## Lie Group/Quantum Mathematics SeminarOrganizers Lisa Carbone, Yi-Zhi
Huang, Jim
Lepowsky and Siddhartha Sahi.
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 webmaster@math.rutgers.edu. 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.pdf file of the slides of the talk.
**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**Donald Richards, Pennsylvania State University**Title**Integral Transform Methods in Goodness-of-Fit Testing for the Wishart Distributions**Time/place**3/8/2019, Friday, 12:00 in Hill 705**Abstract**This talk is based on joint work with Elena Hadjicosta.In recent years, random data consisting of positive definite (symmetric) matrices have appeared in several areas of applied research, e.g., diffusion tensor imaging, wireless communication systems, synthetic aperture radar, and volatility models in finance. Given a random sample of such matrices, we wish to test whether the data are drawn from a given statistical population. In this talk, we apply the Hankel transform of matrix argument to develop goodness-of-fit tests for the Wishart distributions. The asymptotic distribution of the test statistic is derived in terms of the integrated square of a Gaussian random field, and an explicit formula is obtained for the corresponding covariance operator. The eigenfunctions of the covariance operator are determined explicitly, and the eigenvalues are shown to satisfy certain interlacing properties. Throughout this work, the Bessel functions of matrix argument of Herz (1955) and the zonal polynomials of James (1964) play a crucial role. Also, our results raise the possibility of extending to the Wishart distributions many statistical properties of the classical, one-dimensional gamma distributions.
**Speaker**Daniel Nakano, University of Georgia**Title**On BBW Parabolics for Simple Classical Lie Superalgebras**Time/place**3/29/2019, Friday, 12:00 in Hill 705**Abstract**Let g be a (simple) classical Lie superalgebra over the complex numbers. In this talk I will discuss the developments over the past 10 years that have led to effective methods to systematically study these algebras via the construction of detecting subalgebras. The detecting subalgebras play an important role in the theory. They completely detect the cohomology relative to the even subalgebra, and provide a structural interpretation of combinatorial invariants that were defined by Kac and Wakimoto.Recently, the speaker with his collaborators have constructed parabolic subalgebras, b, (like Borel subalgebras) where the detecting subalgebras can be viewed as the Levi component. By comparing the cohomology of g and b, we discovered an important relationship with the Poincare series of an ambient complex reflection group via the Bott-Borel-Weil theorem. Furthermore, these Poincare series describe the higher sheaf cohomology groups of the trivial line bundle over G/B where g=Lie G and b=Lie B. At the end of the talk, applications will be given to verifying the conjecture due to Boe, Kujawa and Nakano on the realization of support varieties for g. This talk represents joint work with D. Grantcharov, N. Grantcharov and J. Wu.
**Speaker**Nitu Kitchloo, Johns Hopkins University**Title**Stability for Kac-Moody Groups**Time/place**4/5/2019, Friday, 12:00 in Hill 705**Abstract**In the class of Kac-Moody groups, one can extend all the exceptional families of compact Lie groups yielding infinite families (E_n, F_n, G_n), as well as other infinite families. These families often pass through the Affine or loop-groups of compact Lie groups. We will show that these exceptional families stabilize in a homotopical sense and that the (co)homology of their classifying spaces behave well (in a precise sense) for all but a finite set of primes that is determined by the family and not the individual groups in the family. I will also speculate on the underlying structure that might be present in these families.
**Speaker**Thomas Creutzig, University of Alberta**Title**Glueing Vertex Algebras**Time/place**4/12/2019, Friday, 12:00 in Hill 705**Abstract**The aim of this talk is to explain why certain types of extensions of vertex algebras are possible if and only if equivalences of associated braided tensor categories hold. This result is due to joint work with Shashank Kanade and Robert McRae and it will be motivated via its many nice consequences as e.g. rigidity and fusion rules of categories of W-algebra modules, constructing vertex algebras for S-duality and many more.
**Speaker**Lisa Carbone, Rutgers University University**Title**Imaginary reflections and automorphisms of the monster Lie algebra**Time/place**4/19/2019, Friday, 12:00 in Hill 705**Abstract**We describe a `hidden symmetry' of the root lattice of the monster Lie algebra m, namely a reflection with respect to an imaginary simple root, that preserves the bilinear form and root multiplicities. This symmetry appears only after the infinite root lattice is specialized to the unique even 2-dimensional unimodular Lorentzian lattice II_{1,1}. When composed with reflection in the unique real simple root, this symmetry extends to the Cartan involution on m. We describe an uncountable family of involutions on m and their associated eigenspaces, known as Cartan pairs, and we examine these involutions as elements of Aut(m).
**Speaker**Juan Villarreal, Virginia Commonwealth University**Title**The Witten Genus and sheaves of vertex algebras**Time/place**4/26/2019, Friday, 12:00 in Hill 705**Abstract**In this talk I want to introduce the Witten genus. We can think on the Witten genus as a loop generalization of the Atiyah Singer index theorem for Dirac operators. However, the idea to formalize a Dirac operator on loop spaces has been always a little subtle. Different approaches have been done, and always it has been considered that a vertex algebraic description of this construction must be possible. I want to show a construction using some sheaves of vertex algebras.
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