# Algebra & Combinatorics Seminar

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

### Spring 2019

 Hassain M. (IISc Mathematics) Jan 4, 2019 Representation growth of Special Compact Linear Groups of Order Two

Abstract. Let $\mathfrak{O}$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Then the abscissa of convergence of representation zeta function of Special Linear group $\mathrm{SL}_2(\mathfrak{O})$ is $1.$ The case $p\neq 2$ is already known in the literature. For $p=2$ we need more tools to prove the result. In this talk I will discuss the difference between those cases and give an outline of the proof for $p=2.$

Let $\mathfrak{p}$ be the maximal ideal of $\mathfrak{O}$ and $|\mathfrak{O}/\mathfrak{p}|=q.$ It is already shown in literature that for $r \geq 1,$ the group algebras $\mathbb C[\mathrm{GL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb C[\mathrm{GL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. Also for $2\nmid q,$ the group algebras $\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. In this talk I will also show that if $2\mid q$ and $\mathrm{char}(\mathfrak{O})=0$ then, the group algebras, $\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{2\ell})]$ and $\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{2\ell}))]$ are not isomorphic for $\ell > \mathrm{e}$, where $\mathrm{e}$ is the ramification index of $\mathfrak{O}.$

 Apoorva Khare (IISc Mathematics) Jan 11, 2019 From word games to a Polymath project

Abstract. Consider the following three properties of a general group $G$:

1. Algebra: $G$ is abelian and torsion-free.
2. Analysis: $G$ is a metric space that admits a "norm", namely, a translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$ for all $g \in G$ and integers $n$.
3. Geometry: $G$ admits a length function with "saturated" subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g \in G$.

While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a "norm".

We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and if time permits, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint – as D.H.J. PolyMath – with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)

 Sanjay Ramassamy (École Normale Supériure, Paris, France) Jan 18, 2019 Extensions of partial cyclic orders, boustrophedons and polytopes (11 am – note: unusual time)

Abstract. While the enumeration of linear extensions of a given poset is a well-studied question, its cyclic counterpart (enumerating extensions to total cyclic orders of a given partial cyclic order) has been subject to very little investigation. In this talk I will introduce some classes of partial cyclic orders for which this enumeration problem is tractable. Some cases require the use of a multidimensional version of the classical boustrophedon construction (a.k.a. Seidel–Entringer–Arnold triangle). The integers arising from these enumerative questions also appear as the normalized volumes of certain polytopes.

This is partly joint work with Arvind Ayyer (Indian Institute of Science) and Matthieu Josuat-Vergès (Laboratoire d'Informatique Gaspard Monge / CNRS).

 Ramesh Gangolli (University of Washington, Seattle, USA) Jan 18, 2019 Some unpublished work of Harish-Chandra (speaking in / subsumed by the Eigenfunctions Seminar)

 Niranjan Balachandran (IIT, Bombay) Jan 30, 2019 (note: two talks, unusual day)

 Talk 1: The Polynomial Method in Combinatorics – Two illustrative theorems 2:30 pm

Abstract. Since Dvir proved the Finite Kakeya Conjecture in 2008, the Polynomial Method has become a new and powerful tool and a new paradigm for approaching extremal questions in combinatorics (and other areas too). We shall take a look at the main philosophical principle that underlies this method via two recent (2017) theorems. One is the upper bound for Cap-sets by Ellenberg–Gijswijt, and the other, a function field analogue of a theorem of Sarkozy, due to Ben Green.

 Talk 2: Bisections and bicolorings of hypergraphs 4 pm

Abstract. Given a hypergraph $\mathcal{H}$ with vertex set $[n]:=\{1,\ldots,n\}$, a bisecting family is a family $\mathcal{A}\subseteq\mathcal{P}([n])$ such that for every $B\in\mathcal{H}$, there exists $A\in\mathcal{A}$ with the property $|A\cap B|-|A\cap\overline{B}|\in\{-1,0,1\}$. Similarly, for a family of bicolorings $\mathcal{B}\subseteq \{-1,1\}^{[n]}$ of $[n]$ a family $\mathcal{A}\subseteq\mathcal{P}([n])$ is called a System of Unbiased Representatives for $\mathcal{B}$ if for every $b\in\mathcal{B}$ there exists $A\in\mathcal{A}$ such that $\sum_{x\in A} b(x) =0$.

The problem of optimal families of bisections and bicolorings for hypergraphs originates from what is referred to as the problem of Balancing Sets of vectors, and has been the source for a few interesting extremal problems in combinatorial set theory for about 3 decades now. We shall consider certain natural extremal functions that arise from the study of bisections and bicolorings and bounds for these extremal functions. Many of the proofs involve the use of polynomial methods (not the Polynomial Method, though!).
(Joint work with Rogers Mathew, Tapas Mishra, and Sudebkumar Prasant Pal)

 Ved Datar (IISc Mathematics) Feb 1, 2019 TBA (speaking in the Eigenfunctions Seminar; no Alg–Comb today)

 Mohan R. (Indian Statistical Institute, Bangalore) Feb 8, 2019 TBA

Abstract. TBA

 Subhajit Ghosh (IISc Mathematics) Feb 15, 2019 TBA

Abstract. TBA

 Samarpita Ray (IISc Mathematics) Feb 22, 2019 TBA

Abstract. TBA

 Parthanil Roy (Indian Statistical Institute, Bangalore) Mar 1, 2019 TBA (speaking in the Eigenfunctions Seminar; no Alg–Comb today)

 Xavier Viennot (CNRS and LaBRI, France) Mar 8, 2019 TBA

Abstract. TBA

 TBA Mar 15, 2019 TBA

Abstract. TBA

 Sanchayan Sen (IISc Mathematics) Mar 22, 2019 TBA (speaking in the Eigenfunctions Seminar; no Alg–Comb today)

 Projesh Nath Choudhury (IISc Mathematics) Mar 29, 2019 TBA

Abstract. TBA

 Sudip Bera (IISc Mathematics) Apr 5, 2019 TBA

Abstract. TBA

 Apoorva Khare and Gautam Bharali (IISc Mathematics) Apr 12, 2019 TBA (speaking in the Eigenfunctions Seminar; no Alg–Comb today)

### Autumn 2018

 Apoorva Khare (IISc Mathematics) Sep 7, 2018 The Diamond Lemma in ring theory

Abstract. I will give a gentle introduction to the Diamond Lemma. This is a useful technique to prove that certain "PBW-type" bases exist of algebras given by generators and relations. In particular, we will see the PBW theorem for usual Lie algebras.

 R. Venkatesh (IISc Mathematics) Sep 14, 2018 Graphs vs. Lie algebras

Abstract. Let $G$ be a finite simple graph. The Lie algebra $\mathfrak{g}$ of $G$ is defined as follows: $\mathfrak{g}$ is generated by the vertices of $G$ modulo the relations $[u, v]=0$ if there is no edge between the vertices $u$ and $v$. Many properties of $\mathfrak{g}$ can be obtained from the properties of $G$ and vice versa. The Lie algebra $\mathfrak{g}$ of $G$ is naturally graded and the graded dimensions of the Lie algebra $\mathfrak{g}$ of $G$ have some deep connections with the vertex colorings of $G$. In this talk, I will explain how to get the generalized chromatic polynomials of $G$ in terms of graded dimensions of the Lie algebra of $G$. We will use this connection to give a Lie theoretic proof of of Stanley's reciprocity theorem of chromatic polynomials.

 Bharatram Rangarajan (Hebrew University, Jerusalem, Israel) Sep 26, 2018 Expanders: From Graphs to Complexes (note: unusual day)

Abstract. The aim of this talk is to give a high-level overview of the theory of expander graphs and introduce motivations and possible approaches to generalizing it to higher dimensions. I shall begin with three perspectives on expansion in graphs- discrepancy, isoperimetry and mixing time, and show a qualitative equivalence of these notions in defining expansion for graphs. Next I shall briefly discuss upper and lower bounds on expansion, and sketch the Lubotzky-Phillips-Sarnak construction of Ramanujan graphs. Finally, I hope to motivate high-dimensional expanders using two interesting topics- the overlapping problem, and the threshold problem.

 Amritanshu Prasad (IMSc, Chennai) Sep 28, 2018 The words that describe symmetric polynomials (speaking in / subsumed by the Eigenfunctions Seminar)

 G.V.K. Teja (IISc Mathematics) Oct 5, 2018 Patterns in recurring decimals

Abstract. We will study recurrence patterns in decimal expansions of rational numbers (in any integer base for this talk). After making some initial observations, we will compute the length of the repeating part of any fraction. We conclude by explaining this result over a Euclidean domain.

 Pooja Singla (IISc Mathematics) Oct 12, 2018 Gelfand's criterion and multiplicity one results

Abstract. We will describe Gelfand's criterion for the commutativity of associative algebras and discuss some of its applications towards the multiplicity one theorems for the representations of finite groups.

 Arvind Ayyer (IISc Mathematics) Oct 19, 2018 Factorization theorems for classical group characters (2:30 pm – note: unusual time)

Abstract. Characters of classical groups appear in the enumeration of many interesting combinatorial problems. We show that, for a wide class of partitions, and for an even number of variables of which half are reciprocals of the other half, Schur functions (i.e., characters of the general linear group) factorize into a product of two characters of other classical groups. Time permitting, we will present similar results involving sums of two Schur functions. All the proofs will involve elementary applications of ideas from linear algebra.

This is joint work with Roger Behrend.

 Dhruv Ranganathan (IAS (Princeton), MIT (Boston), USA; CMI (Chennai); Cambridge, UK) Oct 31, 2018 Tropical geometry of moduli spaces (speaking in / subsumed by the Eigenfunctions Seminar)

(Note: The following Seminar on Nov 2 may be of related interest to those attending this talk.)

 Dhruv Ranganathan (IAS (Princeton), MIT (Boston), USA; CMI (Chennai); Cambridge, UK) Nov 2, 2018 Curve counting and tropical geometry

Abstract. The counts of algebraic curves in projective space (and other toric varieties) has been intensely studied for over a century. The subject saw a major advance in the 1990s, due to groundbreaking work of Kontsevich in the 1990s. Shortly after, considerations from high energy physics led to an entirely combinatorial approach to these curve counts, via piecewise linear embeddings of graphs, pioneered by Mikahlkin. I will give an introduction to the surrounding ideas, outlining new results and new proofs that the theory enables. Time permitting I will discuss generalizations, difficulties, and future directions for the subject.

(Organizer's note: It may help to attend the preceding Eigenfunctions Seminar on Oct 31, before this talk.)

 K.N. Raghavan (IMSc, Chennai) Nov 9, 2018 A refinement of the Littlewood–Richardson rule (speaking in / subsumed by the Eigenfunctions Seminar)

 Abhishek Banerjee (IISc Mathematics) Nov 16, 2018 Rings with several objects

Abstract. An ordinary ring may be seen as a preadditive category with just one object. This leads to the powerful analogy, first formulated explicitly by Mitchell in 1975, that a small preadditive category should be seen as a "ring with several objects". We will trace the history and development of the category of modules over a preadditive category.

 Katharina Jochemko (KTH, Stockholm, Sweden) Dec 12, 2018 Combinatorial positive valuations (note: unusual day)

Abstract. Valuations are a classical topic in convex geometry. The volume plays an important role in many structural results, such as Hadwiger's famous characterization of continuous, rigid-motion invariant valuations on convex bodies. Valuations whose domain is restricted to lattice polytopes are less well-studied. The Betke–Kneser Theorem establishes a fascinating discrete analog of Hadwiger's Theorem for lattice-invariant valuations on lattice polytopes in which the number of lattice points – the discrete volume – plays a fundamental role. In this talk, we explore striking parallels, analogies and also differences between the world of valuations on convex bodies and those on lattice polytopes with a focus on positivity questions and links to Ehrhart theory.

 Thorsten Heidersdorf (Max-Planck-Institut für Mathematik, Bonn, Germany) Dec 20, 2018 Tensor product decomposition for the general linear supergroup $GL(m|n)$ (11 am – note: unusual time and day)

Abstract. Let $\mathfrak{gl}(n)$ denote the Lie algebra of the general linear group $GL(n)$. Given two finite dimensional irreducible representations $L(\lambda), L(\mu)$ of $\mathfrak{gl}(n)$, its tensor product decomposition $L(\lambda) \otimes L(\mu)$ is given by the Littlewood-Richardson rule. The situation becomes much more complicated when one replaces $\mathfrak{gl}(n)$ by the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. The analogous decomposition $L(\lambda) \otimes L(\mu)$ is largely unknown. Indeed many aspects of the representation theory of $\mathfrak{gl}(m|n)$ are more akin to the study of Lie algebras and their representations in prime characteristic or to the BGG category $\mathcal{O}$. I will give a survey talk about this problem and explain why some approaches don't work and what can be done about it. This will give me the chance to speak about a) the character formula for an irreducible representation $L(\lambda)$, b) Deligne's interpolating category $Rep(GL_t)$ for $t \in \mathbb{C}$ and c) the process of semisimplification of a tensor category.