The Algebra & Combinatorics Seminar has traditionally met on Fridays
in Lecture Hall LH1 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 
(LH1 –
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 padic groups and quantum groups at roots of
unity (Joint with the Number
Theory Seminar) 
(LH1 –
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 nonuniqueness of Whittaker functionals.
Motivated on the one hand by the recent theory of Weyl group multiple
Dirichlet series, and on the other by the socalled "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 nonuniqueness 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 
(LH1 –
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
zerodivisor 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 $L(m,n)$.



Bharatram
Rangarajan
(Einstein Institute of Mathematics, Hebrew University of Jerusalem,
Israel) 
Jan 10, 2024 
"Almost"
Representations and Group Stability 
(LH1 –
4 pm, Wed)


Abstract.
Consider the following natural robustness question: is an
almosthomomorphism 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 nonsofic and nonhyperlinear groups, the group
Connes embedding problem and the recent breakthrough result MIP*=RE,
apart from property testing and errorcorrecting 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 
(LH1 –
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 Hodgede 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 
(LH3 –
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 
(LH3 –
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 multivariate 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 multicoloring 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 
(LH3 –
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 socalled 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 
(LH3 – 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 representationtheoretic
proof was given by Lepowsky and Wilson about four decades ago.
Nowadays, 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 
(LH1 –
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 
(LH3 –
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 (BarIlan University, RamatGan,
Israel) 
Nov 15, 2023 
Aldoustype
spectral gap results for the complete monomial group 
(LH1 – 3
pm, Wed)


Abstract.
Let us consider the continuoustime 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 continuoustime 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 
(LH1 – 3
pm, Fri)


Abstract.
Let $\mathfrak g$ be a symmetrizable KacMoody 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.


Abstract.
The Alpha invariant of a complex Fano manifold was introduced by Tian to
detect its Kstability, 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 FrobeniusAlpha invariant. This analog is obtained by
replacing "log canonical threshold" with "Fpure 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 FrobeniusAlpha
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 Fsignature.

Sutanay Bhattacharya (University of California, San
Diego, USA) 
Aug 21, 2023 
The lattice of nilHecke algebras over reflection
groups 
(LH1 – 11:30
am, Mon)


Abstract.
Associated to every reflection group, we construct a lattice of quotients
of its braid monoidalgebra, which we term nilHecke algebras, obtained
by killing all "sufficiently long" braid words, as well as some integer
power of each generator. These include usual nilCoxeter algebras,
nilTemperley–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 finitedimensional nilHecke algebras for all reflection groups
$W$. These include the usual nilCoxeter algebras for $W$ of finite type,
their "fully commutative" analogues for $W$ of FCfinite 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 nilHecke algebras in the aforementioned cases. (Joint with
Apoorva Khare.)

2020–23
2019–20
2018–19
