# IMSc Algebraic Combinatorics Seminar

## Upcoming talks

• Inna Entova (Ben Gurion University), Thursday 25 June 2020, 8:30pm.
Title: To be announced.
• Deniz Kus (Ruhr-University Bochum), Thursday 11 June 2020, 8:30pm.
Title: Quiver varieties and their combinatorial crystal structure.
Abstract: The aim of this talk is to describe combinatorially the crystal operators on the geometric realization of crystal bases in terms of irreducible components of quiver varieties. As a consequence of this description one can extend the geometric description to an affine crystal isomorphic to a Kirillov-Reshetikhin crystal. The underlying combinatorics is in terms of Auslander Reiten quivers.
Zoom meeting ID 875 9226 8649

## Past talks

• Jeanne Scott, (Universidad de los Andes) Thursday 4 June 2020, 8:30pm.
Title: What's the right notion of content for the Young-Fibonacci lattice?
Abstract: The Young-Fibonacci lattice $\Bbb{YF}$ is a ranked lattice invented by R. Stanley as an example of a differential poset; a nice consequence of this feature is that saturated chains (which a fixed top) are counted by a generalized hook-length formula. In 1994 S. Okada showed that $\Bbb{YF}$ is also the branching poset for a tower of complex semi-simple algebras $\frak{F}(n)$, each having a simple Coxeter-like presentation. The representation theory of these algebras strongly parallels the story of the symmetric groups $S(n)$ --- in particular each element w of rank $\mathrm{rk}(w) = n$ in the $\Bbb{YF}$ lattice corresponds to an irreducible representation $V(w)$ of $\frak{F}(n)$ whose basis is indexed by saturated chains in the $\Bbb{YF}$ lattice ending at $w$. Furthermore there is a theory of $\Bbb{YF}$-Schur functions obeying a Littlewood-Richardson rule whose structure coefficients coincide with the induction product multiplicities for representations of the Okada algebras. As in any tower of semi-simple algebras with a simple braching poset, we may define the Gelfand-Tsetlin algebra $\mathrm{GT}(n)$ as the (maximal) commutative subalgebra of $\frak{F}(n)$ generated by the centers $Z\frak{F}(1), Z\frak{F}(2), \dots, Z\frak{F}(n)$. The problem I would like to address is how to find (additive) Jucys-Murphy elements, namely an infinite sequence of elements $J(n)$ such that:
(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.
• Amritanshu Prasad (IMSc, Chennai), Thursday 21 May 2020, 8:30pm.
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.
• Nate Harman (The University of Chicago), Wednesday 13 May 2020, 8:30pm.
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.
• Mike Zabrocki (York University), Wednesday 6 May 2020, 8:30pm.
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.
• G. Arunkumar (IISER Mohali), Wednesday 29 April 2020, 11:00am.
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.
• Arvind Ayyer (IISc Bangalore), Wednesday 22 April 2020, 3:00pm.
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
• Anne Schilling (UC Davis), Wednesday 15 April 2020, 9:30 am.
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.
• Anne Schilling (UC Davis), Wednesday 8 April 2020, 9:30 am.
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).
• Jyotirmoy Ganguly (IMSc, Chennai), Wednesday 1 April 2020, 2:00 pm.
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
• Sridhar P Narayanan (IMSc, Chennai), Friday 27 March 2020, 11:00 am.
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.