**Title.**TBA

**Abstract.**TBA

Zoom Meeting ID: 893 5344 7324; Password: Littlewood.

Zoom Meeting ID: 868 7916 4965; Password: Littlewood.

Video recording, slides.

Video recording, notes from the talk.

Video recording , Slides.

Video Recording.

Meeting ID: 869 5914 1402.

Video recording, Slides.

Video recording, slides.

Video recording, Slides.

Video recording.

- A. O. Morris, The spin representation of the symmetric group, Proc. London Math. Soc. (3), 12 (1962).
- J. R. Stembridge, Shifted tableaux and the projective representations of the symmetric groups. Adv. in Math. 74 (1989).

"Let G be a finite simple group of order divisible by the prime p, and suppose that G has a p-Steinberg character. Does it follow that G is a semisimple group of Lie type in characteristic p?"

This motivates us to define Quasi p-Steinberg character for finite groups. An irreducible character of a finite group G is called quasi p-Steinberg for a prime p dividing order of G if it is non zero on every p-regular element of G. In this talk, we discuss the existence of quasi p-Steinberg Characters of Symmetric as well as Alternating groups and their double covers. On the way, we also answer a question, similar to Feit, asked by Dipendra Prasad. This is based on ongoing work with Pooja Singla.1. Humphreys, J. E. The Steinberg representation,1987.

2. W. Feit, Extending Steinberg Characters,1993.

3. M. R. Darafsheh, p-Steinberg Characters of Alternating and Projective Special Linear Groups 1995.

Video recording.

Slides, Video

Zoom meeting ID: 873 0780 6787.

Slides, Video.

Kronecker coefficients are structural constants for the category $\mathrm{Rep}(S_n)$ of finite-dimensional representations of the symmetric group; namely, given three irreducible representations $\mu, \tau, \lambda$ of $S_n$, the Kronecker coefficient $\mathrm{Kron}( \lambda, \mu, \tau)$ is the multiplicity of $\lambda$ inside $\mu \otimes \tau$. The study of Kronecker coefficients has been described as "one of the main problems in the combinatorial representation theory of the symmetric group", yet very little is known about them.

I will define a "stable" version of the Kronecker coefficients (due to Murnaghan), which generalizes both Kronecker coefficientsand Littlewood-Richardson coefficients (structural constants for general linear groups).

It turns out that the stable Kronecker coefficients appear naturally as structural constants in the Deligne categories $\mathrm{Rep}(S_t)$, which are interpolations of the categories $\mathrm{Rep}(S_n)$ to complex $t$. I will explain this phenomenon, and show that the categorical properties of $\mathrm{Rep}(S_t)$ allow us not only to recover known properties of the stable Kronecker coefficients, but also obtain new identities.

This is a report on my project from 2014.

Video recording, Slides.Slides, Video recording

Video recording.

(1) each $J(n)$ resides in $\mathrm{GT}(n)$

(2) $J(1), \dots, J(n)$ generate $\mathrm{GT}(n)$

(3) the sum $J(1) + \cdots + J(n)$ resides in $Z\frak{F}(n)$

(4) each $J(k)$ acts diagonally on the irreducible representation $V(w)$ and its eigenvalue, with respect to a basis vector indexed by a saturated chain $u(0) \lhd \cdots \lhd u(n)$, depend only on the covering relation $u(k-1) \lhd u(k)$ in $\Bbb{YF}$.

This local eigenvalue $c(u \lhd v)$ is called the content of covering relation $u \lhd v$ with respect to the choice of Jucys-Murphy generators. Keep in mind that there are many different systems of elements $J(n)$ satisfying properties (1), (2), (3), and (4). However, not any assignment of covering weights $c(u \lhd v)$ can be realized as contents for such a system. Indeed a necessary condition requires that two saturated chains coincide if and only if the corresponding sequences of covering weights are equal; see recent work of S. Doty et. al. Since the Jucys-Murphy problem is under-determined it is natural to use the tower of symmetric groups $S(n)$ together with its branching poset, the Young lattice $\Bbb{Y}$, as a guide to impose further constraints. For example, one might try determine a system of Jucys-Murphy elements by forcing the attending system of contents to satisfy a specialization formula for the $\Bbb{YF}$-Schur functions in analogy with the principal specialization of classical Schur functions. This is work in progress.

Video, Notes from the talk.

**Title:** *A Timed Version of the Plactic Monoid*.

**Abstract:**
Lascoux and Schutzenberger introduced the plactic monoid as a tool to prove the Littlewood-Richardson rule.
The plactic monoid is the quotient of the free monoid on an ordered alphabet modulo Knuth relations.
In this talk I will explain how their theory can be generalized to *timed words*, which are words where each letter occurs for a positive amount of time rather than discretely.
This generalization gives an organic approach to piecewise-linear interpolations of correspondences involving semi-standard Young tableaux.
This talk is based on the arxiv preprint arXiv:1806.04393.

Video recording, sildes.

**Title:** *Intermediate Algebraic Structure in the Restriction Problem*.

**Abstract:**
The restriction problem refers to understanding in a combinatorial sense the decomposition of an irreducible representation of GL_n as a representation of S_n. In this talk I will discuss some of the intermediate algebraic structures that arise when studying this problem which constrain the symmetric group representations that appear and (hopefully) give some insight into the general problem. Things I will mention include:
Representation stability, the rook monoid, the group of monomial matrices, and a certain subalgebra of the universal enveloping algebra which seems to have interesting combinatorial properties.

Slides, Video recording.

**Title:** *Symmetric Group Characters as Symmetric Functions*.

**Abstract:**
I will present a basis of the symmetric functions whose evaluations are irreducible
characters of the symmetric group in the same way that the evaluations of Schur
functions are irreducible characters of the general linear group. These symmetric
functions are related to character polynomials (that go back to a paper of
Frobenius in 1904) but they have the advantage that we are able to use the Hopf
structure of the symmetric functions to compute with them. In addition, they
indicate that the combinatorics of Kronecker coefficients is governed by multiset
tableaux. We use this basis to give a combinatorial interpretation for the tensor
products of the form
$$\chi^{(n-|\lambda|,\lambda)} \otimes \chi^{(n-a_1,a_1)}
\otimes \chi^{(n-a_r,a_2)} \otimes \cdots \otimes \chi^{(n-a_r,a_r)}$$
where $\lambda$ is a partition and $a_1, a_2, \ldots, a_r$ are non-negative
integers.
This is joint work with Rosa Orellana.

slides, video recording.

**Title:** *Chromatic Symmetric Function of Graphs from Borcherds Lie Algebra*.

**Abstract:**
Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominators of $\mathfrak g$ and this gives a Lie theoretic proof of Stanley’s expression for chromatic symmetric function in terms of power sum symmetric functions.
Also, this gives an expression for the chromatic symmetric function of $G$ in terms of root multiplicities of $\mathfrak g$. We prove a modified Weyl denominator identity for Borcherds algebras which is an extension of the celebrated classical Weyl denominator identity and this plays an important role in the proof our results. The absolute value of the linear coefficient of the chromatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our main theorem, we prove that certain coefficients appearing in the above said expression of chromatic symmetric function is equal to the chromatic discriminant of $G$. Also, we find a connection between the Weyl denominators and the $G$-elementary symmetric functions. Using this connection, we give a Lie-theoretic proof of non-negativity of coefficients of $G$-power sum symmetric functions.

slides and Video recording.

**Title:** *Combinatorics of an exclusion process driven by an asymmetric tracer*.

**Abstract:**
We consider an exclusion process on a periodic one-dimensional lattice where all particles perform simple symmetric exclusion except for a _tracer particle_, which performs partially asymmetric exclusion with forward and backward rates p and q respectively. This process and its variants have been investigated starting with Ferrari, Goldstein and Lebowitz (1985) motivated by questions in statistical physics. We prove product formulas for stationary weights and exact formulas for the nonequilibrium partition function in terms of combinatorics of set partitions. We will also compute the current, and the density profile as seen by the test particle. Time permitting, we will illustrate the ideas involved in performing asymptotic analysis. This talk is based on the preprint arXiv:2001.02425.

Slides, Video

**Title:** *Crystal for stable Grothendieck polynomials, part 2*.

**Abstract:**
This will be a continuation of last week's talk.

We introduce a new crystal on decreasing factorizations on 321-avoiding
elements in the 0-Hecke monoid which we call ⋆-crystal. This crystal is a
K-theoretic generalization of the crystal on decreasing factorizations in
the symmetric group of the first and last author. We prove that under the
residue map the ⋆-crystal intertwines with the crystal on set-valued
tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also
define a new insertion from decreasing factorization in the 0-Hecke monoid
to pairs of (transposes of) semistandard Young tableaux and prove several
properties about this new insertion, in particular its relation to the
Hecke insertion and the uncrowding algorithm. The new insertion also
intertwines with the crystal operators.

Video recording.

**Title:** *Crystal for stable Grothendieck polynomials, part 1.*.

**Abstract:**
We introduce a new crystal on decreasing factorizations on 321-avoiding
elements in the 0-Hecke monoid which we call ⋆-crystal. This crystal is a
K-theoretic generalization of the crystal on decreasing factorizations in
the symmetric group of the first and last author. We prove that under the
residue map the ⋆-crystal intertwines with the crystal on set-valued
tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also
define a new insertion from decreasing factorization in the 0-Hecke monoid
to pairs of (transposes of) semistandard Young tableaux and prove several
properties about this new insertion, in particular its relation to the
Hecke insertion and the uncrowding algorithm. The new insertion also
intertwines with the crystal operators.

Video recording
, and
Slides (for parts 1 and 2).

**Title:** *The First and Second Stiefel-Whitney Classes of Representations of the Symmetric Group*.

**Abstract:**
A representation $(\pi,V)$ of $S_n$ can be regarded as a homomorphism to an
orthogonal group $\mathrm{O}(V)$.
One can define Stiefel-Whitney classes $w_i(\pi)$ of these representations
as members of the cohomology groups $H^i(S_n, \mathbb{Z}/2\mathbb{Z})$ , for $0\leq i\leq \mathrm{deg} \pi$.
Here $\mathbb{Z}/2\mathbb{Z}$ is trivial as an $S_n$ module.
In this talk we will compute the first and second Stiefel-Whitney classes of $\pi$ in terms of character values. This is joint work with Steven Spallone.

Video Recording

**Title:** *Restricting representations of S_n to Sylow 2-subgroups*.

**Abstract:** I will discuss a recent result on the decomposition of irreducible representations of the Symmetric group to a Sylow 2-subgroup (char 0) and discuss some interesting consequences of the same.

**Venue:** To be held via Zoom, Meeting ID: 164 038 297.