The Algebra & Combinatorics Seminar has traditionally met on Fridays
from 3–4 pm, in Lecture Hall LH1 of the IISc Mathematics
Department – or online in some cases. The organizers are R.
Venkatesh and Apoorva Khare.

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
