Algebra & Combinatorics Seminar:   2020–23

Abstract. I will discuss how the inverse problem of recovering conductances in an electrical network from its response matrix can be solved using an automorphism of the positive Grassmannian called the twist.

Abstract. A "buckyball" or "fullerene" is a trivalent graph embedded in the sphere, all of whose rings have length 5 or 6. The term originates from the most famous buckyball, "Buckminsterfullerene," a molecule composed of 60 carbon atoms. In this talk, I will explain why there are exactly 1203397779055806181762759 buckyballs with 10000 carbon atoms.

Abstract. The Virasoro algebra, which can be realized as a central extension of (complex) polynomial vector fields on the unit circle, plays a key role in the representation theory of affine Lie algebras, as it acts on almost every highest weight module for the affine Lie algebra. This remarkable phenomenon eventually led to constructing the affine-Virasoro algebra, which is a semi-direct product of the affine Lie algebra and the Virasoro algebra with a common extension. The representation theory of the affine-Virasoro algebra has been studied extensively and is an extremely well-developed classical object.

In this talk, we shall consider a natural higher-dimensional analogue of the affine-Virasoro algebra, popularly known as the full toroidal Lie algebra in the literature and henceforth classify the irreducible Harish-Chandra modules over this Lie algebra. As a by-product, we also obtain the classification of all possible irreducible Harish-Chandra modules over the higher-dimensional Virasoro algebra, thereby proving Eswara Rao's conjecture (conjectured in 2004). These directly generalize the well-known result of O. Mathieu for the classical Virasoro algebra and also the recent work of Billig–Futorny for the higher rank Witt algebra.

Abstract. Consider a finite group $G$ and a prime number $p$ dividing the order of $G$. A $p$-regular element of $G$ is an element whose order is coprime to $p$. An irreducible character $\chi$ of $G$ is called a quasi $p$-Steinberg character if $\chi(g)$ is nonzero for every $p$-regular element $g$ in $G$. The quasi $p$-Steinberg character is a generalization of the well-known $p$-Steinberg character. A group, which does not have a non-linear quasi $p$-Steinberg character, can not be a finite group of Lie type of characteristic $p$. Therefore, it is natural to ask for the classification of all non-linear quasi $p$-Steinberg characters of any finite group $G$. In this joint work with Digjoy Paul and Pooja Singla, we classify quasi $p$-Steinberg characters of all finite complex reflection groups.

Abstract. Cauchy's determinantal identity (1840s) expands via Schur polynomials the determinant of the matrix $f[{\bf u}{\bf v}^T]$, where $f(t) = 1/(1-t)$ is applied entrywise to the rank-one matrix $(u_i v_j)$. This theme has resurfaced in the 2010s in analysis (following a 1960s computation by Loewner), in the quest to find polynomials $p(t)$ with a negative coefficient that entrywise preserve positivity. A key novelty here has been an application of Schur polynomials, which essentially arise from the expansion of $\det(p[{\bf u}{\bf v}^T])$, to positivity.

In the first half of the talk, I will explain the above journey from matrix positivity to determinantal identities and Schur polynomials; then go beyond, to the expansion of $\det(f[{\bf u}{\bf v}^T])$ for all power series $f$. (Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.) In the second half, joint with Siddhartha Sahi, I will explain how to extend the above determinantal identities to (a) any subgroup $G$ of signed permutations; (b) any character of $G$, or even complex class function; (c) any commutative ground ring $R$; and (d) any power series over $R$.

Abstract. In 1976 Bernstein, Gelfand, and Gelfand introduced Category $\mathcal{O}$ for a semi-simple Lie algebra $\mathfrak{g}$. This is roughly the smallest sub-category of $\mathfrak{g}$-mod containing the Verma modules and such that the simple modules have projective covers. After work of Beilinson–Bernstein and Beilinson–Ginzburg–Soergel it became clear that the the good homological properties of this category were due to the fact that it can be identified with a category of perverse sheaves on the flag variety $G/B$.

In this talk I will show how this story fits into the physics of 3d mirror symmetry. This leads to conjectural 2-categorifications of category $\mathcal{O}$ that can be computed explicitly for $\mathfrak{g} = \mathfrak{sl}_2$.

Abstract. The geometry, and the (exposed) faces, of $X$ a "Root polytope" or "Weyl polytope" over a complex simple Lie algebra $\mathfrak{g}$, have been studied for many decades for various applications, including by Satake, Borel–Tits, Casselman, and Vinberg among others. This talk focuses on two recent combinatorial analogues to these classical faces, in the discrete setting of weight-sets $X$.

Chari et al [Adv. Math. 2009, J. Pure Appl. Algebra 2012] introduced and studied two combinatorial subsets of $X$ a root system or the weight-set ${\rm wt} V$ of an integrable simple highest weight $\mathfrak{g}$-module $V$, for studying Kirillov–Reshetikhin modules over the specialization at $q=1$ of quantum affine algebras $U_q(\hat{\mathfrak{g}})$ and for constructing Koszul algebras. Later, Khare [J. Algebra 2016] studied these subsets under the names "weak-$\mathbb{A}$-faces" (for subgroups $\mathbb{A}\subseteq (\mathbb{R},+)$) and "$212$-closed subsets". For two subsets $Y\subseteq X$ in a vector space, $Y$ is said to be $212$-closed in $X$, if $y_1+y_2=x_2+x_2$ for $y_i\in Y$ and $x_i\in X$ implies $x_1,x_2\in Y$.

In finite type, Chari et al classified these discrete faces for $X$ root systems and ${\rm wt} V$ for all integrable $V$, and Khare for all (non-integrable) simple $V$. In the talk, we extend and completely solve this problem for all highest weight modules $V$ over any Kac–Moody Lie algebra $\mathfrak{g}$. We classify, and show the equality of, the weak faces and $212$-closed subsets in the three prominent settings of $X$: (a) ${\rm wt} V$ $\forall V$, (b) the hull of ${\rm wt} V$ $\forall V$, (c) ${\rm wt} \mathfrak{g}$ (consisting of roots and 0). Moreover, in the case of (a) (resp. of (b)), such subsets are precisely the weights falling on the exposed faces (resp. the exposed faces) of the hulls of ${\rm wt} V$.

Abstract. Let $\mathfrak g$ be a Borcherds–Kac–Moody Lie superalgebra (BKM superalgebra in short) with the associated graph $G$. Any such $\mathfrak g$ is constructed from a free Lie superalgebra by introducing three different sets of relations on the generators: (1) Chevalley relations, (2) Serre relations, and (3) Commutation relations coming from the graph $G$. By Chevalley relations we get a triangular decomposition $\mathfrak g = \mathfrak n_+ \oplus \mathfrak h \oplus \mathfrak n_{-}$, and each root space $\mathfrak g_{\alpha}$ is either contained in $\mathfrak n_+$ or $\mathfrak n_{-}$. In particular, each $\mathfrak g_{\alpha}$ involves only the relations (2) and (3). In this talk, we will discuss the root spaces of $\mathfrak g$ which are independent of the Serre relations. We call these roots free roots of $\mathfrak g$. Since these root spaces involve only commutation relations coming from the graph $G$ we can study them combinatorially using heaps of pieces and construct two different bases for these root spaces of $\mathfrak g$.

Abstract. The intersection theory of the Grassmannian, known as Schubert calculus, is an important development in geometry, representation theory and combinatorics. The Quot scheme is a natural generalization of the Grassmannian. In particular, it provides a compactification of the space of morphisms from a smooth projective curve C to the Grassmannian. The intersection theory of the Quot scheme can be used to recover Vafa-Intriligator formulas, which calculate explicit expressions for the count of maps to the Grassmannian subject to incidence conditions with Schubert subvarieties.

The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with symplectic (or symmetric) bilinear form. I will present explicit formulas for certain intersection numbers of the symplectic and the orthogonal analogue of Quot schemes. Furthermore, I will compare these intersection numbers with the Gromov–Ruan–Witten invariants of the corresponding Grassmannians.

Abstract. We shall discuss Legendre Pairs, an interesting combinatorial object related to the Hadamard conjecture. We shall demonstrate the exceptional versatility of Legendre Pairs, as they admit several different formulations via concepts from disparate areas of Mathematics and Computer Science. We shall mention old and new results and conjectures within the past 20+ years, as well as potential future avenues for investigation.

Abstract. $\mathrm{Per}_n $ is an affine algebraic curve, defined over $\mathbb Q$, parametrizing (up to change of coordinates) degree-$2$ self-morphisms of $\mathbb P^1$ with an $n$-periodic ramification point. The $n$-th Gleason polynomial $G_n$ is a polynomial in one variable with $\mathbb Z$-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-$2$ self-morphisms of $\mathbb C$ with an $n$-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is $\mathrm{Per}_n$ connected? (2) Is $G_n$ irreducible over $\mathbb Q$?

We show that if $G_n$ is irreducible over $\mathbb Q$, then $\mathrm{Per}_n$ is irreducible over $\mathbb C$, and is therefore connected. In order to do this, we find a $\mathbb Q$-rational smooth point of a projective completion of $\mathrm{Per}_n$. This $\mathbb Q$-rational smooth point represents a special degeneration of degree-$2$ morphisms, and as such admits an interpretation in terms of tropical geometry.

(This talk will be pitched at a broad audience.)

Abstract. Given a bipartite graph $G$ (subject to a constraint), the "cross-ratio degree" of G is a non-negative integer invariant of $G$, defined via a simple counting problem in algebraic geometry. I will discuss some natural contexts in which cross-ratio degrees arise. I will then present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings. Finally, time permitting, I may discuss the tropical side of the story.

Abstract. Let $q$ be a prime power and define $(n)_q:=1+q+q^2+\cdots+q^{n-1}$, for a non-negative integer $n$. Let $B_q(n)$ denote the set of all subspaces of $\mathbb{F}_q^n$, the $n$-dimensional $\mathbb{F}_q$-vector space of all column vectors with $n$ components. Define a $B_q(n)\times B_q(n)$ complex matrix $M_{q,n}$ with entries given by \begin{equation} M_{q,n}(X,Y):= \begin{cases} 1&\text{ if }Y\subseteq X, \dim(Y)=\dim(X)-1,\\ q^{\dim(X)}&\text{ if }X\subseteq Y, \dim(Y)=\dim(X)+1,\\ 0&\text{ otherwise.} \end{cases} \end{equation} We think of $M_{q,n}$ as a $q$-analog of the adjacency matrix of the $n$-cube. We show that the eigenvalues of $M_{q,n}$ are \begin{equation} (n-k)_q-(k)_q\text{ with multiplicity }\binom{n}{k}_q,\quad k=0,1,\dots,n, \end{equation} and we write down an explicit canonical eigenbasis of $M_{q,n}$. We give a weighted count of the number of rooted spanning trees in the $q$-analog of the $n$-cube.

This talk is based on a joint work with M. K. Srinivasan.

Abstract. This talk is based on the work of Stark and Terras (Zeta functions of Finite graphs and Coverings I, II, III). In this talk we start with an introduction to zeta functions in various branches of mathematics. Our focus is mainly on zeta functions on finite undirected connected graphs. We obtain an analogue of the prime number theorem, but for graphs, using the Ihara Zeta Function. We also introduce edge and path zeta functions and show interesting results.

Abstract. The polynomial method is an ever-expanding set of algebraic techniques, which broadly entails capturing combinatorial objects by algebraic means, specifically using polynomials, and then employing algebraic tools to infer their combinatorial features. While several instances of the polynomial method have been part of the combinatorialist's toolkit for decades, development of this method has received more traction in recent times, owing to several breakthroughs like (i) Dvir's solution (2009) to the finite-field Kakeya problem, followed by an improvement by Dvir, Kopparty, Saraf, and Sudan (2013), (ii) Guth and Katz (2015) proving a conjecture by Erdös on the distinct distances problem, (iii) solutions to the capset problem by Croot, Lev, and Pach (2017), and Ellenberg and Gijswijt (2017), to name a few.

One of the ways to employ the polynomial method is via the classical algebraic objects – (affine) Zariski closure, (affine) Hilbert function, and Gröbner basis. Owing to their applicability in several areas like computational complexity, combinatorial geometry, and coding theory, an important line of enquiry is to understand these objects for 'structured' sets of points in the affine space. In this talk, we will be mainly concerned with Zariski closures of symmetric sets of points in the Boolean cube.

Firstly, we will look at a combinatorial characterization of Zariski closures of all symmetric sets, and its application to some hyperplane and polynomial covering problems for the Boolean cube, over any field of characteristic zero. We will also briefly look at Zariski closures over fields of positive characteristic, although much less is known in this setting. Secondly, we will see a simple illustration of a 'closure statement' being used as a technique for proving bounds on the complexity of approximating Boolean functions by polynomials. We will conclude with some open questions on Zariski closures motivated by problems on these two fronts.

Some parts of this talk will be based on the works:

1. https://arxiv.org/abs/2107.10385
2. https://arxiv.org/abs/2111.05445
3. https://arxiv.org/abs/1910.02465

Abstract. I will explain a generalisation of the constructions Quillen used to prove that the $K$-groups of rings of integers are finitely generated. It takes the form of a 'rank' spectral sequence, converging to the homology of Quillen's $Q$-construction on the category of coherent sheaves over a Noetherian integral scheme, and whose $E^1$ terms are given by homology of Steinberg modules. Computing its $d^1$ differentials is a challenge, which can be approached through the universal modular symbols of Ash–Rudolph.

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.

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.

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.

Abstract. Postnikov defined the totally nonnegative Grassmannian as the part of the Grassmannian where all Plücker 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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.