WAYTA Seminar in Representation Theory and Mathematical Physics

University of Amsterdam / Korteweg-de Vries Institute for Mathematics

General Information: The What Are You Thinking About? (WAYTA) Seminar is a biweekly in person mathematics seminar held at the Korteweg-de Vries Institute for Mathematics in Amsterdam focusing on representation theory, mathematical physics and related areas. The goal of the seminar is to allow researchers at or near the KdVI to discuss their most recent research interests, or give introductory talks about topical research areas. If you are interested in giving a talk in the seminar, please contact one of the organizers.

Organizers: Valentin Buciumas and Mikhail Isachenkov and Jasper Stokman. If you would like to be on the mailing list, please email Misha at m dot isachenkov at gmail dot com.

Time and Place: The WAYTA seminar will take place in person at KdVI in the seminar room F3.20 (unless otherwise noted). If the speaker connects via Zoom, we will still be in F3.20 projecting the talk. The seminar will usually take place on Friday starting at 3pm (Amsterdam time!) and last for 1h or 1:30h. After 4pm/4:30pm there is room for a more informal continuation with part of the audience, which might well continue at some point at the Polder.

Talk Schedule -- Fall 2022

Date Time Speaker Affiliation Title
October 14 3pm Chun-Ju Lai Academia Sinica Quasi-hereditary covers, Hecke subalgebras and quantum wreath product
Abstract: The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover via category O for certain rational Cherednik algebras due to Ginzburg-Guay-Opdam-Rouquier. It was proved in type A that this category O can be realized using q-Schur algebra, but this realization problem remains open beyond types A/B/C. An essential step for type D is to study Hu's Hecke subalgebra, which deforms from a wreath product that is not a Coxeter group. In this talk, I'll talk about a new theory allowing us to take the 'quantum wreath product' of an algebra by a Hecke algebra. Our wreath product produces the Ariki-Koike algebra as a special case, as well as new 'Hecke algebras' of wreath products between symmetric groups. We expect them to play a role in answering the realization problem for complex reflection groups. This is a joint work with Dan Nakano and Ziqing Xiang.
October 28 3pm Christian Korff University of Glasgow The boson-fermion correspondence, Hecke characters and the six-vertex model
Abstract: The boson-fermion correspondence plays an important role in representation theory and mathematical physics. It is a module isomorphism between two representations of the Heisenberg algebra, one on the fermionic and the other on the bosonic Fock space. I will discuss its connection with Macdonald's characteristic map which is an isomorphism between the ring of class functions of symmetric groups and the ring of symmetric functions, which is identified with the bosonic Fock space in the correspondence. Macdonald's map has been generalised by Wang and Wang to the Iwahori-Hecke algebras and using the boson-fermion correspondence I discuss how the irreducible characters of Hecke algebras can be computed as partition functions of the asymmetric six-vertex model at the free fermion point. This will lead to a new fermionic description of the (half) vertex operators for Hall-Littlewood functions defined on the bosonic side by Jing.
November 11 3pm Jules Lamers Institut de Physique Théorique, CEA/Saclay A new spin on elliptic difference operators
Abstract: In this talk I'll discuss some new results and conjectures about elliptic difference operators for versions of Ruijsenaars models where the particles have spins. This is based on work in progress with Oleg Chalykh (U Leeds) and Rob Klabbers (Humboldt U Berlin).
November 18 3pm Eugenia Rosu Leiden University Higher weight Schwartz forms
Abstract: In their seminal work, Kudla and Millson have constructed a class of Schwartz forms corresponding to special cycles on Shimura varieties. I will talk about work in progress on extending the Kudla-Millson construction of Schwartz forms to higher weight analogues and their expected geometric interpretation.
December 2 3pm Hadewijch De Clercq Universiteit Gent Graphical calculus for quantum vertex operators
Abstract: Graphical calculus provides a diagrammatic framework for performing topological computations with morphisms in strict ribbon categories. This amounts to a functorial identification of such morphisms with oriented diagrams colored by a ribbon category, such as the category of finite-dimensional representations of a quantum group. In this talk I will explain how the graphical calculus can be extended to a larger category of quantum group representations, encompassing the q-analog of the BGG category O. In particular, this extended framework allows to graphically represent quantum vertex operators on Verma modules, as well as morphisms depending on a dynamical parameter, such as dynamical R-matrices. I will demonstrate the potential of this approach by graphically deriving certain q-difference equations for twisted trace functions of N-point quantum vertex operators. These include the dual q-KZB and dual Macdonald equations first obtained by Etingof and Varchenko, as well as some generalizations. This talk is based on joint work with Jasper Stokman and Nicolai Reshetikhin.
December 16 3pm Dmitry Noshchenko Universiteit van Amsterdam A tale of two quivers, or new faces of the pentagon identity for the quantum dilogarithm function
Abstract: We discuss some recent insights on symmetric quivers, their motivic generating series and the quantum dilogarithm identities, which have a strong connection to physics. We also provide a new proof of integrality of motivic Donaldson-Thomas invariants for symmetric quivers and its slight generalization. Based on a joint work with Jakub Jankowski, Piotr Kucharski, Hélder Larraguível, and Piotr Sułkowski, arXiv:2212.04379.

Old Talks (2021-2022)

Date Time Speaker Affiliation Title
October 7 4pm Valentin Buciumas Korteweg-de Vries Institute for Mathematics Quantum-metaplectic dualities via the Whittaker model
Abstract: The classical Satake isomorphisms is one of the stepping stones of the Langlands programme. It states that the spherical Hecke algebra of a p-adic reductive group can be identified with the Grothendieck ring of the category of representations of the dual group. In this setup, one may study the Whittaker module, which contains information about unramified Whittaker functions and show that it is isomorphic to the spherical Hecke algebra. Understanding a certain basis of the Whittaker module produces the Casselman-Shalika formula. In this talk I will present some ideas on how to generalise this setup to the setting of metaplectic p-adic groups where the category of representations of the quantum group at a root of unity will appear.
October 28 4pm Arno Kret University of Amsterdam Construction of Galois representations for GSp_2n and GSO_2n
Abstract: We discuss the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups and general special orthogonal groups over totally real number fields under the local hypothesis that there is a Steinberg component.
November 11 4pm Yuan Miao Institute for Theoretical Physics, University of Amsterdam Onsager, Temperley-Lieb and more from Clifford
Abstract: I will explain how (a quotient of) Onsager algebra and affine Temperley-Lieb algebra emerge from ``Generalised Clifford algebra''. The construction can be extended to more general case, related to the recent development of generalised Onsager algebra. I will give several concrete examples of motivated from the physics of exactly solvable model. These examples can be considered as a (new) generalisation of the Ising model.
December 2 4pm Mikhail Isachenkov University of Amsterdam Crossing equations and Rankin-Selberg identities
This talk is online due to the current lockdown measures in the Netherlands.
Abstract: I will show how a specific type of Rankin-Selberg identity for automorphic forms on a real reductive group is analogous to the conformal crossing equation for correlation functions in CFT. We will discuss some examples and implications of this fact. The talk is based on my work in preparation.
December 16 4pm Jules Lamers Institut de Physique Théorique, CEA/Saclay ansWers And questions - Yangians eT spin chAins
Abstract: As several of you know I'm interested in quantum-integrable long-range spin chains. These quantum-mechanical models for magnetism have a rich representation-theoretic structure. In this talk I will focus on the 'isotropic' level and explore the connection between the Yangian of gl_2 and spin chains.
I will briefly review the role of the Yangian in the context of the Heisenberg XXX spin chain.
Then I'll introduce the Haldane--Shastry spin chain and explain what is known about its Yangian symmetry. In particular I will use affine Schur--Weyl duality between the degenerate affine Hecke algebra and the Yangian to derive the Yangian generators for Haldane--Shastry that were conjectured by Ha, Haldane and others nearly 30 years ago.
Finally I will turn to a correspondence, part established in collaboration with Rob Klabbers (Humboldt U) and part conjectural, between Yangian irreps for Haldane--Shastry and solutions to the Bethe-ansatz equations of Heisenberg. I will end with some puzzles that keep me busy at the moment.
February 10 4pm Tamás Görbe University of Groningen Elliptic Racah polynomials
Abstract: Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known q-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit. Joint work with Jan Felipe van Diejen.
February 17 4pm Guus Regts University of Amsterdam Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite regular tree
Abstract: In this talk I will discuss recent joint work with Ferenc Bencs, David de Boer and Pjotr Buys in which we determine the critical temperature for uniqueness of the Gibbs measure of the anti-ferromagnetic Potts model on the infinite regular tree (also known as the Bethe lattice). This confirms a folklore conjecture.
March 3 4pm Marcel Vonk University of Amsterdam Matrix models: asymptotics, transesseries, theta functions and all that
Abstract: Matrix integrals are very nice toy models for physicists, having a level of difficulty in between ordinary integrals and path integrals - and in contrast to the latter, being perfectly well-defined. Like the path integrals that appear in quantum field theory, matrix integrals can be approximated using (asymptotic) perturbative expansions, but the true interesting physics is hidden in their nonperturbative content. This leads us to the realm of Écalle's theory of resurgence, where concepts like alien derivatives and transseries play an important role. Fascinating links with modular forms and theta functions also show up.
March 17 4pm Pablo Zadunaisky Jacobs University, Bremen Schur-Weyl and Gelfand-Tsetlin and Vershik-Okounkov
Abstract: Set g = gl(n,C) and V = Cn its natural representation. It is a classical result that the d-fold tensor product of V decomposes as g-module into a direct sum of irreducible components of highest weight w = (w1, ... , wn) a partition of n, each appearing with multiplicity given by the number of standard Young tableaux of shape w. This can be proved by a purely combinatorial argument about weights, or using representation theory of the symmetric group and Schur-Weyl duality. In the 50's Gelfand and Tsetlin gave a construction of a distinguished basis for any irreducible representation of g. In 2008, Vershik and Okounkov gave an analogous construction for irreducible representations of the symmetric group. In both cases the basis arises as the set of common eigenvectors of a maximal commutative subalgebra of the enveloping algebra of g and the group algebra of the symmetric group, respectively. Our [unachieved] aim is to see these bases inside the d-fold tensor product of V, and discuss some intermediate results. This is being thought about in collaboration with Joanna Meinel from Bonn University.
March 24 4pm Miranda Cheng University of Amsterdam 3d Manifolds, Log VOAs and Quantum Modular Forms
Abstract: The q-series 3-manifold invariants provide new insights and computational tools in 3-manifold topology, 3d SQFT, and M-theory compactifications. In this talk I will survey the relation between these q-series 3-manifold invariants, (logarithmic) VOAs, and quantum modular forms.
April 7 4pm Jean-Sébastien Caux University of Amsterdam Dynamics of many-body quantum systems: Integrability, from Newton's cradle to Gibbs' grave
Abstract: Recent years have demonstrated that integrability can be used to compute many physical properties of experimentally-accessible magnetic or cold atomic systems. Besides their rich equilibrium dynamics, such systems also host relaxation and equilibration behaviour which cannot be simply described by traditional textbook methods, and can lead to long-lived non-thermal equilibrium states. This talk will provide an overview of recent work in this area, and introduce some new methods to treat quenched and driven systems in various contexts.
April 14 4pm Erik Koelink Radboud Universiteit Matrix spherical functions and matrix orthogonal polynomials
Abstract: We will discuss some generalities for matrix spherical functions on compact symmetric spaces. After imposing additional conditions on the related multiplicities, the matrix spherical polynomials can be related to matrix orthogonal polynomials. We will discuss an explicit example in order to see how the radial part of the Casimir operator plays a role in determining the properties of the matrix orthogonal polynomials.
April 28 4pm Jiandi Zou Technion - Israel Institute of Technology Classification of irreducible representations of a Kazhdan-Patterson covering group of GL(r)
Abstract: The local Langlands correspondence for a general linear group over a non-archimedean local field is known for a while, which gives a bijection between the set of equivalence classes of irreducible representations of GL(r) and the set of equivalence classes of r-dimensional Weil-Deligne representations. To establish such a bijection, we first need to do that for cuspidal representations, which is indeed the most crucial and difficult step. Then, the problem reduces to classifying all the irreducible representations of GL(r) via cuspidal ones, which is due to Bernstein-Zelevinsky and Zelevinsky. Their methods are also adapted and improved by others including Tadic, Lapid-Minguez, Minguez-Sécherre to classify irreducible representations of an inner form of GL(r). In this talk, I will focus on explaining some key points in the proof of Zelevinsky classification, then I will explain how to adapt it to classify all the irreducible representations of a certain covering group of GL(r), i.e. a Kazhdan-Patterson covering group. This is a joint work with Erez Lapid and Eyal Kaplan.
May 12 4pm Max Gurevich Technion - Israel Institute of Technology In between finite and p-adic groups in type A
Abstract: Using the Bruhat decomposition, a general linear group over a p-adic field may be thought of as a "quantum affine" version of a finite group of permutations. I am currently exploring some possible implications of this analogy on the spectral properties of the two groups, and would like to present two specific points of view on standing problems in representation theory. For one, restriction of an irreducible smooth representation to its finite counterpart gives a flexible definition of the notion of the wavefront set - an invariant of arithmetic significance which is often approached using microlocal analysis. From another perspective, the class of cyclotomic Hecke algebras is a natural interpolation between the finite and p-adic groups. I will show how the class of RSK representations (developed with Erez Lapid) serves as a uniform bridge between the Langlands classification for the p-adic group and the classical Specht construction of the finite domain.
June 16 5pm Andrea Appel University of Parma Schur-Weyl dualities for quantum affine symmetric pairs
Abstract: In the work of Kang, Kashiwara, Kim, and Oh, the Schur-Weyl duality between quantum affine algebras and affine Hecke algebras is extended to certain Khovanov-Lauda-Rouquier (KLR) algebras, whose defining combinatorial datum is given by the poles of the normalised R-matrix on a set of representations. In this talk, I will review their construction and introduce a "boundary" analogue, consisting of a Schur-Weyl duality between a quantum symmetric pair of affine type and a modified KLR algebras arising from a (framed) quiver with a contravariant involution. With respect to the Kang-Kashiwara-Kim-Oh construction, the extra combinatorial datum we take into account is given by the poles of the normalised K-matrix of the quantum symmetric pair.
June 24 4pm Alexandr Garbali University of Melbourne Shuffle algebras and integrability
Abstract: I will discuss Feigin-Odesskii shuffle algebras and their connections with integrable models. The main example will be the trigonometric shuffle algebra. This algebra is related to the quantum toroidal algebra of gl_1 and is useful for studying the associated XXZ type integrable model.