_n is n). The Coxeter number for sp_{2n} is 2n. We will propose a method to determine which indecomposable tilting modules for sp_{2n} have non-zero dimension, when n< L< 2n+1. The answer appears to be related to Kazhdan-Lusztig cells in the anti-spherical Hecke module, which we hope will indicate what to expect in the much more complicated case when L< n+1.

David Rose
Abstract: In his seminal 1996 paper, Kuperberg gives presentations for the categories of finite-dimensional representations of quantum groups associated to rank 2 simple complex Lie algebras (as braided pivotal categories). Such presentations underly various invariants in low-dimensional topology; in particular, they serve as a “foundation” for link homology theories. Kuperberg also poses the following problem: to find analogous descriptions of these categories for quantum groups associated with higher rank Lie algebras. In 2012, Cautis-Kamnitzer-Morrison solved this problem in type A using skew Howe duality, a technique that does not immediately extend to give a solution in other types.

In this talk, we will present a solution to Kuperberg’s problem in type C. Our proof combines results on pivotal categories and quantum group representations with diagrammatic/topological analogues of theorems concerning reduced expressions in the symmetric group. Time permitting, we’ll discuss some future directions. This work is joint with Elijah Bodish, Ben Elias, and Logan Tatham.

Dinakar Muthiah
Abstract: Affine Grassmannians are objects of central interest in geometric representation theory. For example, the geometric Satake correspondence tells us that their singularities carry representation theoretic information. In fact, it suffices to work with affine Grassmannian slices, which retain all of this information.

Recently, Braverman, Finkelberg, and Nakajima showed that affine Grassmannian slices arise as Coulomb branches of certain quiver gauge theories. Remarkably, their construction works in Kac-Moody type as well. Their work opens the door to studying affine Grassmannians for Kac-Moody groups. Unfortunately, it is difficult at present to do any explicit geometry with the Coulomb branch definition. For example, a basic feature is that affine Grassmannian slices embed into one another. However, this is not apparent from the Coulomb branch definition. In this talk, I will explain why these embeddings are necessarily subtle. Nonetheless, I will show a way to construct the embeddings using fundamental monopole operators. This is joint work in progress with Alex Weekes.

Tommaso Botta
Abstract: The notion of stable envelopes of a symplectic resolution, developed by Okounkov and his coauthors in the last decade, lies at the heart of the geometric approach to the representation theory of quantum groups and q-difference equations. Nakajima quiver varieties form a rich family of symplectic resolutions, whose geometry governs the representation theory of Kac-Moody Lie algebras and, via stable envelopes, their q-deformations. In this talk, I will introduce an inductive formula that produces the stable envelopes of an arbitrary Nakajima variety, taking as input the stable envelopes of two other Nakajima varieties with smaller dimension and framing vectors. Some explicit examples will be also discussed. This formula is a wide generalisation earlier results inherited form the theory of weight functions. Time permitting, I will also discuss some connections with cohomological Hall algebras (CoHa) and Cherkis bow varieties, which are object of ongoing research.

Ivan Loseu
Abstract: Let O be a nilpotent orbit in a semisimple Lie algebra over the complex numbers. Then it makes sense to talk about filtered quantizations of O, these are certain associative algebras that necessarily come with a preferred homomorphism from the universal enveloping algebra. Assume that the codimension of the boundary of O is at least 4, this is the case for all birationally rigid orbits (but six in the exceptional type), for example. In my talk I will explain a geometric classification of faithful irreducible Harish-Chandra modules over quantizations of O, concentrating on the case of canonical quantizations — this gives rise to modules that could be called unipotent. The talk is based on a joint paper with Shilin Yu (in preparation).

Josh Kiers
Abstract: Demazure modules are certain B-submodules of irreducible G-modules, where B is a Borel subgroup of a semisimple Lie group G. They arise as cohomology groups of line bundles on Schubert varieties, and their characters are described by the Demazure character formula. We will consider the weight polytopes associated to these characters, presenting basic formulas and results on them. We will show how the polytopes shed light on the supports of the characters, at least in classical types. This gives an alternate proof of a conjecture of Monical, Tokcan, and Yong, first proven by Fink, Meszaros, and St. Dizier.

Shiliang Gao
Abstract: The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. A. Klyachko connected eigenvalues of sums of Hermitian matrices to the saturated LR-cone and established defining linear inequalities. We prove analogues for the saturated NL-cone. This is based on work with Gidon Orelowitz, Nicolas Ressayre and Alexander Yong; see arxiv.org/abs/2005.09012, arxiv.org/abs/2009.09904, and arxiv.org/abs/2107.03152.

Cristian Lenart
Abstract: I present a combinatorial Chevalley formula for an arbitrary weight in the equivariant K-theory of semi-infinite flag manifolds, which are certain affine versions of finite flag manifolds G/B. The formula is expressed in terms of the so-called quantum alcove model. One application is a Chevalley formula in the equivariant quantum K-theory of G/B. Another application is that the so-called quantum Grothendieck polynomials represent Schubert classes in the (non-equivariant) quantum K-theory of the type A flag manifold. Both applications solve longstanding conjectures. Other results include the Chevalley formula for partial flag manifolds G/P and related combinatorics of the quantum alcove model. This is joint work with Takafumi Kouno, Satoshi Naito, and Daisuke Sagaki. The talk will be largely self-contained.

Prakash Belkale
Abstract: Local systems are sheaves which describe the behavior of solutions of differential equations. A local system is rigid if local monodromy determines global monodromy. We give a construction which produces irreducible complex rigid local systems on a punctured Riemann sphere via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices (which can be inductively determined) of these polytopes give (all) possible unitary irreducible rigid local systems.

We note that these polytopes are generalizations of the classical Littlewood-Richardson cones of algebraic combinatorics. Answering a question of Nicholas Katz, we show that there are no irreducible rigid local systems on a punctured Riemann sphere of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing n, when n is a prime number.