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.

2021 Spring–Zoom Meeting ID: 975 6220 3148

Date Speaker Affiliation Title
March 26 Alex Weeks University of British Columbia TBA
March 19 Tsao-Hsien Chen University of Minnesota TBA
March 5 Anne Dranowski IAS TBA
Feb 19 Charlotte Chan MIT Flag varieties and representations of p-adic groups
Jan 29 Yau Wing Li MIT Endoscopy for affine Hecke categories

 Charlotte Chan Flag varieties and representations of p-adic groups Abstract: Deligne–Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called “depth-zero” supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint work with Masao Oi.

 Yau Wing Li Endoscopy for affine Hecke categories Abstract: Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H. It is consistent with the Langlands functoriality conjecture.

2020 Spring

Date Speaker Affiliation Title
Feb 21 Dennis Tseng Harvard Equivariant degenerations of plane curve orbits
Feb 14 Rekha Biswal MPIM Macdonald polynomials and level two Demazure modules for affine sl_{n+1}

 Rekha Biswal Macdonald polynomials and level two Demazure modules for affine sl_{n+1} Abstract: An important result due to Sanderson and Ion says that characters of level one Demazure modules are specialized Macdonald polynomials. In this talk, I will introduce a new class of symmetric polynomialsindexed by a pair of dominant weights of sl_{n+1} which is expressed as linear combination of specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating the characters of higher level Demazure modules. Using representation theory, we will see that these new family of polynomials interpolate between characters of level one and level two Demazure modules for affine sl_{n+1} and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey Wand.

 Dennis Tseng Equivariant degenerations of plane curve orbits Abstract: In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.

2019 Fall

Date Speaker Affiliation Title
November 22 4:15-5:15pm Gurbir Dhillon Stanford The tamely ramified Fundamental Local Equivalence
November 22 3-4pm Vikraman Balaji Chennai Mathematical Institute Torsors on semistable curves and the problem of degenerations
November 15 Alex Yong University of Illinois at Urbana-Champaign The A.B.C.D’s of Schubert calculus
November 1 Cris Negron UNC Chapel Hill Modularization of quantum groups and some conformal field theory
October 25 Andrey Smirnov UNC Chapel Hill Elliptic stable envelope for Hilbert scheme of points on C^2
September 27 Richard Rimanyi UNC Chapel Hill Elliptic classes of Schubert varieties

 Gurbir Dhillon The tamely ramified Fundamental Local Equivalence Let G be an almost simple algebraic group with Langlands dual G’. Gaitsgory conjectured that affine Category O for G at a noncritical level should be equivalent to Whittaker D-modules on the affine flag variety of G’ at the dual level. We will provide motivation and background for this conjecture, which is some form of geometric Satake for quantum groups. We have proven this conjecture when the level is appropriately integral with Justin Campbell, and the general case is work in progress with Sam Raskin.

 Vikraman Balaji Torsors on semistable curves and the problem of degenerations Let G be an almost simple, simply connected algebraic group G over the field of complex numbers. In this talk I answer two basic questions in the classification of G-torsors on curves. The first one is to construct a at degeneration of the moduli stack G-torsors on a smooth projective curve when the curve degenerates to an irreducible nodal curve. Torsors for a generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools. The second question is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible nodal curve. The absence of obvious analogues of torsion-free sheaves in the setting of G-torsors makes the question interesting. This also leads to the construction of a proper separated coarse space for G-torsors on an irreducible nodal curve.

 Alex Yong The A.B.C.D’s of Schubert calculus We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory. This is joint work with Colleen Robichaux and Harshit Yadav.

 Cris Negron Modularization of quantum groups and some conformal field theory I will discuss recent work on constructing small quantum groups at even order roots of unity. (Recall that the small quantum group for a given simple Lie algebra is a characteristic 0, q-analog of its corresponding restricted enveloping algebra.) Our investigations are inspired by a conjectured equivalence of categories between representations for small quantum sl_2, at a certain even order parameter q, and representations for the so-called triplet conformal field theory. I will elaborate on this conjecture and explain how its resolution necessitates the introduction of certain “new” quantum groups, which are obtained from “old” quantum groups via deformation. No familiarity with quantum groups or conformal field theory will be assumed, and all relevant notions will be defined in the talk.

 Andrey Smirnov Elliptic stable envelope for Hilbert scheme of points on C^2 In this talk I describe an explicit formula for elliptic stable envelope of torus fixed points on the Hilbert scheme of points in C^2. In K-theoretic limit we obtain new combinatorial formulas for Schur, rational Schur and Macdonald polynomials. In particular, we obtain explicit combinatorial formula for the coefficients of the Kostka matrix.

 Richard Rimanyi Elliptic classes of Schubert varieties Assigning characteristic classes to singular varieties is an effective way of studying the enumerative properties of the singularities. Initially one wants to consider the so-called fundamental class in H, K, or Ell, but it turns out that in Ell such class is not well defined. However, a deformation of the notion of fundamental class (under the name of Chern-Schwartz-MacPherson class in H, motivic Chern class in K) extends to Ell, due to Borisov-Libgober. We will introduce a twisted version of the elliptic class, and show its relation to Tarasov-Varchenko weight functions and Okounkov’s stable envelopes. (Based on results with A. Weber, and on a work in progress with S. Kumar and A. Weber.)

2019 Spring

Date Speaker Affiliation Title
April 12 Oliver Pechenik University of Michigan Crystal structures for symmetric Grothendieck polynomials
March 29 Matt Hogancamp USC Serre duality for Khovanov-Rozansky homology
March 22 Arnav Tripathy Harvard A geometric model for complex analytic equivariant elliptic cohomology
March 15 Justin Allman USNA Annapolis Interpolating quantum dilogarithm identities, the topological viewpoint
March 1 Andrzej Weber University of Warsaw, Poland Characteristic classes of Schubert varieties and Hecke-type algebras
Jan 25 Spencer Leslie Duke Parity sheaves and Smith theory
Jan 18 3:30pm Sergei Gukov Caltech Hidden algebraic structures in topology

 Oliver Pechenik Crystal structures for symmetric Grothendieck polynomials The symmetric Grothendieck polynomials representing Schubert classes in the K-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type A crystal structure on these tableaux. Applications include a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For rectangular shapes, we give a new interpretation of Lascoux polynomials (K-analogues of Demazure characters) by constructing a K-theoretic analogue of crystals with an appropriate analogue of a Demazure subcrystal. (Joint work with Cara Monical and Travis Scrimshaw.)

 Matt Hogancamp Serre duality for Khovanov-Rozansky homology I will discuss recent joint work with Gorsky, Mellit, and Nagane in which we consider a monoidal version of Serre duality for the category of Soergel bimodules in type A, in which the role of the Serre functor is played by the Rouquier complex associated to the full twist braid. This is a lift of (the type A special case of) a result of Mazorchuk-Stroppel (2008) and Beilinson-Bezrukavnikov-Mirkovic (2004), which states that the action of the full twist is the Serre functor on the BGG category O, and as a result we obtain a new “topological” proof of this fact. I will conclude by discussing consequences for Khovanov-Rozansky link homology.

 Arnav Tripathy A geometric model for complex analytic equivariant elliptic cohomology A long-standing question in the study of elliptic cohomology or topological modular forms has been the search for geometric cocycles. Such cocycles are crucial for applications in both geometry and, provocatively, for the elliptic frontier in representation theory. I will explain joint work with D. Berwick-Evans which turns Segal’s physically-inspired suggestions into rigorous cocycles for the case of equivariant elliptic cohomology over the complex numbers, with some focus on the role of supersymmetry on allowing for the possibility of rigorous mathematical definition. As time permits, I hope to indicate towards the end how one might naturally extend these ideas to higher genus.This talk is joint with the Physically Inspired Mathematics Seminar

 Justin Allman Interpolating quantum dilogarithm identities, the topological viewpoint Quantum dilogarithm identities have a rich history and connections to partition counting, BPS spectra in quantum field theories, stability conditions for quiver representations, Poincare series of cohomological Hall algebras, cluster algebras/categories, and Donaldson–Thomas invariants. In this talk we describe a family of factorization formulas for the combinatorial Donaldson–Thomas invariant for an acyclic quiver. A quantum dilogarithm identity originally due to Reineke, and again established by Rimanyi by counting dimensions of quiver loci, gives two extremal cases of our formulation in the Dynkin case. We establish interpolating factorizations explicitly with a dimension counting argument by defining stratifications of the space of representations for the quiver and calculating Betti numbers of strata in the corresponding equivariant cohomology algebras.

 Adrzej Weber Characteristic classes of Schubert varieties and Hecke-type algebras Abstract: We study various types of cohomological invariants of Schubert varieties in the generalized flag variety G/B. According to Bernstein-Gelfand-Gelfand the fundamental classes in cohomology can be computed via the action of the nil-Hecke algebra. It was shown by Aluffi and Mihalcea that the Chern-Schwartz-MacPherson classes are obtained by the action of the group ring of the Weyl group. Similarly, the motivic Chern classes in K-theory are related to the classical Hecke algebra, as announced by Aluffi-Mihalcea-Schürmann-Su. We will concentrate on the equivariant elliptic classes in the sense of Borisov-Libgober, which depend on an additional parameter – an auxiliary line bundle. We show that these classes are related to a Hecke-type elliptic algebra. The proof uses basic properties of the canonical divisor of the Schubert varieties and its Bott-Samelson resolutions. As a corollary we show that for G=GL_n the Borisov-Libgober elliptic classes are represented by Rimanyi-Tarasov-Varchenko elliptic weight function.

 Spencer Leslie Parity sheaves and Smith theory Abstract: Parity sheaves are a topologically-defined class of sheaves on a variety with many important connections to modular representation theory. In this talk, we discuss connections between parity sheaves on a variety X endowed with the action of a finite cyclic group of order p and parity sheaves on the fixed-point set. For this we use Smith theory, which gives a localization functor for actions of finite cyclic groups. The key idea is to define a good notion of parity sheaf in an intermediate localized category. In the context of the geometric Satake equivalence, this gives a geometric construction of the Frobenius-contraction functor on tilting modules.

 Sergei Gukov Hidden algebraic structures in topology Abstract: The goal of this talk is to give an exposition, from several different angles, of new connections that seem to emerge between low-dimensional topology and abstract algebra. While the general picture for such connections — which will be presented toward the end — is rooted in physics, each particular connection can be formulated as a concrete and verifiable statement or, in some cases, even as a theorem. For example, we will see how Kirby moves of 4-manifolds can be realized as equivalences of vertex operator algebras, some of which were known previously and some are new. In the opposite direction, we will see how starting with purely algebraic questions that involve, say, classification of modular tensor categories and the Witt group, we will be naturally led to questions in topology of 3-manifolds and smooth 4-manifolds.This talk is joint with physically inspired mathematics seminar

2018 Fall

Date Speaker Affiliation Title
Dec 07 3:15pm Peter Fiebig University of Erlangens Sheaves on the alcoves and modular representations
Nov 16 Alejandro Ginory Rutgers The Verlinde Formula and Twisted Affine Lie Algebras
Nov 09 3:30pm Changjian Su Toronto Motivic Chern classes, K-theoretic stable basis and Iwahori invariants of principal series
Oct 5 Michael Strayer UNC Minuscule Kac-Moody settings unified by new poset coloring properties
Sep 21 Yiqiang Li SUNY at Buffalo Quiver varieties and symmetric pairs
Sep 07 Sami Assaf University of Southern California Nonsymmetric Macdonald polynomials and Demazure characters
Aug 31 Richárd Rimányi UNC Chapel Hill Motivic characteristic classes, Hall algebras, and DT type identities

Peter Fiebig
Sheaves on the alcoves and modular representations
Abstract: The main topic is the problem of determining the characters of reductive algebraic groups over a field of positive characteristic. Since Williamson gave countless (unexpected) examples of characters that differ from the character formula given by George Lusztig in 1980 (which holds for large enough characteristics), the community is hoping for a revised formula, or an effective algorithm, that provides a solution in the general case. For this, one surely needs a much better understanding on the torsion phenomena occurring for small primes. In my talk I first want to give an understandable introduction to the problem, and then I want to introduce a new category related to modular representation theory that hopefully will help us understand the problem in greater depth.

 Alejandro Ginory The Verlinde Formula and Twisted Affine Lie Algebras Abstract: In the category of integrable highest weight modules for affine Lie algebras, the usual tensor product fails to preserve the so-called level (i.e., the scalar action of the canonical central element). For untwisted affine Lie algebras, a product structure called the fusion product makes the subcategory of modules at a fixed level into a braided monoidal category (in particular, with non-negative integral structure constants). In this talk, I will discuss a fusion product with integral structure constants on the space of characters of twisted affine Lie algebras. Surprisingly, in the A^(2)_{2r} case and for certain natural quotients for the other twisted cases, these algebras have negative structure constants “half” the time, (depending on the parity of the level). We will discuss these and other new features in the twisted cases, and their representation-theoretic meaning.

 Changjian Su Motivic Chern classes, K-theoretic stable basis and Iwahori invariants of principal series Abstract: Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump–Nakasuji–Naruse about certain transition matrix between these two bases. We will first relate the Iwahori invariants to Maulik–Okounkov’s stable envelopes and Brasselet–Schurmann–Yokura’s motivic Chern classes for the Langlands dual groups. Then the conjecture follows from a K-theoretic generalization of Kumar’s smoothness criterion for the Schubert varieties. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.

 Michael Strayer Minuscule Kac-Moody settings unified by new poset coloring properties Abstract: R.M. Green axiomatically defined full heap posets and used them to build elegant minuscule-like doubly infinite representations of many affine Kac-Moody algebras. Two other classes of colored posets, namely minuscule and d-complete posets, have also been used in many applications to Lie theory. For instance, the d-complete posets correspond to “dominant lambda-minuscule” elements of Kac-Moody Weyl groups. We present new poset coloring properties that unify the above three classes of colored posets. They can be used to characterize various Kac-Moody (sub)algebra representations. These include the full heap representations of Green, the minuscule representations of semisimple Lie algebras, and the Demazure modules of dominant minuscule Weyl group elements. We give Dynkin diagram-indexed classifications of the colored posets that satisfy the two most important sets of these properties.

 Yiqiang Li Quiver varieties and symmetric pairs Abstract: To a simply-laced Dynkin diagram, one can attach a complex simple Lie algebra, say g, and a class of Nakajima (quiver) varieties. The latter provides a natural home for a geometric representation theory of the former. If the algebra g is further equipped with an involution, it leads to a so-called symmetric pair (g,k), where k is the fixed-point subalgebra under involution. In this talk, I’ll present a recent study of fixed-point loci of Nakajima varieties under certain involutions and provide bridges at several levels between symmetric pairs and Nakajima varieties.

 Sami Assaf Nonsymmetric Macdonald polynomials and Demazure characters Abstract: Macdonald introduced symmetric functions in two parameters that simultaneously generalize Hall—Littlewood symmetric functions and Jack symmetric functions. Opdam and Macdonald independently introduced nonsymmetric polynomial versions of these that Cherednik then generalized to any root system. Ion showed that these nonsymmetric Macdonald polynomials with one parameter specialized to 0 arise as characters for affine Demazure modules. Recently, I used the Haglund—Haiman—Loehr combinatorial formula for nonsymmetric Macdonald polynomials in type A to show that, in fact, the specialized nonsymmetric Macdonald polynomials are graded sums of finite Demazure characters in type A. In this talk, I’ll present joint work with Nicolle Gonzalez where we construct an explicit Demazure crystal for specialized nonsymmetric Macdonald polynomials, giving rise to an explicit formula for the Demazure expansion in terms of certain lowest weight elements. Connecting back with the symmetric case, this gives a refinement of the Kostka—Foulkes polynomials defined by the Schur expansion of Hall—Littlewood symmetric functions. This talk assumes no prior knowledge of Macdonald polynomials, Demazure characters, or crystals.

 Richárd Rimányi Motivic characteristic classes, Hall algebras, and DT type identities Abstract: The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are fine characteristic classes of singular varieties in cohomology and K theory—and their theory overlaps with the theory of Okounkov’s stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_\beta$, $C^o_\beta$ parameterized by positive roots $\beta$. We prove an identity in a deformed version of Kontsevich-Soibelman’s Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of $c^o_\beta$, $C^o_\beta$ classes formally depending on a stability function Z, does not depend on Z. This identity—which encodes infinitely many identities among rational functions in infinitely many variables—has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the $c^o_\beta$, $C^o_\beta$ classes as certain commutators in the CoHA, KHA.

2018 Spring

Date Speaker Affiliation Title
April 20 Daping Weng Yale University Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian
April 13 Natalia Kolokolnikova University of Geneva K-theoretic Thom polynomial and the rationality of the singularities of the A2 loci
April 12 Nicolas Ressayre Universite Claude Bernard Lyon 1 On the tensor semigroup of affine Kac-Moody Lie algebras
April 6 Simon Salamon King’s College Wolf spaces and Fano contact manifolds
March 22 Chiara Damiolini Rutgers Twisted conformal blocks
Feb 23 Baiying Liu Purdue University On the local converse theorem for GL_n
Feb 9 Johannes Flake Rutgers Dirac cohomology, Hopf-Hecke algebras, and infinitesimal Cherednik algebras

 Natalia Kolokolnikova K-theoretic Thom polynomial and the rationality of the singularities of the A2 loci Abstract: I will discuss the definitions of two K-theoretic invariants of the singularity loci, prove that they are not always equal and tell how this problem is connected to the study of the rationality of the singularities of the singularity loci. I will prove that the singularities of the A2 loci are rational in some very specific cases, but are not rational in general. Daping Weng Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Abstract: Fix two positive integers $a$ and $b$. Scott showed that a homogeneous coordinate ring of the Grassmannian $Gr_{a, a+b}$ has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to integer multiples of the fundamental weight $w_a$. By proving the Fock-Goncharov cluster duality conjecture for the Grassmannian using a sufficient condition found by Gross, Hacking, Keel, and Kontsevich, we obtain bases parametrized by plane partitions for these irreducible representations. As an application, we use these bases to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen. Nicolas Ressayre On the tensor semigroup of affine Kac-Moody Lie algebras. Abstract: In this talk, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra g. Let P+ be the set of dominant integral weights. For λ ∈ P+, L(λ) denotes the irreducible, integrable, highest weight representation of g with highest weight λ. Consider the tensor cone Γ(g):={(λ1,λ2,μ)∈P+3 |∃N >1 L(Nμ)⊂L(Nλ1)⊗L(Nλ2)}. If g is finite dimensional, Γ(g) is a polyhedral convex cone described by Belkale-Kumar by an explicit finite list of inequalities. In general, Γ(g) is nor polyhedral, nor closed. We will describe the closure of Γ(g) by an explicit countable family of linear inequalities, when g is untwisted affine. This solves a Brown-Kumar’s conjecture in this case. Simon Salamon Wolf spaces and Fano contact manifolds Abstract: The correspondence between Riemannian symmetric spaces with holonomy a subgroup of Sp(n)Sp(1) and complex homogeneous spaces with a holomorphic contact structure was discovered by Joseph Wolf in 1965, yet the possibility of non-homogeneous manifolds with positive curvature subscribing to this model remains open. I shall explain what is known about positive quaternion-Kaehler manifolds and their Fano twistor spaces, in the light of recent results of Buczynski-Wisniewski-Weber on torus actions on contact Fano manifolds. Chiara Damiolini Twisted conformal blocks Abstract: Let G be a simple and simply connected algebraic group over a field. We can attach to a n-tuple of representations of G the sheaf of conformal blocks: a vector bundle on M_{g,n} whose fibres are identified with global sections of a certain line bundle on the stack of G-torsors. We generalize the construction of conformal blocks to the case in which G is replaced by a group H defined over curves in terms of covering data. In this case the associated conformal blocks define a sheaf on a Hurwitz space and have properties analogous to the classical case. Baiying Liu On the local converse theorem for GL_n Abstract: In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field. This is a joint work with Prof. Herve Jacquet. I will also briefly talk about extensions of the local converse theorem to the setting of l-adic families, which is a joint work with Gilbert Moss. Johannes Flake Dirac cohomology, Hopf-Hecke algebras, and infinitesimal Cherednik algebras Abstract: Dirac operators have played an important role in the representation theory of reductive Lie groups, but also of various Hecke algebras. We show how these cases can be studied uniformly, using smash products of Hopf algebras, PBW deformations, and superalgebras. We prove a generalized version of a result known as Vogan’s conjecture in certain special cases, which relates Dirac cohomology with central characters. We discuss infinitesimal Cherednik algebras as a novel special case, and we obtain partial results on the classification of the class of algebras to which our theory applies.