IISc Alg Comb 2018-19

Algebra & Combinatorics Seminar:   2020–22

The Algebra & Combinatorics Seminar meets on Fridays from 3–4 pm, in Lecture Hall LH-1 of the IISc Mathematics Department – or online in the Autumn 2021 semester. The organizers are R. Venkatesh and Apoorva Khare.

This semester, the Algebra & Combinatorics Seminar is subsumed by the ARCSIN Seminar$\dots$ for the most part. Here are the (few) exceptions.

Madhusudan Manjunath (IIT, Bombay) May 17, 2022
Combinatorial Brill–Noether theory via lattice points and polyhedra (LH-1 – 2 pm, Tue)

Abstract. We start by considering analogies between graphs and Riemann surfaces. Taking cue from this, we formulate an analogue of Brill–Noether theory on a finite, undirected, connected graph. We then investigate related conjectures from the perspective of polyhedral geometry.

Amritanshu Prasad (IMSc, Chennai) May 6, 2022
Splitting subspaces and the Touchard–Riordan formula (LH-1 – 3 pm, Fri)

Abstract. Let $T$ be a linear endomorphism of a $2m$-dimensional vector space. An $m$-dimensional subspace $W$ is said to be $T$-splitting if $W$ intersects $TW$ trivially.

When the underlying field is finite of order $q$ and $T$ is diagonal with distinct eigenvalues, the number of splitting subspaces is essentially the the generating function of chord diagrams weighted by their number of crossings with variable $q$. This generating function was studied by Touchard in the context of the stamp folding problem. Touchard obtained a compact form for this generating function, which was explained more clearly by Riordan.

We provide a formula for the number of splitting subspaces for a general operator $T$ in terms of the number of $T$-invariant subspaces of various dimensions. Specializing to diagonal matrices with distinct eigenvalues gives an unexpected and new proof of the Touchard–Riordan formula.

This is based on joint work with Samrith Ram.

Jacob P. Matherne (Universität Bonn, Germany) Mar 25, 2022
Singular Hodge theory for combinatorial geometries (LH-1 – 3 pm, Fri)

Abstract. Here are two problems about hyperplane arrangements.

Problem 1: If you take a collection of planes in $\mathbb{R}^3$, then the number of lines you get by intersecting the planes is at least the number of planes. This is an example of a more general statement, called the "Top-Heavy Conjecture", that Dowling and Wilson conjectured in 1974.

Problem 2: Given a hyperplane arrangement, I will explain how to uniquely associate a certain polynomial (called its Kazhdan–Lusztig polynomial) to it. These polynomials should have nonnegative coefficients.

Both of these problems were formulated for all matroids, and in the case of hyperplane arrangements they are controlled by the Hodge theory of a certain singular projective variety, called the Schubert variety of the arrangement. For arbitrary matroids, no such variety exists; nonetheless, I will discuss a solution to both problems for all matroids, which proceeds by finding combinatorial stand-ins for the cohomology and intersection cohomology of these Schubert varieties and by studying their Hodge theory. This is joint work with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.

Maitreyee Kulkarni (Universität Bonn, Germany) Mar 25, 2022
A combinatorial model for totally nonnegative partial flag varieties (LH-1 – 2:30 pm, Fri)

Abstract. Postnikov defined the totally nonnegative Grassmannian as the part of the Grassmannian where all Pl├╝cker coordinates are nonnegative. This space can be described by the combinatorics of planar bipartite graphs in a disk, by affine Bruhat order, and by a host of other combinatorial objects. In this talk, I will recall some of this story, then talk about in progress joint work, together with Chris Fraser and Jacob Matherne, which hopes to extend this combinatorial description to more general partial flag varieties.

This semester, the Algebra & Combinatorics Seminar is subsumed by the ARCSIN Seminar.

This semester, the Algebra & Combinatorics Seminar is subsumed by the ARCSIN Seminar.

This semester, the Algebra & Combinatorics Seminar is online on Microsoft Teams.

Pooja Singla (IIT Kanpur) Jan 8, 2021
Projective representations of discrete nilpotent groups (3 pm, Fri)

Abstract. The study of projective representations of a group has a long history starting from the work of Schur. Two essential ingredients to study the group's projective representations are describing its Schur multiplier and representation group. In this talk, we describe these for the discrete Heisenberg groups. We also include a few general results regarding projective representations of finitely generated discrete nilpotent groups. This talk is based on the joint work with Sumana Hatui and E.K. Narayanan.

Sudhanshu Shekhar (IIT Kanpur) Dec 18, 2020
Multiplicities in Selmer groups and root numbers for Artin twists (3 pm, Fri)

Abstract. Given a Galois extension of number fields $K/F$ and two elliptic curves $A$ and $B$ with equivalent residual Galois representation mod $p$, for an odd prime $p$, we will discuss the relation between the $p$-parity conjecture of $A$ twisted by $\sigma$ and that of $B$ twisted by $\sigma$ for an irreducible, self dual, Artin representation $\sigma$ of the Galois group of $K/F$.

This is a joint work with Somnath Jha and Tathagata Mandal.

Subsumed by the Discussion Meeting on Representation Theory (IISc Mathematics) Dec 11, 2020

Ravindranathan Thangadurai (Harish-Chandra Research Institute, Allahabad) Dec 7, 2020
On the simultaneous approximation of algebraic numbers (moved to unusual day: 3 pm, Mon)

Abstract. In 2004, Corvaja and Zannier proved an extension of Roth's theorem on rational approximation of algebraic numbers. With a collaboration of Dr. Veekesh Kumar, we proved a simultaneous version of Corvaja and Zannier's result. These results are applications of a strong form of the Subspace Theorem. In this talk, we shall discuss the motivation of Corvaja and Zannier's result and our generalization.

Arvind Ayyer (IISc Mathematics) Dec 4, 2020
Toppleable permutations, excedances and acyclic orientations (3 pm, Fri)

Abstract. Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (arXiv:1612.06816) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. For the most part of the talk, we will explain the main ideas in showing that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor \}$. Time permitting, we will give some ideas showing that this is also the number of acyclic orientations with unique sink of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.
This is joint work with P. Tetali (GATech) and D. Hathcock (CMU), and is available at arXiv:2010.11236

Sanoli Gun (IMSc, Chennai) Nov 20, 2020
Distinguishing newforms by their Hecke eigenvalues (3 pm, Fri)

Abstract. In this talk, we will discuss some of the existing techniques for distinguishing newforms. We will also report on a recent joint work with Kumar Murty and Biplab Paul.

Shifra Reif (Bar-Ilan University, Israel) Nov 13, 2020
Denominator identities for the periplectic Lie superalgebra p(n) (3 pm, Fri)

Abstract. We present the denominator identities for the periplectic Lie superalgebras and discuss their relations to representations of $\mathbf{p}(n)$ and $\mathbf{gl}(n)$. Joint work with Crystal Hoyt and Mee Seong Im.

Michael J. Schlosser (University of Vienna) Nov 6, 2020
A weight-dependent inversion statistic and Catalan numbers (3 pm, Fri)

Abstract. We introduce a weight-dependent extension of the inversion statistic, a classical Mahonian statistic on permutations. This immediately gives us a new weight-dependent extension of $n!$. By restricting to $312$-avoiding permutations our extension happens to coincide with the weighted Catalan numbers that were considered by Flajolet in his combinatorial study of continued fractions. We show that for a specific choice of weights the weighted Catalan numbers factorize into a closed form, hereby yielding a new $q$-analogue of the Catalan numbers, different from those considered by MacMahon, by Carlitz, or by Andrews. We further refine the weighted Catalan numbers by introducing an additional statistic, namely a weight-dependent extension of Haglund's bounce statistic, and obtain a new family of bi-weighted Catalan numbers that generalize Garsia and Haiman's $q,t$-Catalan numbers and appear to satisfy remarkable properties. This is joint work with Shishuo Fu.

Deniz Kus (Ruhr-University Bochum, Germany) Oct 23, 2020
Polytopes, truncations of representations and their characters (3 pm, Fri)

Abstract. Generators and relations of graded limits of certain finite-dimensional irreducible representations of quantum affine algebras have been determined in recent years. For example, the representations in the Hernandez-Leclerc category corresponding to cluster variables appear to be certain truncations of representations for current algebras and tensor products are related to the notion of fusion products. In this talk, we will discuss some known results on this topic and study the characters of arbitrary truncated representations.

R. Venkatesh (IISc Mathematics) Oct 16, 2020
Fusion product decomposition of $\mathfrak{g}$-stable affine Demazure modules (3 pm, Fri)

Abstract. The affine Demazure modules are the Demazure modules that occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We call them $\mathfrak{g}$-stable if they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a $\mathfrak{g}$-stable affine Demazure module is isomorphic to the fusion (tensor) product of smaller $\mathfrak{g}$-stable affine Demazure modules, thus completing the main theorems of Chari et al. (J. Algebra, 2016) and Kus et al. (Represent. Theory, 2016). We obtain a new combinatorial proof for the key fact that was used in Chari et al. (op cit.), to prove the decomposition of $\mathfrak{g}$-stable affine Demazure modules. Our proof for this key fact is uniform, avoids the case-by-case analysis, and works for all finite-dimensional simple Lie algebras.

Manish Mishra (IISER Pune) Oct 9, 2020
Regular Bernstein blocks (3 pm, Fri)

Abstract. Let $G$ be a connected reductive group defined over a non-archimedean local field $F$. The category $R(G)$ of smooth representations of G(F) has a decomposition into a product of indecomposable subcategories called Bernstein blocks and to each block is associated a non-negative real number called Moy–Prasad depth. We will begin with recalling all this basic theory. Then we will focus the discussion on 'regular' blocks. These are 'most' Bernstein blocks when the residue characteristic of $F$ is suitably large. We will then talk about an approach of studying blocks in $R(G)$ by studying a suitably related depth-zero block of certain other groups. In that context, I will explain some results from a joint work with Jeffrey Adler. One of them being that the Bernstein center (i.e., the center of a Bernstein block) of a regular block is isomorphic to the Bernstein center of a depth-zero regular block of some explicitly describable another group. I will give some applications of such results.

Ignazio Longhi (IISc Mathematics) Sep 25, 2020
Densities on Dedekind domains, completions and Haar measure (3 pm, Fri)

Abstract. A traditional way of assessing the size of a subset X of the integers is to use some version of density. An alternative approach, independently rediscovered by many authors, is to look at the closure of X in the profinite completion of the integers. This for example gives a quick, intuitive solution to questions like: what is the probability that an integer is square-free? Moreover, in many cases, one finds that the density of X can be recovered as the Haar measure of the closure of X. I will discuss some things that one can learn from this approach in the more general setting of rings of integers in global fields. This is joint work with Luca Demangos.

Apoorva Khare (IISc Mathematics) Sep 18, 2020
Totally positive matrices, Pólya frequency sequences, and Schur polynomials
(Joint with the APRG Seminar)
(3 pm, Fri)

Abstract. I will discuss totally positive/non-negative matrices and kernels, including Polya frequency (PF) functions and sequences. This includes examples, history, and basic results on total positivity, variation diminution, sign non-reversal, and generating functions of PF sequences (with some proofs). I will end with applications of total positivity to old and new phenomena involving Schur polynomials.

Charanya Ravi (Universität Regensburg, Germany) Sep 8, 2020
Algebraic K-theory of varieties with group actions (unusual day: 3 pm, Tue)

Abstract. Cohomology theories are one of the most important algebraic invariants of topological spaces and this has inspired the definition of several different cohomology theories in algebraic geometry. In this talk, we focus on algebraic K-theory, which is one such classical cohomological invariant of algebraic varieties. After motivating and introducing this notion, we discuss several fundamental properties of algebraic K-theory of varieties with algebraic group actions. Well-known examples of varieties with group actions include toric varieties and flag varieties.

Sandeep Varma (TIFR, Mumbai) Sep 4, 2020
Some Bernstein projectors for $SL_2$ (3 pm, Fri)

Abstract. Let $G$ be the group $SL_2$ over a finite extension $F$ of $\mathbb{Q}_p$, $p$ odd. I will discuss certain distributions on $G(F)$, belonging to what is called its Bernstein center (I will explain what this and many other terms in this abstract mean), supported in a certain explicit subset of $G(F)$ arising from the work of A. Moy and G. Prasad. The assertion is that these distributions form a subring of the Bernstein center, and that convolution with these distributions has very agreeable properties with respect to orbital integrals. These are 'depth $r$ versions' of results proved for general reductive groups by J.-F. Dat, R. Bezrukavnikov, A. Braverman and D. Kazhdan.

Bharathwaj Palvannan (National Center for Theoretical Sciences, Taiwan) Aug 28, 2020
Codimension two cycles in Iwasawa theory (3 pm, Fri)

Abstract. In classical Iwasawa theory, one studies a relationship called the Iwasawa main conjecture, between an analytic object (the p-adic L-function) and an algebraic object (the Selmer group). This relationship involves codimension one cycles of an Iwasawa algebra. The topic of higher codimension Iwasawa theory seeks to generalize this relationship. We will describe a result in this topic using codimension two cycles, involving an elliptic curve with supersingular reduction. This is joint work with Antonio Lei.