IISc Alg Comb 2023-24

Algebra & Combinatorics Seminar:   2023–24

The Algebra & Combinatorics Seminar has traditionally met on Fridays in Lecture Hall LH-1 of the IISc Mathematics Department – or online in some cases. The organizers are Apoorva Khare and R. Venkatesh.

Ratheesh T V (IMSc Chennai) Feb 23, 2024
Monomial expansions for $q$-Whittaker polynomials (LH-1 – 3 pm, Fri)

Abstract. We consider the monomial expansion of the $q$-Whittaker polynomials given by the fermionic formula and via the inv and quinv statistics. We construct bijections between the parametrizing sets of these three models which preserve the $x$- and $q$-weights, and which are compatible with natural projection and branching maps. We apply this to the limit construction of local Weyl modules and obtain a new character formula for the basic representation of $\widehat{\mathfrak{sl}_n}$.

Manish Patnaik (University of Alberta, Edmonton, Canada) Feb 9, 2024
Whittaker functions on covers of p-adic groups and quantum groups at roots of unity (Joint with the Number Theory Seminar) (LH-1 – 2 pm, Fri)

Abstract. Since the work of Kubota in the late 1960s, it has been known that certain Gauss sum twisted (multiple) Dirichlet series are closely connected to a theory of automorphic functions on metaplectic covering groups. The representation theory of such covering groups was then initiated by Kazhdan and Patterson in the 1980s, who emphasized the role of a certain non-uniqueness of Whittaker functionals.

Motivated on the one hand by the recent theory of Weyl group multiple Dirichlet series, and on the other by the so-called "quantum" geometric Langlands correspondence, we explain how to connect the representation theory of metaplectic covers of $p$-adic groups to an object of rather disparate origin, namely a quantum group at a root of unity. This gives us a new point of view on the non-uniqueness of Whittaker functionals and leads, among other things, to a Casselman–Shalika type formula expressed in terms of (Gauss sum) twists of "$q$"-Littlewood–Richardson coefficients, objects of some combinatorial interest.

Joint work with Valentin Buciumas.

Rameez Raja (NIT Srinagar) Jan 24, 2024
Some combinatorial structures realized by commutative rings (LH-1 – 2:30 pm, Wed)

Abstract. There are many ways to associate a graph (combinatorial structure) to a commutative ring $R$ with unity. One of the ways is to associate a zero-divisor graph $\Gamma(R)$ to $R$. The vertices of $\Gamma(R)$ are all elements of $R$ and two vertices $x, y \in R$ are adjacent in $\Gamma(R)$ if and only if $xy = 0$. We shall discuss Laplacian of a combinatorial structure $\Gamma(R)$ and show that the representatives of some algebraic invariants are eigenvalues of the Laplacian of $\Gamma(R)$. Moreover, we discuss association of another combinatorial structure (Young diagram) with $R$. Let $m, n \in \mathbb{Z}_{>0}$ be two positive integers. The Young's partition lattice $L(m,n)$ is defined to be the poset of integer partitions $\mu = (0 \leq \mu_1 \leq \mu_2 \leq \cdots \leq \mu_m \leq n)$. We can visualize the elements of this poset as Young diagrams ordered by inclusion. We conclude this talk with a discussion on Stanley's conjecture regarding symmetric saturated chain decompositions (SSCD) of $L(m,n)$.

Gopal Prasad (University of Michigan, Ann Arbor, USA) Jan 12, 2024
Fake Projective Planes (Joint with the Number Theory Seminar) (LH-1 – 12 pm, Fri)

Bharatram Rangarajan (Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel) Jan 10, 2024
"Almost" Representations and Group Stability (LH-1 – 4 pm, Wed)

Abstract. Consider the following natural robustness question: is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This classical question of stability goes all the way back to Turing and Ulam, and can be posed for different target groups, and different notions of distance. Group stability has been an active line of study in recent years, thanks to its connections to major open problems like the existence of non-sofic and non-hyperlinear groups, the group Connes embedding problem and the recent breakthrough result MIP*=RE, apart from property testing and error-correcting codes.

In this talk, I will survey some of the main results, techniques, and questions in this area.

Sridhar Venkatesh (University of Michigan, Ann Arbor, USA) Dec 28, 2023
The Du Bois complex and some associated singularities (LH-1 – 11:30 am, Thu)

Abstract. For a smooth variety $X$ over $\mathbb{C}$, the de Rham complex of $X$ is a powerful tool to study the geometry of $X$ because of results such as the degeneration of the Hodge-de Rham spectral sequence (when $X$ is proper). For singular varieties, it follows from the work of Deligne and Du Bois that there is a substitute called the Du Bois complex which satisfies many of the nice properties enjoyed by the de Rham complex in the smooth case. In this talk, we will discuss some classical singularities associated with this complex, namely Du Bois and rational singularities, and some recently introduced refinements, namely $k$-Du Bois and $k$-rational singularities. This is based on joint work with Wanchun Shen and Anh Duc Vo.

Ravindra Girivaru (University of Missouri, St. Louis, USA) Dec 20, 2023
Matrix factorisations of polynomials (LH-3 – 4 pm, Wed)

Abstract. A matrix factorisation of a polynomial $f$ is an equation $AB = f \cdot {\rm I}_n$ where $A,B$ are $n \times n$ matrices with polynomial entries and ${\rm I}_n$ is the identity matrix. This question has been of interest for more than a century and has been studied by mathematicians like L.E. Dickson. I will discuss its relation with questions arising in algebraic geometry about the structure of subvarieties in projective hypersurfaces.

R. Venkatesh (IISc Mathematics) Dec 13, 2023
A simple proof for the characterization of chordal graphs using Horn hypergeometric series (LH-3 – 11:30 am, Wed)

Abstract. Let $G$ be a finite simple graph (with no loops and no multiple edges), and let $I_G(x)$ be the multi-variate independence polynomial of $G$. In 2021, Radchenko and Villegas proved the following interesting characterization of chordal graphs, namely $G$ is chordal if and only if the power series $I_G(x)^{-1}$ is Horn hypergeometric. In this talk, I will give a simpler proof of this fact by computing $I_G(x)^{-1}$ explicitly using multi-coloring chromatic polynomials. This is a joint work with Dipnit Biswas and Irfan Habib.

Duncan Laurie (University of Oxford, UK) Dec 8, 2023
The structure and representation theory of quantum toroidal algebras (LH-3 – 3 pm, Fri)

Abstract. Quantum toroidal algebras are the next class of quantum affinizations after quantum affine algebras, and can be thought of as "double affine quantum groups". However, surprisingly little is known thus far about their structure and representation theory in general.

In this talk we'll start with a brief recap on quantum groups and the representation theory of quantum affine algebras. We shall then introduce and motivate quantum toroidal algebras, before presenting some of the known results. In particular, we shall sketch our proof of a braid group action, and generalise the so-called Miki automorphism to the simply laced case.

Time permitting, we shall discuss future directions and applications including constructing representations of quantum toroidal algebras combinatorially, written in terms of Young columns and Young walls.

Shashank Kanade (University of Denver, USA) Dec 1, 2023
A glimpse into the world of Rogers–Ramanujan identities (LH-3 – 11 am, Fri)

Abstract. I will give a gentle introduction to the combinatorial Rogers–Ramanujan identities. While these identities are over a century old, and have many proofs, the first representation-theoretic proof was given by Lepowsky and Wilson about four decades ago. Now-a-days, these identities are ubiquitous in several areas of mathematics and physics. I will mention how these identities arise from affine Lie algebras and quantum invariants of knots.

Sagar Shrivastava (TIFR, Mumbai) Nov 28, 2023
Branching multiplicity of symplectic groups as $SL_2$ representations (LH-1 – 11:30 am, Tue)

Abstract. Branching rules are a systematic way of understanding the multiplicity of irreducible representations in restrictions of representations of Lie groups. In the case of $GL_n$ and orthogonal groups, the branching rules are multiplicity free, but the same is not the case for symplectic groups. The explicit combinatorial description of the multiplicities was given by Lepowsky in his PhD thesis. In 2009, Wallach and Oded showed that this multiplicity corresponds to the dimension of the multiplicity space, which was a representation of $SL_2(=Sp(2))$. In this talk, we give an alternate proof of the same without invoking any partition function machinery. The only assumption for this talk would be the Weyl character formula.

Rekha Biswal (NISER, Bhubaneswar) Nov 17, 2023
Ideals in enveloping algebras of affine Kac–Moody algebras (LH-3 – 11:30 am, Fri)

Abstract. In this talk, I will discuss about the structure of ideals in enveloping algebras of affine Kac–Moody algebras and explain a proof of the result which states that if $U(L)$ is the enveloping algebra of the affine Lie algebra $L$ and "$c$" is the central element of $L$, then any proper quotient of $U(L)/(c)$ by two sided ideals has finite Gelfand–Kirillov dimension. I will also talk about the applications of the result including the fact that $U(L)/(c-\lambda)$ for non zero $\lambda$ is simple. This talk is based on joint work with Susan J. Sierra.

Subhajit Ghosh (Bar-Ilan University, Ramat-Gan, Israel) Nov 15, 2023
Aldous-type spectral gap results for the complete monomial group (LH-1 – 3 pm, Wed)

Abstract. Let us consider the continuous-time random walk on $G\wr S_n$, the complete monomial group of degree $n$ over a finite group $G$, as follows: An element in $G\wr S_n$ can be multiplied (left or right) by an element of the form

  • $(u,v)_G:=(\mathbf{e},\dots,\mathbf{e};(u,v))$ with rate $x_{u,v}(\geq 0)$, or
  • $(g)^{(w)}:=(\dots,\mathbf{e},g,\mathbf{e},\dots;\mathbf{id})$ with rate $y_w\alpha_g\; (y_w \gt 0,\;\alpha_g=\alpha_{g^{-1}}\geq 0)$,
such that $\{(u,v)_G,(g)^{(w)}:x_{u,v} \gt 0,\;y_w\alpha_g \gt 0,\;1\leq u \lt v \leq n,\;g\in G,\;1\leq w\leq n\}$ generates $G\wr S_n$. We also consider the continuous-time random walk on $G\times\{1,\dots,n\}$ generated by one natural action of the elements $(u,v)_G,1\leq u \lt v\leq n$ and $g^{(w)},g\in G,1\leq w\leq n$ on $G\times\{1,\dots,n\}$ with the aforementioned rates. We show that the spectral gaps of the two random walks are the same. This is an analogue of the Aldous' spectral gap conjecture for the complete monomial group of degree $n$ over a finite group $G$.

V. Sathish Kumar (IMSc, Chennai) Sep 22, 2023
Unique factorization for tensor products of parabolic Verma modules (LH-1 – 3 pm, Fri)

Abstract. Let $\mathfrak g$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra $\mathfrak h$. We prove that unique factorization holds for tensor products of parabolic Verma modules. We prove more generally a unique factorization result for products of characters of parabolic Verma modules when restricted to certain subalgebras of $\mathfrak h$. These include fixed point subalgebras of $\mathfrak h$ under subgroups of diagram automorphisms of $\mathfrak g$. This is joint work with K.N. Raghavan, R. Venkatesh and S. Viswanath.

Swaraj Pande (University of Michigan, Ann Arbor, USA) Aug 30, 2023
A Frobenius version of Tian's Alpha invariant (LH-1 – 3:30 pm, Wed)
(Joint with the Geometry & Topology Seminar)

Abstract. The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing "log canonical threshold" with "F-pure threshold", a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature.

Sutanay Bhattacharya (University of California, San Diego, USA) Aug 21, 2023
The lattice of nil-Hecke algebras over reflection groups (LH-1 – 11:30 am, Mon)

Abstract. Associated to every reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, obtained by killing all "sufficiently long" braid words, as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley–Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal.

Motivated by the classical work of Coxeter (1957) and the Broue–Malle–Rouquier freeness conjecture, and continuing beyond the previous work of Khare, we attempt to obtain a classification of the finite-dimensional nil-Hecke algebras for all reflection groups $W$. These include the usual nil-Coxeter algebras for $W$ of finite type, their "fully commutative" analogues for $W$ of FC-finite type, three exceptional algebras (of types $F_4$,$H_3$,$H_4$), and three exceptional series (of types $B_n$ and $A_n$, two of them novel). We further uncover combinatorial bases of algebras, both known (fully commutative elements) and novel ($\overline{12}$-avoiding signed permutations), and classify the Frobenius nil-Hecke algebras in the aforementioned cases. (Joint with Apoorva Khare.)