Algebra & Combinatorics Seminar:   2018–19

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 $\mathfrak{O}.$

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.)

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).

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.)

Abstract. We say a unital ring $R$ has the Invariant Basis Number (IBN) property in case, for each pair of positive integers $i,j$ if the left $R$-modules $R^i$ and $R^j$ are isomorphic, then $i=j$. The first examples of non IBN rings were studied by William Leavitt in the 1950s and he defined (what are now known as) Leavitt algebras which are 'universal' with non IBN property. In 2004 the algebraic structures arising from directed (multi)graphs known as Leavitt path algebras (LPA for short) were initially developed as algebraic analogues of graph $C^*$ algebras. LPAs generalize a particular class of Leavitt algebras.
During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in $C^*$-algebras, and symbolic dynamicists as well. The goal of this talk is to introduce the notion of Leavitt path algebras and to present some results on LPAs arising from weighted Cayley graphs of finite cyclic groups.

Abstract. Valiant (1979) showed that unless P = N P, there is no polynomial-time algorithm to compute the number of perfect matchings of a given graph – even if the input graph is bipartite. Earlier, the physicist Kasteleyn (1963) introduced the notion of a special type of orientation of a graph, and we refer to graphs that admit such an orientation as Pfaffian graphs. Kasteleyn showed that the number of perfect matchings is easy to compute if the input graph is Pfaffian, and he also proved that every planar graph is Pfaffian. The complete bipartite graph $K_{3,3}$ is the smallest graph that is not Pfaffian. In general, the problem of deciding whether a given graph is Pfaffian is not known to be in N P.

Special types of minors, known as conformal minors, play a key role in the theory of Pfaffian orientations. In particular, a graph is Pfaffian if and only if each of its conformal minors is Pfaffian. It was shown by Little (1975) that a bipartite graph $G$ is Pfaffian if and only if $G$ does not contain $K_{3,3}$ as a conformal minor (or, in other words, if and only if $G$ is $K_{3,3}$-free); this places the problem of deciding whether a bipartite graph is Pfaffian in co – N P. Several years later, a structural characterization of $K_{3,3}$-free bipartite graphs was obtained by Robertson, Seymour and Thomas (1999), and independently by McCuaig (2004), and this led to a polynomial-time algorithm for deciding whether a given bipartite graph is Pfaffian.

Norine and Thomas (2008) showed that, unlike the bipartite case, it is not possible to characterize all Pfaffian graphs by excluding a finite number of graphs as conformal minors. In light of everything that has been done so far, it would be interesting to even identify rich classes of Pfaffian graphs (that are nonplanar and nonbipartite).

Inspired by a theorem of Lovasz (1983), we took on the task of characterizing graphs that do not contain $K_4$ as a conformal minor – that is, $K_4$-free graphs. In a joint work with U. S. R. Murty (2016), we provided a structural characterization of planar $K_4$-free graphs. The problem of characterizing nonplanar $K_4$-free graphs is much harder, and we have evidence to believe that it is related to the problem of recognizing Pfaffian graphs. In particular, we conjecture that every graph that is $K_4$-free and $K_{3,3}$-free is also Pfaffian. The talk will be mostly self-contained. I will assume only basic knowledge of graph theory. For more details, see: https://onlinelibrary.wiley.com/doi/full/10.1002/jgt.21882.

Abstract. We investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of 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 matrix for this shuffle. We mainly use the representation theory of alternating group. We show that the mixing time is of order $\left(n-\frac{3}{2}\right)\log n$ and prove that there is a total variation cutoff for this shuffle.

Abstract. An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchell, an arbitrary small preadditive category may be understood as a "ring with several objects". In this respect, for a Hopf algebra H, an H-category will denote an "H-module algebra with several objects" and a co-H-category will denote an "H-comodule algebra with several objects". Modules over such Hopf categories were first considered by Cibils and Solotar. In this talk, we present a study of cohomology in such module categories.

In particular, we consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants.

(Joint work with Abhishek Banerjee and Mamta Balodi.)

Abstract. The theory of orthogonal polynomials started with analytic continued fractions going back to Euler, Gauss, Jacobi, Stieltjes... The combinatorial interpretations started in the late 70's and is an active research domain. I will give the basis of the theory interpreting the moments of general (formal) orthogonal polynomials, Jacobi continued fractions and Hankel determinants with some families of weighted paths. In a second part I will give some examples of interpretations of classical orthogonal polynomials and of their moments (Hermite, Laguerre, Jacobi, ...) with their connection to theoretical physics.

Abstract. The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems strongly related to the moments of some classical orthogonal polynomials (Hermite, Laguerre, Askey–Wilson). The partition function has been interpreted with various combinatorial objects such as permutations, alternative and tree-like tableaux, etc. We introduce a new one called "Laguerre heaps of segments", which seems to play a central role in the network of bijections relating all these interpretations.

Abstract. We describe how a completion of the factorial row by Suslin led to showing some ideals in a polynomial ring are set-theoretic complete intersections.

Abstract. The study of projective modules along the lines initiated by J.-P. Serre, H. Bass is outlined; the impetus given by A. Suslin in that direction is described, and recent progress made along the vision of A. Suslin on it by Fasel–Rao–Swan is shared briefly.

Abstract. 'Holomorphic eta quotients' are certain explicit classical modular forms on suitable Hecke subgroups of the full modular group. We call a holomorphic eta quotient $f$ 'reducible' if for some holomorphic eta quotient $g$ (other than $1$ and $f$), the eta quotient $f/g$ is holomorphic. An eta quotient or a modular form in general has two parameters: weight and level. We shall show that for any positive integer $N$, there are only finitely many irreducible holomorphic eta quotients of level $N$. In particular, the weights of such eta quotients are bounded above by a function of $N$. We shall provide such an explicit upper bound. This is an analog of a conjecture of Zagier which says that for any positive integer $k$, there are only finitely many irreducible holomorphic eta quotients of weight $k/2$ which are not integral rescalings of some other eta quotients. This conjecture was established in 1991 by Mersmann. We shall sketch a short proof of Mersmann's theorem and we shall show that these results have their applications in factorizing holomorphic eta quotient. In particular, due to Zagier and Mersmann's work, holomorphic eta quotients of weight $1/2$ have been completely classified. We shall see some applications of this classification and we shall discuss a few seemingly accessible yet longstanding open problems about eta quotients.

This talk will be suitable also for non-experts: We shall define all the relevant terms and we shall clearly state the classical results which we use.

Abstract. In this talk we prove that the solution of a general linear recurrence with constant coefficients can be interpreted as the determinant of some suitable matrix using a purely combinatorial method. As a consequence of our approach, we give combinatorial proofs of some recent identities due to Sury and McLaughlin in a unified way.

Abstract. In this talk I consider crossed product C*-algebras of higher dimensional noncommutative tori with actions of cyclic groups. I will discuss K-theory of those C*-algebras and some applications.

Abstract. Representations of Symmetric groups $S_n$ can be considered as homomorphisms to the orthogonal group $\mathrm{O}(d,\mathbb{R})$, where $d$ is the degree of the representation. If the determinant of the representation is trivial, we call it achiral. In this case, its image lies in the special orthogonal group $\mathrm{SO}(V)$. It is called chiral otherwise. The group $\mathrm{O}(V)$ has a non-trivial topological double cover $\mathrm{Pin}(V)$. We say the representation is spinorial if it lifts to $\mathrm{Pin}(V)$. We obtained a criterion for whether the representation is spinorial in terms of its character. We found similar criteria for orthogonal representations of Alternating groups and products of symmetric groups. One can use these results to count the number of spinorial irreducible representations of $S_n$, which are parametrized by partitions of $n$. We say a partition is spinorial if the corresponding irreducible representation of $S_n$ is spinorial. In this talk, we shall present a summary of these results and count for the number of odd-dimensional, irreducible, achiral, spinorial partitions of $S_n$. We shall also prove that almost all the irreducible representations of $S_n$ are achiral and spinorial. This is joint work with my supervisor Dr. Steven Spallone.

Abstract. In this talk, we will introduce the notion of an embedding of a quadratic space (in an associative algebra). Familiar examples of embeddings are given by Complex Numbers, Quaternions, Octonions, Clifford Algebras, and Suslin Matrices. When there are two embeddings of the same quadratic space, then we can gather information about one embedding using the other. This is the main theme of the talk. To illustrate this, we will see the connection between Suslin Matrices and Clifford Algebras. In one direction, we are able to give a simple description of Clifford Algebras using matrices. Conversely, we can now explain some (seemingly) accidental properties of Suslin matrices in a conceptual way.

Abstract. Each finite dimensional irreducible representation $V$ of a simple Lie algebra $L$ admits a filtration induced by a principal nilpotent element of $L$. This, so-called, Brylinski or Brylinski–Kostant filtration, can be restricted to the dominant weight spaces of $V$, and the resulting Hilbert series is very interesting $q$-analogs of weight multiplicity, first defined by Lusztig.

This picture can be extended to certain infinite-dimensional Lie algebras $L$ and to irreducible highest weight, integrable representations $V$. We focus on the level $1$ vacuum modules of special linear affine Lie algebras. In this case, we show how to produce a basis of the dominant weight spaces that is compatible with the Brylinski filtration. Our construction uses the so-called $W$-algebra, a natural vertex algebra associated to $L$.

This is joint work with Sachin Sharma (IIT Kanpur) and Suresh Govindarajan (IIT Madras).

Abstract. Consider the tensor product of two irreducible finite dimensional representations of a simple Lie algebra. The submodules generated by a tensor product of extremal vectors of the two components are called Kostant–Kumar submodules. These are parametrized by a double coset space of the Weyl group.

Littelmann's path model is a very general combinatorial model for representations, which encompasses many classical constructs such as Young tableaux and Lakshmibai–Seshadri chains. The path model for the full tensor product is simply the set of concatenations of paths of the individual components. We describe a way to associate a Weyl group element (rather, a double coset) to each such concatenated path and thereby obtain a path model for Kostant–Kumar submodules.

Finally, we recall the many descriptions of Demazure modules, which may be viewed as the analog of the above picture for single paths (rather than concatenations). In this case, the Weyl group element associated to a path admits different descriptions in different path models in terms of statistics such as initial direction, minimal standard lifts and Kogan faces of Gelfand–Tsetlin polytopes. Along the way, we mention some relations to jeu-de-taquin and the Schutzenberger involution.

This is based on joint work with Mrigendra Singh Kushwaha and KN Raghavan.

Abstract. Schur classified polynomial representations of $GL(n)$ using the commuting actions of the $d$th symmetric group and $GL(n)$ on the $d$-fold tensor power of $n$-dimensional space. We set out to investigate what happens when the symmetric group in this picture is shrunk to the alternating group. This led to a simple interpretation and clearer understanding of the Koszul duality functor on the category of polynomial representations of $GL(n)$.

This is based on joint work with T. Geetha and Shraddha Srivastava (https://arxiv.org/abs/1902.02465).

Abstract. We construct certain representations of affine Hecke algebras and Weyl groups, which depend on several auxiliary parameters. We refer to these as "metaplectic" representations, and as a direct consequence we obtain a family of "metaplectic" polynomials, which generalizes the well-known Macdonald polynomials.

Our terminology is motivated by the fact that if the parameters are specialized to certain Gauss sums, then our construction recovers the Kazhdan-Patterson action on metaplectic forms for $GL(n)$; more generally it recovers the Chinta-Gunnells action on $p$-parts of Weyl group multiple Dirichlet series.

This is joint work with Jasper Stokman and Vidya Venkateswaran.

Abstract. Integer partitions have been an interesting combinatorial object for centuries but we still have a lot left to understand. In this talk we will see an uncommon way to view partitions using 'Abaci' and use this to illustrate partition division (defining t-cores and t-quotients of a given integer partition). I am planning to go in more detail with a special kind of partition called t-core (partitions) and discuss different ways to enumerate related objects.

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.

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.

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.

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.

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.

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.

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

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.)

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.

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.

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.