with Sridhar Narayanan, Digoy Paul and Shraddha Srivastava,
to appear in the Proceedings of Group Algebras, Representations and Computation
(ICTS, Oct. 2019).

We construct the
polynomial induction functor, which is the right adjoint to the
restriction functor from the category of polynomial representations
of a general linear group to the category of representations of its
Weyl group. This construction leads to a representation-theoretic
proof of Littlewood's plethystic formula for the multiplicity of an
irreducible representation of the symmetric group in such a
restriction. The unimodality of certain bipartite partition
functions follows.

with Jyotirmoy Ganguly and Steven Spallone, EJC,
27(2):P2.1, Apr 2020.

Fix a partition
$\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and positive integer
$d$. For each $n\geq k$, let $\chi^\lambda_\mu$ denote the value of
the irreducible character $\chi^\lambda$ of $S_n$, corresponding to
a partition $\lambda$ of $n$, at a permutation with cycle type
$(\mu_1,\dotsc,\mu_m,1^{n-k})$. We show that the proportion of
partitions $\lambda$ of $n$ such that $\chi^\lambda_\mu$ is
divisible by $d$ approaches $1$ as $n$ approaches infinity.

with Sridhar Narayanan, Digoy Paul and Shraddha Srivastava,
submitted Jan 2020.

Character polynomials are used to study the restriction of a
polynomial representation of a general linear group to its subgroup
of permutation matrices. A simple formula is obtained for computing
inner products of class functions given by character polynomials.
Character polynomials for symmetric and alternating tensors are
computed using generating functions with Eulerian factorizations.
These are used to compute character polynomials for Weyl modules,
which exhibit a duality. By taking inner products of character
polynomials for Weyl modules and character polynomials for Specht
modules, stable restriction coefficients are easily computed.
Generating functions of dimensions of symmetric group invariants in
Weyl modules are obtained. Partitions with two rows, two columns,
and hook partitions whose Weyl modules have non-zero vectors
invariant under the symmetric group are characterized. A
reformulation of the restriction problem in terms of a restriction
functor from the category of strict polynomial functors to the
category of finitely generated FI-modules is obtained.

with T. Geetha and Shraddha Srivastava, Pacific J. Math.,
306(1):153--184, June 2020.

We introduce the alternating Schur algebra $AS_F(n,d)$ as the
commutant of the action of the alternating group $A_d$ on the
$d$-fold tensor power of an $n$-dimensional $F$-vector space. When
$F$ has characteristic different from $2$, we give a basis of
$AS_F(n,d)$ in terms of bipartite graphs, and a graphical
interpretation of the structure constants. We introduce the
abstract Koszul duality functor on modules for the even part of any
$\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is
$\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra
$S_F(n,d)$ as its even part. This leads to an approach to Koszul
duality for $S_F(n,d)$-modules that is amenable to combinatorial
methods. We characterize the category of $AS_F(n,d)$-modules in
terms of $S_F(n,d)$-modules and their Koszul duals. We use the
graphical basis of $AS_F(n,d)$ to study the dependence of the
behavior of derived Koszul duality on $n$ and $d$.

with Arvind Ayyer and Steven Spallone, JCTA, 150:208-232,
2017.

We give a closed formula for the number of partitions λ of n
such that the corresponding irreducible representation
V_{λ} of Sn has non-trivial determinant. We determine how
many of these partitions are self-conjugate and how many are hooks.
This is achieved by characterizing the 2-core towers of such
partitions. We also obtain a formula for the number of partitions
of n such that the associated permutation representation of
S_{n} has non-trivial determinant.

with Arvind Ayyer and Steven Spallone, Séminaire Lotharingien
de Combinatoire, volume 75, article B75g, June 2016.

We show that the subgraph
induced in Young's graph by the set of partitions with an odd
number of standard Young tableaux is a binary tree. This tree
exhibits self-similarities at all scales, and has a simple
recursive description.