Lie Group/Quantum Math Seminar

Lie Group/Quantum Mathematics Seminar

Organizers Lisa Carbone, Yi-Zhi Huang, Jim Lepowsky and Siddhartha Sahi.

Time Friday, 12:00 to 1:00 pm.

Place Hill 705 (via Zoom for Fall, 2020 and Spring, 2021; see below for the Zoom link and passcode for Spring, 2021).

Starting from Spring, 2008, the Lie Group Seminar and Quantum Mathematics Seminar have merged together to a single seminar called the Lie Group/Quantum Mathematics Seminar. The information on seminar talks can also be found in the Seminars & Colloquia Calendar page in the department. For the Lie Group/Quantum Mathematics seminar in previous semesters, see this page. For talks in the Quantum Mathematics Seminar from Spring, 1998 to Fall, 2007, see this page. For a few years before 2008, the Quantum Mathematics Seminar shared the time and place with the Algebra Seminar. For talks in both the Algebra and Quantum Mathematics Seminars in these few semesters, see the page for the Previous Rutgers Algebra Seminars. For all the seminars and colloquia in the department, see the Seminars & Colloquia Calendar page.

Spring, 2021

All seminar talks will be online using zoom this semester.

Zoom link: https://rutgers.zoom.us/j/93921465287

Meeting ID: 939 2146 5287

Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module

Most of the zoom talks will be recorded and will be placed in the YouTube channel Rutgers Lie Groups Quantum Math Seminar

  • Speaker Lisa Carbone, Rutgers University and Institute for Advanced Study, School of Natural Sciences
    • Title Complete pro-unipotent automorphism group for the monster Lie algebra
    • Time 1/22/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract We construct a complete pro-unipotent group of automorphisms for a completion of the monster Lie algebra. We also construct an analog of the exponential map and Adjoint representation. This gives rise to some useful identities involving imaginary root vectors.

  • Speaker Hao Li, SUNY-Albany
    • Title Arc spaces, vertex algebras and principal subspaces
    • Time 1/29/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract Arc spaces were originally introduced in algebraic geometry to study singularities. More recently they show in connections to vertex algebras. There is a closed embedding from the singular support of a vertex algebra V into the arc space of associated scheme of V. We call a vertex algebra "classically free" if this embedding is an isomorphism. In this introductory survey talk, we will first introduce arc spaces and some of its backgrounds. Then we will provide several examples of classically free vertex algebras including Feigin-Stoyanovsky principal subspaces, and explain their applications in differential algebras, $q$-series identities, etc. In particular, we will show the classically freeness of principal subspaces of type A at level 1 by using a method of filtrations and identities from quantum dilogarithm or quiver representations. As a result, we obtain new presentations and graded dimensions of the principal subspaces of type A at level 1, which can be thought of as the continuation of previous works by Calinescu, Lepowsky and Milas. The classically freeness of some principal subspaces which possess free fields realisation will also be discussed. Most of the talk is based on the joint work with A. Milas.

  • Speaker Lilit Martirosyan, University of North Carolina, Wilmington
    • Title Braided rigidity for path algebras (joint work with Hans Wenzl)
    • Time 2/5/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups Bn for all n in N. We say that such representations are rigid if they are determined by the path algebra and the representations of B2. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of G2 satisfies the rigidity condition, provided B3 generates End(V^{⊗3}). We obtain a complete classification of ribbon tensor categories with the fusion rules of g(G2) if this condition is satisfied.
    • Slides pdf file.
    • Archive paper arXiv:2001.11440 and arXiv:1609.08440
    • YouTube video Braided rigidity for path algebras

  • Speaker Emily Sergel, Rutgers University
    • Title Positivity of interpolation polynomials
    • Time 2/12/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract The interpolation polynomials are a family of inhomogeneous symmetric polynomials characterized by simple vanishing properties. In 1996, Knop and Sahi showed that their top homogeneous components are Jack polynomials. For this reason these polynomials are sometimes called interpolation Jack polynomials, shifted Jack polynomials, or Knop-Sahi polynomials.

      We prove Knop and Sahi's main conjecture from 1996, which asserts that, after a suitable normalization, the interpolation polynomials have positive integral coefficients. This result generalizes Macdonald's conjecture for Jack polynomials that was proved by Knop and Sahi in 1997. Moreover, we give a combinatorial expansion for the interpolation polynomials that exhibits the desired positivity property.

      This is joint work with Y. Naqvi and S. Sahi.
    • Slides pdf file.
    • Archive paper arXiv:2104.08598
    • YouTube video Positivity of interpolation polynomials

  • Speaker Antun Milas, SUNY-Albany
    • Title Graph q-series, graph schemes, and 4d/2d correspondences
    • Time 2/19/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract To any graph with n nodes we associate two n-fold q-series, with single and double poles, closely related to Nahm's sum associated to a positive definite symmetric bilinear form.

      Quite remarkably series with "double poles" sometimes capture Schur's indices of 4d N = 2 superconformal field theories (SCFTs) and thus, under 2d/4d correspondence, they give new character formulas of certain vertex operator algebras.

      If poles are simple, they arise in algebraic geometry as Hilbert-Poincare series of "graph" arc algebras. These q-series are poorly understood and seem to exhibit peculiar modular transformation behavior.

      In this talk, we explain how these "counting" functions arise in different areas of mathematics and physics. This talk will be fairly accessible, assuming minimal background. No familiarity with concepts like vertex algebras and 4d N=2 SCFT is needed.

  • Speaker Lisa Carbone, Rutgers University and Institute for Advanced Study, School of Natural Sciences
    • Title Imaginary root strings and Chevalley-Steinberg group commutators for hyperbolic Kac--Moody algebras
    • Time 3/12/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract Let L be a symmetrizable hyperbolic Kac--Moody algebra. We show that any root string in the direction of an imaginary root is infinite and we show that bi-infinite roots strings in the direction of an imaginary root can occur. For L hyperbolic of rank 2, we classify all the possible root strings that can occur. When L is also symmetric, we describe how imaginary root strings determine the Chevalley-Steinberg group commutators for imaginary roots in a complete Kac--Moody group associated to L and we give a recursive method for determining them.

      This is joint work with T. Coelho, J. Fonseca, J. Meng, S. H. Murray, F. Thurman and S. Zhu. This work arose from collaborative discussions in my graduate course "Topics in Algebra" in Spring 2020.

  • Speaker Jason Saied, Rutgers University
    • Title Combinatorial formula for SSV polynomials
    • Time 3/26/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract Macdonald polynomials are homogeneous polynomials that generalize many important representation-theoretic families of polynomials, such as Jack polynomials, Hall-Littlewood polynomials, affine Demazure characters, and Whittaker functions of GL_r(F) (where F is a non-Archimedean field). They may be constructed using the basic representation of the corresponding double affine Hecke algebra (DAHA): a particular commutative subalgebra of the DAHA acts semisimply on the space of polynomials, and the (nonsymmetric) Macdonald polynomials are the simultaneous eigenfunctions. In 2018, Sahi, Stokman, and Venkateswaran constructed a generalization of this DAHA action, recovering the metaplectic Weyl group action of Chinta and Gunnells. As a consequence, they discovered a new family of polynomials, called SSV polynomials, that generalize both Macdonald polynomials and Whittaker functions of metaplectic covers of GL_r(F). We will give a combinatorial formula for these SSV polynomials in terms of alcove walks, generalizing Ram and Yip's formula for Macdonald polynomials.
    • YouTube video Combinatorial formula for SSV polynomials

  • Speaker Chiara Damiolini, Rutgers University
    • Title Geometric properties of sheaves of coinvariants and conformal blocks
    • Time 4/2/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract One method to study the moduli space of stable pointed curves is via the study of vector bundles on them as they can yield interesting maps to projective spaces. An effective way to produce such vector bundles is through representations of vertex operator algebras: more precisely attached to n simple modules over a vertex opearator algebra of CohFT type, we can construct sheaves of coinvariants over the space of stable n-pointed curves. This generalizes the construction of coinvariants associated with representations of affine Lie algebras. In this talk I will focus on some geometric properties of these sheaves, especially on global generation. Investigating this property we can see phenomena that did not occur for coinvariants associated with affine Lie algebra representations. This is based on joint work with A. Gibney and N. Tarasca and ongoing work with A. Gibney.
    • Slides pdf file.
    • YouTube video Geometric properties of sheaves of coinvariants and conformal blocks.

  • Speaker Haisheng Li, Rutgers University at Camden
    • Title Deforming vertex algebras by module and comodule actions of vertex bialgebras
    • Time 4/9/2021, Friday, 12:00-12:45 pm (Eastern Time)
    • Location Zoom link above
    • Abstract Previously, we introduced a notion of vertex bialgebra and a notion of module vertex algebra for a vertex bialgebra, and gave a smash product construction of nonlocal vertex algebras. Here, we introduce a notion of right comodule vertex algebra for a vertex bialgebra. Then we give a construction of quantum vertex algebras from vertex algebras with a right comodule vertex algebra structure and a compatible (left) module vertex algebra structure for a vertex bialgebra. As an application, we obtain a family of deformations of the lattice vertex algebras. This is based on a joint work with Naihuan Jing, Fei Kong, and Shaobin Tan.
    • Slides pdf file.

  • Speaker Corina Calinescu, New York City College of Technology and CUNY Graduate Center
    • Title Principal subspaces of standard modules for twisted affine Lie algebras
    • Time 4/16/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract This talk is an overview of results about principal subspaces for twisted affine Lie algebras. In joint works with J. Lepowsky, A. Milas, M. Penn and C. Sadowski, we studied the principal subspaces of certain standard modules for the twisted affine Lie algebra of type A, by using vertex algebraic methods. Their graded dimensions are given in connection with q-series identities.

  • Speaker Christopher Sadowski, Ursinus College
    • Title Permutation orbifold subalgebras of Virasoro vertex algebras
    • Time 4/23/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract In this talk, we examine certain permutation orbifold subalgebras (fixed-point subalgebras) of tensor powers of Virasoro vertex operator algebras under certain group actions. In particular, we determine the strong generators of these subalgebras and point out isomorphisms between these subalgebras at certain central charges and W-algebras. The search for these strong generators makes heavy use of Mathematica. In this talk, we will demonstrate exactly how such computations are performed with live examples. One of the primary aims of this talk is to demonstrate the usefulness of computers in the study of examples of vertex operator algebras, and to promote the use of computer algebra software to derive results that would be difficult to obtain by hand.

      This talk is based on joint work with Antun Milas and Michael Penn.

  • Speaker Abid Ali, Rutgers University
    • Title Eisenstein Series on Arithmetic Quotients of Rank 2 Kac--Moody Groups Over Finite Fields
    • Time 4/30/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract Let G be a rank 2 Kac-Moody group over a finite field. The group G comes equipped with a data (X, K), where X is the Tits building of G and K is the standard parabolic subgroup of the negative BN-pair. By using this data and the Iwasawa decomposition of G, we define Eisenstein series on the quotient graph K/X, which is induced by a character defined on the set of vertices of X.

      In this seminar, I will discuss this Eisenstein series, its convergence and meromorphic continuation.

      This is a joint work with Lisa Carbone (Rutgers), Kyu-Hwan Lee (UConn) and Dongwen Liu (Zhejiang) (With an appendix by Paul Garrett (UMN)).
    • Slides pdf file.
    • YouTube video Eisenstein series on rank 2 Kac-Moody groups over finite fields.

  • Speaker Henrik Gustafsson, Rutgers University
    • Title Multiple Dirichlet series, Whittaker functions and lattice models
    • Time 6/4/2021, Friday, 12:00 (Eastern Time)
    • Location Zoom link above
    • Abstract In this talk I will review recent research joint with Ben Brubaker, Valentin Buciumas and Daniel Bump on certain solvable lattice models and related quantum groups connected to Whittaker functions in non-archimedean representation theory.

      Historically, one motivation for finding such lattice models was to study and compute Weyl group multiple Dirichlet series via Whittaker functions on metaplectic n-covers of GL_r. I will describe how the connections between all these different objects were developed and the benefits of introducing solvable lattice models. Lastly, I will discuss interesting new results for metaplectic Whittaker functions coming from the connection to lattice models that hint of a symmetry between the rank r and the degree n of the cover.

Some additional talks by members of this seminar

  • Speaker Lisa Carbone, Rutgers University and Institute for Advanced Study, School of Natural Sciences
    • Title A Lie group analog for the monster Lie algebra
    • Event Mathematics Colloquium 2020-21, Mathematics Institute, University of Warwick
    • Time 2/26/2021, Friday, 11:00 am (Eastern Time)
    • Location See the Microsoft team link in the web page of Mathematics Colloquium 2020-21 at Warwick above
    • Abstract The monster Lie algebra is an infinite dimensional Lie algebra constructed by Borcherds as part of his program to solve the Conway-Norton conjecture about the representation theory of the Monster finite simple group M. We describe a construction of a Lie group analog for the monster Lie algebra and its relationship with the Monster finite simple group.

  • Speaker Siddhartha Sahi, Rutgers University
    • Title A reduction principle for Fourier coefficients of automorphic forms
    • Event Algebraic Geometry and Representation Theory Seminar, Faculty of Mathematics and Computer Science, The Weizmann Institute of Science
    • Time 4/7/2021, Wednesday, 9:30 am (Eastern Time) or 16:30 (Israel Time)
    • Location See Zoom link in the web page of Algebraic Geometry and Representation Theory Seminar at Weizmann Institute above
    • Abstract We consider a general class of Fourier coefficients for an automorphic form on a finite cover of a reductive adelic group G(A_K), associated to the data of a `Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are `Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a K-distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In follow-up papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of their top Fourier coefficients. This is joint work with Dmitry Gourevitch, Henrik P. A. Gustafsson, Axel Kleinschmidt, and Daniel Persson.
    • Archive paper arXiv:1811.05966

  • Speaker Siddhartha Sahi, Rutgers University
    • Title Quasi-polynomial representations of double affine Hecke algebras and a generalization of Macdonald polynomials
    • Event Solvable Lattice Models Seminar, Stanford University
    • Time 5/5/2021, Wednesday, 5:00 pm (Eastern Time) or 2:00 pm (Pacific Time)
    • Location See Zoom link in the web page of Solvable Lattice Models Seminar above
    • Abstract Macdonald polynomials are a remarkable family of functions. They are a common generalization of many different families of special functions arising in the representation theory of reductive groups, including spherical functions and Whittaker functions.

      In turn, Macdonald polynomials can be understood in terms of a certain representation of Cherednik's double affine Hecke algebra (DAHA), acting on polynomial functions on a torus.

      Whittaker functions admit a natural generalization to the setting of metaplectic covers of reductive p-adic groups, which play a key role in the theory of Weyl group multiple Dirichlet series.

      It turns out that Macdonald polynomials also admit a corresponding generalization, which can be understood in terms of a representation of the DAHA on the space of quasi-polynomial functions on a torus.

      This is joint work with Jasper Stokman and Vidya Venkateswaran.

  • Speaker Siddhartha Sahi, Rutgers University
    • Title Interpolation polynomials, Capelli operators, and Lie superalgebras
    • Event Springfest in honor of Vera Serganova, Research workshop of the Israel Science Foundation
    • Time 5/6/2021, Thursday, 11:30 am (Eastern Time) or 18:30 (Israel Time)
    • Location See Zoom link in the web page of Springfest in honor of Vera Serganova above
    • Abstract The interpolation polynomials are a family of inhomogeneous symmetric polynomials that are characterized by rather simple vanishing (interpolation) conditions. They were introduced by the speaker in connection with joint work with Bertram Kostant on the eigenvalues of generalized Capelli-type operators associated to Jordan algebras. Of particular interest is a one parameter subfamily, which was studied by Friedrich Knop and the speaker, and by Okunkov-Olshanski, and which is closely related to Jack polynomials and Macdonald polynomials.

      We will describe two recent developments in this direction. The first set of results is joint work with Hadi Salmasian and Vera Serganova, which solves the Capelli eigenvalue problem in the setting of Lie superalgebras and Jordan superalgebras. The second is joint work with Yusra Naqvi and Emily Sergel, which proves a long-conjectured positivity property of interpolation polynomials.

  • Speaker Siddhartha Sahi, Rutgers University
    • Title Some properties of the Macdonald kernel and associated integral transforms
    • Event Harmonic Analysis Seminar , Louisiana State University
    • Time 6/1/2021, Tuesday, 4:30 pm (Eastern Time) or 3:30 pm (Central time)
    • Location See Zoom link in the web page of Harmonic Analysis Seminar above
    • Abstract Abstract: Jack polynomials are an important family of symmetric polynomials that depend on a parameter α. For certain values of α they specialize to radial parts of spherical functions on symmetric cones; in particular the values α=2/d, d=1,2,4 correspond to positive definite Hermitian matrices over R, C, H, respectively. I.G. Macdonald has introduced a certain kernel function e(x,y), which is defined as a multivariate power series involving Jack polynomials in two sets of variables x, y∈ R^n. In this paper we establish three key properties of the Macdonald kernel and associated integral transforms. As anticipated by Macdonald, these results allow one to develop a reasonable theory of Fourier and Laplace transforms, and hypergeometric functions, for arbitrary α>0; thereby generalizing classic results of Bochner, Herz, and many others, for symmetric cones. This is joint work with Gestur Olafsson.

A mini-conference with talks given by members of this seminar

    • Conference Name Vertex Operator Algebras and Related Topics
    • Organizers Corina Calinescu, CUNY Graduate Center and New York City College of Technology, and Christopher Sadowski, Ursinus College
    • Time Friday and Saturday, April 9-10, 2021, starting at 1:00 pm (Eastern Time) on April 9
    • Web page Vertex Operator Algebras and Related Topics. (Haisheng Li's April 9 talk above is cross-listed in this conference web page.)

Previous semesters