Randomness and Lie-theoretic Structures

March 4 — 5, 2024

Department of Mathematics, University of Virginia, Charlottesville, VA
Note the related conference just before this one: QUANTUM STRUCTURES IN LIE THEORY, MARCH 1 — 3, 2024 (organized separately)

Fruitful interplay between probability and algebra has led to many new insights in both fields in recent years. In bringing together experts near their intersection, focusing on integrable stochastic models and Lie-theoretic structures, we aim to foster synergy and new collaborations and create opportunities for graduate students and postdocs interested in these areas.


All talks are in Clark Hall 107.

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Monday, March 4, 2024

  • 8.00 • Bagels and Coffee Breakfast
  • 8.55 • Opening Remarks
  • 9.00 - 10.00 • Jimmy He (MIT)
    Symmetries of periodic measures on partitions
    The periodic $q$-Whittaker measure is a probability measure on partitions defined in terms of $q$-Whittaker functions and an additional parameter $u$ known as the winding fugacity. I will explain a hidden distributional symmetry of this model which exchanges the $u$ and $q$ parameters, as well as related results on the periodic Hall-Littlewood measure. As a special case, we recover an identity of Imamura--Mucciconi--Sasamoto. This is joint work with Michael Wheeler.
  • 10.15 - 11.15 • Ben Brubaker (Minnesota)
    The Quantum Group - Partition Function Dictionary
    Quantum group modules give rise to solvable two-dimensional square lattice models. Those lattice models in turn lead to interesting families of partition functions, including special functions in the local representation theory of algebraic groups and in the cohomology of flag varieties. This suggests the possibility of a quantum group (module) - partition function "dictionary" where one can look up special functions based on the required modules, and use the facility of lattice models to uncover their properties. We evaluate the potential for such a dictionary, first reviewing the basic constructions and then detailing examples of entries in the dictionary in joint work with Buciumas, Bump, and Gustafsson, and in work of Aggarwal, Borodin, Petrov, and Wheeler. Does the quantum group connection matter? Or is it more important that these lattice models arise from generalizations of the six-vertex free fermion point?
  • 11.30 - 12.30 • Laura Colmenarejo (NC State)
    Macdonald polynomials: why do we care about them?
    In this talk, we will be talking about Macdonald polynomials and why we care about them. Our motivation will come not only from different areas in mathematics, but also from physics. I will give an overview of some of the results in the literature, and dig more into them depending on the time we have.
  • Lunch — see local hints
  • 2.00 - 4.00 • Participant Talks
    • 2:00 • Weihong Xu (Virginia Tech)
    • 2:20 • Kelvin Rivera-Lopez (Gonzaga University)
    • 2:40 • Hong Chen (Rutgers)
    • 3:05 • Runqiu Xu (UC San Diego)
    • 3:25 • Haihan Wu (UC Davis)
    • 3:45 • Jiaming Xu (University of Wisconsin-Madison)
    • 2:00 • Weihong Xu (Virginia Tech) - Quantum K-theory of IG(2,2n)
      I will talk about explicit computations of the structure of the quantum K-theory ring for the type C co-adjoint flag variety, including a Chevalley formula and the Seidel representation. This is joint work with V. Benedetti and N. Perrin.

    • 2:20 • Kelvin Rivera-Lopez (Gonzaga University) - Up-down chains on permutations and their scaling limits
      An up-down chain is a Markov chain in which each transition can be decomposed into a growth step followed by a reduction step. Generally, these two steps are unrelated, but when they exhibit a certain symmetry, the up-down chain is particularly amenable to analysis. In the first part of this talk, we will discuss a general framework for analyzing these special up-down chains. This approach will mainly be algebraic and will lead to scaling limits. Afterwards, we will discuss an example that is related to a new family of (random) permutons.
      Based on joint work with Valentin Féray.

    • 2:40 • Hong Chen (Rutgers) - Binomial Coefficients and Littlewood--Richardson Coefficients for Interpolation Polynomials
      Interpolation Jack and Macdonald polynomials were introduced by Knop--Sahi and Okounkov in the 90s, defined by certain degree and interpolation conditions. The evaluations of interpolation polynomials are called binomial coefficients. I will talk about some recent work on these binomial coefficients: they are positive and monotone. As an application, we show that this gives a characterization of the containment partial order in terms of Schur positivity or Jack positivity; this result is in parallel with the work of Cuttler--Greene--Skandera--Sra and Khare--Tao, which gave characterizations of two other partial orders, namely, majorization and weak majorization. Time permitting, I will present a new combinatorial formula for the Littlewood--Richardson coefficients and some positivity results. This is joint work with my advisor, Prof. Siddhartha Sahi.

    • 3:05 • Runqiu Xu (UC San Diego) -A Comparison of U(N) and SU(N) Weingarten functions
      U(N) Weingarten function, known as the U(N) link integral, is an essential ingredient in the strong coupling expansion in lattice gauge theory. Although fewer people pay attention to SU(N), the SU(N) Weingarten function is very important and has many applications in lattice Quantum Chromodynamics. In this paper, we present the derivation of the SU(N) Weingarten function and emphasize some details about how it differs from the U(N) Weingarten function. We also derive the 1/N expansion in both cases by counting weakly monotone walks on the Cayley graph of S(d). It shows that the SU(N) large N limit differs from U(N), and we will explore the combinatorial difference further.

    • 3:25 • Haihan Wu (UC Davis) - Dimers and Sp(2n) webs
      The dimer model is a statistical mechanical model that studys random dimer covers (perfect matchings) of a graph. Kasteleyn's theorem computes the number of dimer covers of a graph by calculating the determinant of a modified adjacency matrix. The generalizations of the theorem connect dimers to type A webs. I will talk about further generalizations to the type C case. This talk is based on upcoming joint work with Richard Kenyon.

    • 3:45 • Jiaming Xu (University of Wisconsin-Madison) - Edge universality of Beta-additions through Dunkl operators
      It is well known that the edge limit of Gaussian/Laguerre Beta ensembles is given by Airy(\beta) point process. We prove an universality result that this also holds for a general class of additions of Gaussian and Laguerre ensembles. In order to make sense of Beta-addition, we introduce type A Bessel function as the characteristic function of our matrix ensemble, then extract its moment information through the action of Dunkl operators, a class of differential operators originated from special function theory. Joint work in preparation with David Keating.
  • 4.20 - 5.20 • Jennifer Morse (University of Virginia)
    A rational approach to Macdonald polynomials
    We will discuss classical examples of how Young tableaux and Dyck path combinatorics have been used to answer questions in geometry and representation theory. In the 1980's, these examples inspired a more intricate framework centering around q,t-generalizations of Catalan numbers and Macdonald polynomials. We will talk about questions arising in this contemporary setting and about our approach using symmetric rational functions (called catalanimals) instead of polynomials. Subfamilies of catalanimals truncate to Macdonald polynomials and more general catalanimals enabled us to settle the longest standing conjecture in the area.

    Joint work with Blasiak, Haiman, Pun, and Seelinger

Tuesday, March 5, 2024

  • 8.00 • Bagels and Coffee Breakfast
  • 9.00 - 10.00 • Mark Shimozono (Virginia Tech)
    Centralizer construction of K-homology of affine Grassmannian
    Ginzburg and Peterson independently showed that the equivariant homology of the affine Grassmannian is Hopf-isomorphic to the functions on a family over the Cartan whose zero fiber is the centralizer of a principal nilpotent in Langlands dual type. These are (up to localization) isomorphic to equivariant quantum cohomology of G/B. We discuss a conjectural analogue of the Ginzburg-Peterson centralizer theorem in K-theory. Using ideas from integrable systems, in types A and C we have explicit realizations in symmetric functions Schubert bases, matrix entries, etc. This is joint ongoing work with T. Ikeda, S. Iwao, and K. Yamaguchi.
  • 10.15 - 11.15 • Colin McSwiggen (NYU Courant)
    Large deviations and multivariable special functions
    This talk introduces techniques for using the large deviations of interacting particle systems to study the large-N asymptotics of generalized Bessel functions. These functions arise from a versatile approach to special functions known as Dunkl theory, and they include as special cases most of the spherical integrals that have captured the attention of random matrix theorists for more than two decades. I will give a brief introduction to Dunkl theory and then present a result on the large-N limits of generalized Bessel functions, which unifies several results on spherical integrals in the random matrix theory literature. These limits follow from a large deviations principle for radial Dunkl processes, which are generalizations of Dyson Brownian motion. If time allows, I will discuss some further results on large deviations of radial Heckman-Opdam processes and/or applications to asymptotic representation theory. Joint work with Jiaoyang Huang.
  • 11.30 - 12.30 • Slava Naprienko (UNC Chapel Hill)
    Integrable Lattice Models from Representation Theory of p-adic Groups
    Integrable lattice models usually arise from mathematical physics or from representation theory of quantum algebras. In this talk, I will discuss how integrable lattice models appear in representation theory of p-adic groups. In one instance, we’ll get colored fermionic models, in another — colored bosonic models. This approach provides a representation-theoretical interpretation of many results from integrable lattice models. I hope that this connection can lead to asymptotic representation theory of p-adic groups.
  • Lunch — see local hints
  • 1.45 - 2.45 • Jeffrey Kuan (Texas A&M)
    Universality of dynamic processes using Drinfel'd twisters
    The concept of 'universality' motivates a wide variety of probability and mathematical physics problems, going back to the classical central limit theorem. Most recently, the Kardar--Parisi--Zhang universality class has been proven to have Tracy--Widom fluctuations in the long-time asymptotics. In this talk, I will present a new universality result about the long-time asymptotics of so--called ``dynamic'' processes. The asymptotic fluctuations are related to the Tracy--Widom distribution. The proof will utilize a duality of Markov processes, which is constructed using Drinfel'd twisters of the quantum group U_q(sl_2), viewed as a quasi-triangular quasi-Hopf algebra. The orthogonality of the duality functions allow for an asymptotic analysis.
  • 3.00 - 4.00 • Siddhartha Sahi (Rutgers)
    Lyapunov exponents for random products of real matrices
    We consider probability measures on GL(n,R) that are left-invariant under the orthogonal group O(n,R). For any such measure we consider the following two quantities: (a) the mean of the log of the absolute value of the eigenvalues of the matrices and (b) the Lyapunov exponents of random products of matrices independently drawn with respect to the measure. Our main result is a lower bound for (a) in terms of (b).

    This lower bound was conjectured by Burns-Pugh-Shub-Wilkinson (2001), and special cases were proved by Dedieu-Shub (2002), Avila-Bochi (2003) and Rivin (2005). We give a proof in complete generality by using some results from the theory of spherical functions and Jack polynomials.

    This is joint work with Diego Armentano, Gautam Chinta, and Michael Shub. (Ergodic Theory and Dynamical Systems, to appear)
Venue: Clark Hall 107



  1. Ben Brubaker (Minnesota)
  2. Laura Colmenarejo (North Carolina State University)
  3. Jimmy He (MIT)
  4. Jeffrey Kuan (Texas A&M)
  5. Colin McSwiggen (NYU Courant)
  6. Jennifer Morse (University of Virginia)
  7. Slava Naprienko (UNC Chapel Hill)
  8. Siddhartha Sahi (Rutgers)
  9. Mark Shimozono (Virginia Tech)
UVA Lawn

Registration Deadlines

  • To request financial support and submit a participant talk request: February 10, 2024
  • General registration (please register to help us estimate the number of bagels for breakfast): February 29, 2024

We gratefully acknowledge financial support from