IMSc Algebraic Combinatorics Seminar

Organized by Amritanshu Prasad and S. Viswanath.

Upcoming talks (click to expand)

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Zoom Meeting ID: 893 5344 7324; Password: Littlewood.
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Zoom Meeting ID: 841 8104 1409; Password: Littlewood.
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Zoom Meeting ID: 883 8003 7638; Password: Littlewood.
Title. Filtering Grassmannian Cohomology via k-Schur Functions.
Abstract. This talk concerns the cohomology rings of complex Grassmannians. In 2003, Reiner and Tudose conjectured the form of the Hilbert series for certain subalgebras of these cohomology rings. We build on their work in two ways. First, we conjecture two natural bases for these subalgebras that would imply their conjecture using notions from the theory of k-Schur functions. Second, we formulate an analogous conjecture for Lagrangian Grassmannians.
Zoom Meeting ID: 868 7916 4965; Password: Littlewood.

Past talks (click to expand)

Title.Saturation for refined Littlewood-Richardson coefficients-2
Abstract.The Littlewood-Richardson (LR) coefficients are the multiplicities of irreducible representations occurring in the tensor product of two irreducible polynomial representations of GL_n. To each permutation 'w' in S_n, we associate a 'w-refinement' of the LR coefficients. These correspond to multiplicities in the so-called Kostant-Kumar submodules of the tensor product, or equivalently of multiplicities in "excellent filtrations" of Demazure modules. We prove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231-avoiding, by adapting the proof via hives of the classical saturation conjecture due to Knutson-Tao. This is a report of work-in-progress with Mrigendra Singh Kushwaha and KN Raghavan. This talk will span two seminar days (Oct 1 and 8). In the first part, we describe the setting of the problem and the result. In the second part, we recall the key steps in the Knutson-Tao proof of the saturation conjecture via hives and indicate how it can be adapted to our case.
Video recording, slides.
Title.Saturation for refined Littlewood-Richardson coefficients-1
Abstract.The Littlewood-Richardson (LR) coefficients are the multiplicities of irreducible representations occurring in the tensor product of two irreducible polynomial representations of GL_n. To each permutation 'w' in S_n, we associate a 'w-refinement' of the LR coefficients. These correspond to multiplicities in the so-called Kostant-Kumar submodules of the tensor product, or equivalently of multiplicities in "excellent filtrations" of Demazure modules. We prove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231-avoiding, by adapting the proof via hives of the classical saturation conjecture due to Knutson-Tao. This is a report of work-in-progress with Mrigendra Singh Kushwaha and KN Raghavan. This talk will span two seminar days (Oct 1 and 8). In the first part, we describe the setting of the problem and the result. In the second part, we recall the key steps in the Knutson-Tao proof of the saturation conjecture via hives and indicate how it can be adapted to our case.
Video recording, notes from the talk.
Title.Generating function for the powers in $\text{GL}(n,q)$.
Abstract. Let $M\geq 2$ be any integer. Consider the set $\text{GL}(n,q)^M=\{x^M|x\in \text{GL}(n,q)\}$, which is the set of all $M^{th}$ powers in the group $\text{GL}(n,q)$. In this talk, we will obtain generating functions for (a) the proportion of regular and regular semsimple elements in $\text{GL}(n,q)^M$, assuming $(M,q)=1$,  (b) the proportion of semisimple and all elements which are $M^{th}$ powers when $(M,q)=1$, and $M$ is a power of a prime. Time permitting we will also discuss the other extreme, where we assume $M$ is a prime and $q$ is a power of $M$. This is a joint work with Dr. Anupam Singh.
Video recording , Slides.
Title. On Schur multipliers and projective representations of Heisenberg groups.
Abstract. The study of projective representations has a long history starting with the pioneering work of Schur for finite groups which involves understanding homomorphisms from a group into the projective linear groups. In this study, an important role is played by a group called the Schur multiplier. In this talk, we shall describe the Schur multiplier of the finite as well as infinite discrete Heisenberg groups and their t-variants. We shall discuss the representation groups of these Heisenberg groups and through these give a construction of their finite-dimensional complex projective irreducible representations. This is joint work with Pooja Singla.
Video Recording.
Title. Generalizations of the Selberg integral and combinatorial connections.
Abstract. We'll briefly recall the history of the Selberg Integral and several variants. We'll also go through the proof of some of them like Aomoto's integral before focusing on known and possibly new integrals involving Schur polynomials and Jack polynomials. We shall note the implications that these integrals seem to count (after a suitable normalization) the number of standard young tableaux of skew shapes, before conjecturing the existence of several Naruse-type hook length formulas. Finally we will explain how these integrals arise in number theoretic problems.
Meeting ID: 869 5914 1402.
Title. Pieri rules for polynomials.
Abstract. Schur functions are an amazing basis of symmetric functions originally defined as characters of irreducible modules for of $GL_n$. The Pieri rule for the product of a Schur function and a single row Schur function is a multiplicity-free branching rule with a beautiful combinatorial interpretation in terms of adding boxes to a Young diagram. Key polynomials are an interesting basis of the polynomial ring originally defined as characters of submodules for irreducible $GL_n$ modules under the action of upper triangular matrices. In joint work with Danjoseph Quijada, we give a Pieri rule for the product of a key polynomial and a single row key polynomial. While this formula has signs, it is multiplicity-free and has an interpretation in terms of adding balls to a key diagram, perhaps after dropping some balls down. Time permitting, I’ll give applications to Schubert polynomials where the signs cancel to give a positive Pieri formula.
Video recording, Slides.
Title.Total variation cutoff for random walks on some finite groups
Abstract. This presentation will be on the mixing times for three random walk models. Specifically these are the random walks on the alternating group, the grou\ p of signed permutations and the complete monomial group. The details for the models are given below:
The random walk on the alternating group: We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cyclesof the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrixof this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $\left(n-\frac{3}{2}\right)\log n$ for this shuffle.
The random walk on the group of signed permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n\log n$. Furthermore, we show that a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i,n)$ and $(-i,n)$ for $1\leq i< n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.
The random walk on the complete monomial group: Let $G_1\subseteq\cdots\subseteq G_n \subseteq\cdots $ be a sequence of finite groups with $|G_1|>2$. We study the properties of a random walk on the complete monomial group $G_n\wr S_n$ generated by the elements of the form $(\text{e},\dots,\text{e},g;\text{id})$ and $(\text{e},\dots,\text{e},g^{-1},\text{e},\dots,\text{e},g;(i,n))$ for $g\in G_n,\;1\leq i< n$. We call this the warp-transpose top with random shuffle on $G_n\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixingtime for this shuffle is of order $n\log n+\frac{1}{2}n\log (|G_n|-1)$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n\log n$ if $|G_n|=o(n^{\delta})$ for all $\delta>0$.
Video recording, slides.
Title. Asymptotics of powers in finite reductive groups.
Abstract.Let $G$ be a connected reductive group defined over a finite field $\mathbf F_q$. Fix an integer $M >1$, and consider the power map $x$ going to $x^M$ on G. We denote the image of $G(\mathbf F_q)$ under this map by $G(\mathbf F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbf F_q)$ it contains. We prove that as q tends to infinity, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $GL(n, q)$ and $U(n, q)$.
Video recording, Slides.
Title.Generating Functions for Involutions and Character Degree Sums in Finite Groups of Lie Type.
Abstract. Given a finite group $G$, it is a result of Frobenius and Schur that all complex irreducible representations of $G$ may be defined over the reals if and only if the character degree sum of $G$ is equal to the number of involutions of $G$.  We use this result and generatingfunctionology to study the real representations of finite groups of Lie type, and to obtain some new combinatorial identities.  We will begin with examples of Weyl groups, then discuss joint work with Jason Fulman on finite general linear and unitary groups, and then give more recent results for finite symplectic and orthogonal groups.
Video recording.
Title. Quasi $p$-Steinberg Characters of double covers of Symmetric and Alternating groups.
Abstract. An irreducible character of a finite group $G$ is called Quasi $p$-Steinberg for a prime $p$ if it takes non-zero value on every $p$-regular element of $G$. In this talk, we shall recall some combinatorial aspects of the representation theory of double covers of Symmetric and Alternating groups. Then we discuss the existence of Quasi $p$-Steinberg Characters of those groups. This talk is based on ongoing work with Pooja Singla. Suggested readings:
  1. A. O. Morris, The spin representation of the symmetric group, Proc. London Math. Soc. (3), 12 (1962).
  2. J. R. Stembridge, Shifted tableaux and the projective representations of the symmetric groups. Adv. in Math. 74 (1989).
Video recording.
Title: Quasi p-Steinberg Character for Symmetric, Alternating Groups and their Double Covers
Abstract: Given a finite group of Lie type in characteristic p, Steinberg constructed a distinguished ordinary representation of dimension equals to the cardinality of a Sylow-p-subgroup and whose character, which is now known as p-Steinberg character, vanishes except at p-regular elements. The following question was raised by W. Feit and was answered by M. R. Darafsheh for the alternating group or the projective special linear group:

"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.
References:
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.
Title: Plethystic inversion and representations of the symmetric group, Part 2
Abstract: In this talk we will survey the many instances of plethystic inversion that occur in the representation theory of the symmetric group $S_n$. Perhaps the first such formula is due to Cadogan. The Lie representation of $S_n,$ arising from the free Lie algebra, appears here. We will discuss the equivalence of Cadogan's formula to Thrall's decomposition of the regular representation, and to many other phenomena in a wide variety of contexts. New decompositions of the regular representation will be presented. Some of this material appears in the following papers: arXiv:1803.09368 arXiv:2003.10700. arXiv:2006.01896
Slides, Video

Title: Plethystic inversion and representations of the symmetric group, Part 1
Abstract: In this talk we will survey the many instances of plethystic inversion that occur in the representation theory of the symmetric group $S_n$. Perhaps the first such formula is due to Cadogan. The Lie representation of $S_n,$ arising from the free Lie algebra, appears here. We will discuss the equivalence of Cadogan's formula to Thrall's decomposition of the regular representation, and to many other phenomena in a wide variety of contexts. New decompositions of the regular representation will be presented. Some of this material appears in the following papers: arXiv:1803.09368 arXiv:2003.10700. arXiv:2006.01896
Zoom meeting ID: 873 0780 6787.
Slides, Video.
Title: Deligne categories and stable Kronecker coefficients.
Abstract: In this talk, I will present an application of the theory of Deligne categories to the study of Kronecker coefficients.

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.
Title: Random $t$-cores and hook lengths in random partitions.
Abstract: Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after appropriate rescaling. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally, we extend this result to all modulo classes of $t$ using abacus representations for cores and quotients. This talk is based on the arxiv preprint arXiv:1911.03135.
Slides, Video recording
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.
Video recording.
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.

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.