Fall, 2023

Fall, 2023

  • Speaker Lisa Carbone, Rutgers University
    • Title Integrality of Kac-Moody groups
    • Time/place 9/15/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract Let A be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody algebra g=g(A) over Q. Let V be an integrable highest weight g-module and let V_Z be a Z-form of V. Let G=G(Q) be an associated minimal representation-theoretic Kac--Moody group and let G(Z) be its integral subgroup. By analogy with the finite dimensional case, the integrality question for G is to determine if G(Z)=Stab_G(V_Z), that is, to determine if G(Z) coincides with the subgroup of G that stabilizes the integral lattice V_Z. Integrality of semi-simple algebraic groups G over Q was established by Chevalley in the 1950’s as part of his work on associating an affine group scheme to G(Q) and V_Z. We discuss our approach to this question for Kac-Moody groups and we prove integrality for inversion subgroups U_w of G. Here, for w in W, U_w is the group generated by positive real root groups that are flipped to negative root groups by w^{-1}. There are various applications of this result, including integrality of subgroups of the unipotent subgroup U of G that are generated by commuting real root groups. This is joint work with Abid Ali, Dongwen Liu and Scott H. Murray.

  • Speaker Vidya Venkateswaran, Center for Communications Research at Princeton
    • Title Quasi-polynomials, partial symmetry, and metaplectic Whittaker functions
    • Time/place 9/22/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract In joint work with Siddhartha Sahi and Jasper Stokman, we introduced quasi-polynomial generalizations of (nonsymmetric and symmetric) Macdonald polynomials for arbitrary root systems. In this talk, we discuss some recent results on (anti-) symmetric quasi-polynomials. For GL_r, we show that there is a bijection between (anti-)symmetric quasi-polynomials and partially (anti-)symmetric polynomials. Moreover, under this bijection, the q \rightarrow \infty limit of quasi-polynomial generalizations of (anti-)symmetric Macdonald polynomials map to Macdonald polynomials with prescribed symmetry (as introduced in 2000 by Baker-Dunkl-Forrester), in the same limit. Translating our results to the metaplectic context, we provide a precise statement and proof of a conjecture of Brubaker-Buciumas-Bump-Gustafsson on a ``duality" between metaplectic spherical Whittaker functions and non-metaplectic parahoric Whittaker functions. These results extend to other root systems as well if we restrict to a certain subspace of the quasi-polynomials.
    • Slides pdf file.

  • Speaker Robert McRae, Yau Mathematical Sciences Center at Tsinghua University
    • Title Non-rigid tensor categories for affine sl_2 at admissible levels
    • Time/place 9/29/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract The Kazhdan-Lusztig category KL^k(sl_2) is the category of finite-length modules for affine sl_2 at level k whose composition factors are irreducible highest-weight modules whose highest weights are dominant integral for the finite-dimensional subalgebra sl_2. In this talk, we show that for admissible levels k = ?2 + p/q, where p > 1 and q > 0 are relatively prime integers, KL^k(sl_2) admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang, but that it is not rigid, that is, not every object has a dual. Instead, an object of KL^k(sl_2) is rigid if and only if it is projective and, moreover, KL^k(sl_2) has enough projectives. Most of the indecomposable projective objects are logarithmic modules, which means that the Virasoro L(0) operator acts non-semisimply. We show also that the monoidal subcategory of rigid and projective objects is tensor equivalent to tilting modules for quantum sl_2 at the root of unity e^{pi i/(k+2)}. This leads to a universal property for KL^k(sl_2), which allows us to construct an essentially surjective (but not fully faithful) exact tensor functor from KL^k(sl2) to the category of finite-dimensional weight modules for quantum sl_2 at e^{pi i/(k+2)}. This is joint work with Jinwei Yang.

  • Speaker Angela Gibney, University of Pennsylvania
    • Title Mode Transition Algebras
    • Time/place 10/6/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract This talk is about recent work with Damiolini and Krashen, where we define a new series of associative algebras, which we call mode transition algebras, that give insight into geometry of moduli of curves, and representations of the vertex operator algebras from which they are derived. In this talk I will focus mainly on the VOA side of the story. I'll define the mode transition algebras, and relate our main theorem in this direction, which characterizes how mode transition algebras give information about higher level Zhu algebras. As an application, as I will illustrate, one can give an explicit description of all higher level Zhu algebras for the Heisenberg VOA, proving a conjecture of Addabbo-Barron.

  • Speaker Song Gao, University of Notre Dame
    • Title Coisotropicity of fixed points under torus action on the variety of Lagrangian subalgebras
    • Time/place 10/13/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract I will talk about my recent study of coisotropic subalgebras of Lie bialgebras. Given a complex semisimple Lie algebra \frak{g} with adjoint group G, the set of coisotropic subalgebras of \frak{g} form an algebraic variety, which is called the variety of coisotropic subalgebras. Let H be a fixed maximal torus of G. I will introduce my results on fixed points of H-action on the variety of coisotropic subalgebras. Approaches of toric varieties and algebraic groups will be used.

  • Speaker Andrzej Zuk, Université Paris 7
    • Title From PDEs to groups
    • Time/place 11/3/2023, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
    • Abstract We present a construction which associates to differential equations discrete groups. In order to establish this relation we use automata and random walks on ultra discrete limits.

  • Speaker Darlayne Addabbo, University of Arizona
    • Title Vertex Operators for Imaginary gl_2-subalgebras in the Monster Lie Algebra
    • Time/place 11/10/2023, Friday, 12:10 pm (Eastern Time), Hill 705 and online (Zoom link above)
    • Abstract The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V^\natural\otimes V_{1,1} where V^\natural is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and V_{1,1} is the vertex algebra corresponding to the rank 2 even unimodular lattice II_{1,1}. In this talk, I will discuss the construction of vertex algebra elements that project to bases for subalgebras of m isomorphic to gl_2 and corresponding to imaginary simple roots (1,j) for j>0. The M-action on V^\natural induces an M-action on the set of gl_2 subalgebras corresponding to a fixed imaginary simple root. I will discuss this action and related open questions. (This talk is based on joint work with Lisa Carbone, Elizabeth Jurisich, Maryam Khaqan, and Scott H. Murray).

  • Speaker Daniel Soskin, Lehigh University
    • Title Plucker inequalities and bounded Laurent monomials on the positive loci
    • Time/place 12/1/2023, Friday, 12:10 pm (Eastern Time), Hill 705
    • Abstract Totally positive matrices are matrices in which each minor is positive. Lusztig extended the notion to reductive Lie groups. He also proved that specialization of elements of the dual canonical basis in representation theory of quantum groups at q=1 are totally non-negative polynomials. Thus, it is important to investigate classes of functions on matrices that are positive on totally positive matrices. I will discuss several sources of such functions. One has to do with multiplicative determinantal inequalities (joint work with M.Gekhtman). We extend the problem in this project to the description of bounded Laurent monomials in cluster variables on the positive loci (joint work in progress with M.Gekhtman and Z.Greenberg). Another source deals with certain partial sums of Plucker relations (joint work with P.K.Vishwakarma 2023). The main tools we employed are network parametrization and Temperley-Lieb immanants.

  • Speaker Lisa Carbone, Rutgers University
    • Title Monstrous Lie algebras as Borcherds algebras
    • Time/place 12/8/2023, Friday, 12:10 pm (Eastern Time), Hill 705
    • Abstract
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