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.

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. This seminar also has a page Lie Groups Quantum Mathematics Seminar, maintained by 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 and Colloquia page.

Spring, 2019

  • Speaker Thomas Lam, University of Michigan
    • Title Geometric crystals and Whittaker functions
    • Time/place 1/25/2019, Friday, 12:00 in Hill 705
    • Abstract Kashiwara's crystal graphs provide combinatorial models for finite-dimensional irreducible modules of a complex semisimple Lie algebra. Berenstein and Kazhdan's geometric crystals are algebraic varieties that rationally lift combinatorial crystals, and have found applications for example in mirror symmetry, combinatorics of tableaux, and certain statistical mechanical models.

      I will discuss some of the basics of geometric crystals, and their characters, which turn out to be Archimedean Whittaker functions.

  • Speaker Florencia Orosz Hunziker, Yale University
    • Title Fusion rules for the Virasoro algebra of central charge 25
    • Time/place 2/1/2019, Friday, 12:00 in Hill 705
    • Abstract In 1990 Feigin and Fuchs established a correspondence between the Verma modules for the Virasoro algebras of dual central charges $c$ and $26- c$. In later work, the irreducible quotient module $L(c,0)$ was proved to be a vertex operator algebra called the Virsasoro VOA of central charge $c$. In this talk we will discuss an extension of the Feigin-Fuchs correspondence to the vertex algebra setting for the case $c=1$ and $c=25$. We will prove the the fusion rules for the non-Verma irreducible $L(25,0)$-modules coincide the fusion rules for the non-Verma irreducible $L(1,0)$-modules.

  • Speaker Robert Laugwitz, Rutgers University
    • Title A category interpolating crossed modules over symmetric groups
    • Time/place 2/8/2019, Friday, 12:00 in Hill 705
    • Abstract P. Deligne introduced a remarkable tensor category interpolating the representation theory of symmetric groups, allowing for the natural number of permuted letters to be replaced by any complex number. We compute the monoidal center of this category and obtain a ribbon category that interpolates the category of crossed modules over symmetric groups in a similar way. As an application, interpolations of untwisted Dijkgraaf-Witten invariants of framed ribbon links are obtained.

      This talk is based on joint work with Johannes Flake, RWTH Aachen University.

  • Speaker Yi-Zhi Huang, Rutgers University
    • Title A construction of twisted modules for grading-restricted vertex (super)algebras
    • Time/place 2/15/2019, Friday, 12:00 in Hill 705
    • Abstract We give a general and direct construction of (grading-restricted generalized) twisted modules for a grading-restricted vertex (super)algebra V associated to an automorphism g of V. Even in the case that g is of finite order, finding such a construction has been a long-standing problem in the representation theory of vertex operator algebra and orbifold conformal field theory. Besides twisted vertex operators, one crucial ingredient in this construction is what we call the "twist vertex operators" or "twist fields." Assuming that a grading-restricted vector space W equipped with a set twisted fields and a set of twist fields satisfy a weak commutativity for twisted fields, a generalized weak commutativity for one twisted field and one twist field and a number of other properties that are relatively easy to verify, we define a twisted vertex operator map for W and prove that W equipped with this twisted vertex operator map is a (grading-restricted generalized) g-twisted V-module. As a class of examples, we construct (grading-restricted generlized) twisted moduels for vertex operator algebras associated to affine Lie algebras.

  • Speaker Nicola Tarasca, Rutgers University
    • Title Vertex algebras and moduli spaces of curves
    • Time/place 2/22/2019, Friday, 12:00 in Hill 705
    • Abstract This talk will focus on geometric realizations of vertex algebras. The Virasoro uniformization provides an incarnation of the Virasoro algebra in the tangent space of the Hodge line bundle on moduli of algebraic curves with marked points and local coordinates. This allows to assign to certain representations of the Virasoro algebra a sheaf on moduli of curves together with a projective connection. After reviewing some facts on curves and their moduli spaces, I will discuss the sheaves on moduli of stable curves obtained from coinvariants of modules over conformal vertex algebras, and identify their logarithmic projective connection. This is joint work with Chiara Damiolini and Angela Gibney.

  • Speaker Mark Skandera, Lehigh University
    • Title Total nonnegativity and induced sign characters of the Hecke algebra
    • Time/place 3/1/2019, Friday, 12:00 in Hill 705
    • Abstract Gantmacher's study of totally nonnegative (TNN) matrices in the 1930's eventually found applications in many areas of mathematics. Descending from his work are problems concerning TNN polynomials, those polynomial functions of n^2 variables which take nonnegative values on TNN matrices. Closely related to TNN polynomials are functions in the Hecke algebra trace space whose evaluations at certain Hecke algebra elements yield polynomials in N[q]. In all cases, it would be desirable to combinatorially interpret the resulting nonnegative numbers. In 2017, Kaliszewski, Lambright, and the presenter found the first cancellation-free combinatorial formula for the evaluation of all elements of a basis of V at all elements of a basis of the Hecke algebra. We will discuss a recent improvement upon this result which also advances our understanding of TNN polynomials. This is joint work with Adam Clearwater.

  • Speaker Daniel Nakano, University of Georgia
    • Title
    • Time/place 3/29/2019, Friday, 12:00 in Hill 705
    • Abstract

  • Speaker Thomas Creutzig, University of Alberta
    • Title
    • Time/place 4/12/2019, Friday, 12:00 in Hill 705
    • Abstract

  • Speaker Juan Villarreal, Virginia Commonwealth University
    • Title
    • Time/place 4/26/2019, Friday, 12:00 in Hill 705
    • Abstract

Previous semesters