To appear in the Proceedings of Group Algebras, Representations and Computation, held at ICTS Bangalore in October 2019. (arXiv)

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.

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.

This article is an exposition of Polya's theory of counting
colourings of structures under symmetry. It is based on a lecture
given to the students of Amrita University on 16th September 2019.
For a deeper understanding of the topic, the reader is encouraged
to read the
book of Polya and Read.

Let $G$ be a finite group. We consider the problem of counting
simultaneous conjugacy classes of $n$-tuples and and simultaneous
conjugacy classes of commuting $n$-tuples in $G$. Let
$\alpha_{G,n}$ denote the number of simultaneous conjugacy classes
of $n$-tuples, and $\beta_{G,n}$ the number of simultaneous
conjugacy classes of commuting $n$-tuples in $G$. The generating
functions $A_G(t) = \sum_{n\geq 0} \alpha_{G,n}t^n,$ and $B_G(t) =
\sum_{n\geq 0} \beta_{G,n}t^n$ are rational functions of $t$. We
show that $A_G(t)$ determines and is completely determined by the
class equation of $G$. We show that $\alpha_{G,n}$ grows
exponentially with growth factor equal to the cardinality of $G$,
whereas $\beta_{G,n}$ grows exponentially with growth factor equal
to the maximum cardinality of an abelian subgroup of $G$. The
functions $A_G(t)$ and $B_G(t)$ may be regarded as combinatorial
invariants of the finite group $G$. We study dependencies amongst
these invariants and the notion of isoclinism for finite groups.
Indeed, we prove that the normalized functions $A_G(t/|G|)$ and
$B_G(t/|G|)$ are invariants of isoclinism families. We compute
these normalized functions $A_G(t/|G|)$ and $B_G(t/|G|)$ for
certain finite $p$-groups including all isoclinism families of rank
at most $5$.

Fix a partition $\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and
positive integer $d$. For each $n>k$, let $\chi^\lambda_\mu$
denote the value of the irreducible character of $S_n$ 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.

The alternating Schur algebra $AS_F(n,d)$ is defined 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. It
contains the classical Schur algebra as a subalgebra. When $F$ is a
field of odd characteristic, we find a basis of $AS_F(n,d)$ in
terms of bipartite graphs. We give a combinatorial interpretation
of the structure constants of $AS_F(n,d)$ with respect to this
basis.

What additional structure does being a module for $AS_F(n,d)$
impose on a module for the classical Schur algebra? Our answer to
this question involves the Koszul duality functor, introduced by
Krause for strict polynomial functors. It leads to a simple
interpretation of Koszul duality for modules of the classical Schur
algebra. Krause's work implies that derived Koszul duality functor
is an equivalence when $n\geq d$. Our combinatorial methods prove
the converse.

(arXiv)
To appear in the Indian Journal of Discrete Mathematics.

Every irreducible odd dimensional representation of the $n$th
symmetric or hyperoctahedral group, when restricted to the
$(n-1)$th, has a unique irreducible odd-dimensional constituent.
Furthermore, the subgraph induced by odd-dimensional
representations in the Bratteli diagram of symmetric and
hyperoctahedral groups is a binary tree with a simple recursive
description. We survey the description of this tree, known as the
Macdonald tree, for symmetric groups, from our earlier work. We
describe analogous results for hyperoctahedral groups. A partition
$\lambda$ of n is said to be chiral if the corresponding
irreducible representation $V_\lambda$ of Sn has non-trivial
determinant. We review our previous results on the structure and
enumeration of chiral partitions, and subsequent extension to all
Coxeter groups by Ghosh and Spallone. Finally we show that the
numbers of odd and chiral partitions track each other closely.

We deduce decompositions of natural representations of general
linear groups and symmetric groups from combinatorial bijections
involving tableaux. These include some of Howe's dualities, Gelfand
models, the Schur-Weyl decomposition of tensor space, and
multiplicity-free decompositions indexed by threshold
partitions.

Timed words are words where letters of the alphabet come with
time stamps. We extend the definitions of semistandard tableaux,
insertion, Knuth equivalence, and the plactic monoid to the setting
of timed words. Using this, Greene's theorem is formulated and
proved for timed words, and algorithms for the RSK correspondence
are extended to real matrices.

We give an exposition of Schensted's algorithm to find the
length of the longest increasing subword of a word in an ordered
alphabet, and Greene's generalization of Schensted's results using
Knuth equivalence. We announce a generalization of these results to
timed words.

Notes from a course at the ATM Workshop on Schubert Varieties,
held at The Institute of Mathematical Sciences, Chennai, in
November 2017. Various expansions of Schur functions, the
Lindström-Gessel-Viennot lemma, semistandard Young tableaux,
Schensted's insertion algorithm, the plactic monoid, the RSK
correspondence, and the Littlewood-Richardson rule are
discussed.

We study the subgraph of the Young-Fibonacci graph induced by
elements with odd $f$-statistic (the $f$-statistic of an element
$w$ of a differential graded poset is the number of saturated
chains from the minimal element of the poset to $w$). We show that
this subgraph is a binary tree. Moreover, the odd residues of the
$f$-statistics in a row of this tree equidistibute modulo any power
two. This is equivalent to a purely number theoretic result about
the equidistribution of residues modulo powers of two among the
products of distinct odd numbers less than a fixed number.

We describe the expansion of Gelfand-Tsetlin basis vectors for
representations of alternating groups in terms of Young's
orthogonal basis for representations of symmetric groups.

We give a closed formula for the number of partitions $\lambda$
of $n$ such that the corresponding irreducible representation
$V_\lambda$ of $S_n$ 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.

We show that the Hasse diagram of the subposet of Young's
lattice consisting 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.

We derive functional relationships between spherical generating
functions of graph monoids, right-angled Artin groups and
right-angled Coxeter groups. We use these relationships to express
the spherical generating function of a right-angled Artin group in
terms of the clique polynomial of its defining graph. We also
describe algorithms for computing the geodesic generating functions
of these structures.

Issai Schur, in his doctoral thesis (1901) introduced the Schur
algebra to study the polynomial representation theory of the
general linear group. He described a basis of this algebra and
structure constants. Later, Miguel Mendez (2001) gave a
graph-theoretic description of Schur's basis and computed the
structure constants. In this presentation, we will give a new
graphic interpretation of the basis of Schur algebra and use it to
give a description of the structure constants which is equivalent
to the one given by Mendez.

Let c_{n} denote the number of nodes at a
distance $n$ from the root of a rooted tree. A criterion for
proving the rationality and computing the rational generating
function of the sequence {c_{n}} is described. This
criterion is applied to counting the number of conjugacy classes of
commuting tuples in finite groups and the number of isomorphism
classes of representations of polynomial algebras over finite
fields. The method for computing the rational generating functions,
when applied to the study of point configurations in finite sets,
gives rise to some classical combinatorial results on Bell numbers
and Stirling numbers of the second kind. When applied to the study
of vector configurations in a finite vector space, it reveals a
connection between counting such configurations and Gaussian
binomial coefficients.

We compute the number of orbits of pairs in a finitely generated
torsion module (more generally, a module of bounded order) over a
discrete valuation ring. The answer is found to be a polynomial in
the cardinality of the residue field whose coefficients are
integers which depend only on the elementary divisors of the
module, and not on the ring in question. The coefficients of these
polynomials are conjectured to be non-negative integers.

We describe a basis of the centre of the Schur algebra that
comes from conjugacy classes in the symmetric group via Schur-Weyl
duality. We give a combinatorial description of expansions of these
basis elements in terms of the basis originally used by Schur. The
primitive central idempotents of the Schur algebra can be written
down using this basis and the character table of the symmetric
group in the semisimple case. Along the way we prove a result on
the non-singularity of the submatrix of the character table matrix
of a symmetric group obtained by taking rows and columns indexed by
partitions with at most $n$ parts for any $n$.

Let R be a principal ideal local ring of length two, for
example, the ring $R=\mathbf Z/p^2\mathbf Z$ with $p$ prime. In
this paper we develop a theory of normal forms for similarity
classes in the matrix rings $M_n(R)$ by interpreting them in terms
of extensions of $R[t]$-modules. Using this theory, we describe the
similarity classes in $M_n(R)$ for $n\leq 4$, along with their
centralizers. Among these, we characterize those classes which are
similar to their transposes. Non-self-transpose classes are shown
to exist for all $n>3$. When $R$ has finite residue field of
order $q$, we enumerate the similarity classes and the
cardinalities of their centralizers as polynomials in $q$.
Surprisingly, these polynomials turn out to have non-negative
integer coefficients.

Let $F$ be a non-Archimedean local field and let $E$ be a finite
extension of $F$. Let $G$ be a split semisimple $F$-group. We
discuss how to compare distances on the Bruhat-Tits buildings
$\mathbf B_E$ and $\mathbf B_F$ of $G(E)$ and $G(F)$ respectively.
We also discuss the comparison of volumes on finite volume
arithmetic quotients of the buildings.

A tuple (or subgroup) in a group is said to degenerate to
another if the latter is an endomorphic image of the former. In a
countable reduced abelian group, it is shown that if tuples (or
finite subgroups) degenerate to each other, then they lie in the
same automorphism orbit. The proof is based on techniques that were
developed by Kaplansky and Mackey in order to give an elegant proof
of Ulm's theorem. Similar results hold for reduced countably
generated torsion modules over principal ideal domains. It is shown
that the depth and the description of atoms of the resulting poset
of orbits of tuples depend only on the Ulm invariants of the module
in question (and not on the underlying ring). A complete
description of the poset of orbits of elements in terms of the Ulm
invariants of the module is given. The relationship between this
description of orbits and a very different-looking one obtained by
Dutta and Prasad for torsion modules of bounded order is
explained.

Montashefte für Mathemattik, 167(3-4), pages 333-356,
September 2012. (arXiv), (from journal)

We prove positive characteristic versions of the logarithm laws
of Sullivan and Kleinbock-Margulis and obtain related results in
Metric Diophantine Approximation.

The centralizer algebra of a matrix consists of those matrices
that commute with it. We investigate the basic
representation-theoretic invariants of centralizer algebras, namely
their radicals, projective indecomposable modules, injective
indecomposable modules, simple modules and Cartan matrices. With
the help of our Cartan matrix calculations we determine their
global dimensions. Many of these algebras are of infinite global
dimension.

The Weil representation of the symplectic group associated to a
finite abelian group of odd order is shown to have a
multiplicity-free decomposition. When the abelian group is
p-primary of type λ, the irreducible representations
occurring in the Weil representation are parametrized by a
partially ordered set which is independent of p. As p
varies, the dimension of the irreducible representation
corresponding to each parameter is shown to be a polynomial in
p which is calculated explicitly. The commuting algebra of
the Weil representation has a basis indexed by another partially
ordered set which is independent of p. The expansions of the
projection operators onto the irreducible invariant subspaces in
terms of this basis are calculated. The coefficients are again
polynomials in p. These results remain valid in the more
general setting of finitely generated torsion modules over a
Dedekind domain.

A notion of degeneration of elements in groups is introduced. It
is used to parametrize the orbits in a finite abelian group under
its full automorphism group by a finite distributive lattice. A
pictorial description of this lattice leads to an intuitive
self-contained exposition of some of the basic facts concerning
these orbits, including their enumeration. Given a partition λ, the
lattice parametrizing orbits in a finite abelian p-group of
type λ is found to be independent of p. The order of the
orbit corresponding to each parameter, which turns out to be a
polynomial in p, is calculated. The description of orbits is
extended to subquotients by certain characteristic subgroups. Each
such characteristic subquotient is shown to have a unique maximal
orbit.

Resonance,
15(11):977-987, November 2010 (first part); 15(12):1074-1083,
December 2010 (second part). (arXiv)

In this expository article, we discuss the relation between the
Gaussian binomial and multinomial coefficients and ordinary
binomial and multinomial coefficients from a combinatorial
viewpoint, based on expositions by Butler, Knuth and Stanley.

The Stone-von Neumann-Mackey Theorem for locally compact abelian
groups is proved using the Peter-Weyl theorem and the theory of
Fourier transforms for finite dimensional real vector spaces. A
theorem of Pontryagin and van Kampen on the structure of locally
compact abelian groups (which is evident in any particular case) is
assumed.

Inductive algebras for the irreducible unitary representations
of the universal cover of the group of unimodular two by two
matrices are classified. The classification of homogeneous shift
operators is obtained as a direct consequence. This gives a new
approach to the results of Bagchi and Misra.

with Ilya Shapiro and M. K. Vemuri. Advances in Mathematics, volume 225, pages 2429-2454,
2010. (arXiv), (from journal),
(MathSciNet)

Is every locally compact abelian group which admits a symplectic
self-duality isomorphic to the product of a locally compact abelian
group and its Pontryagin dual? Several sufficient conditions,
covering all the typical applications are found. Counterexamples
are produced by studying a seemingly unrelated question about the
structure of maximal isotropic subgroups of finite abelian groups
with symplectic self-duality (where the original question always
has an affirmative answer).

Journal of the Ramanujan Mathematical
Society, volume 25, issue 2, pages 113-145, June 2010. (arXiv),
(MathSciNet)

This article gives a fairly self-contained treatment of the
basic facts about the Iwahori-Hecke algebra of a split p-adic
group, including Bernstein's presentation, Macdonald's formula, the
Casselman-Shalika formula, and the Lusztig-Kato formula.

A characterization of the maximal abelian sub-algebras of matrix
algebras that are normalized by the canonical representation of a
finite Heisenberg group is given. Examples are constructed using a
classification result for finite Heisenberg groups.

We define a new notion of cuspidality for representations of
GL_{n} over a finite quotient O_{k} of the ring of
integers O of a non-Archimedean local field F using geometric and
infinitesimal induction functors, which involve automorphism groups
G_{λ} of torsion O-modules. When n is a prime, we
show that this notion of cuspidality is equivalent to strong
cuspidality, which arises in the construction of supercuspidal
representations of GL_{n}(F). We show that strongly
cuspidal representations share many features of cuspidal
representations of finite general linear groups. In the function
field case, we show that the construction of the representations of
GL_{n}(O_{k}) for k≥ 2 for all n is
equivalent to the construction of the representations of all the
groups G_{λ}. A functional equation for zeta
functions for representations of GL_{n}(O_{k}) is
established for representations which are not contained in an
infinitesimally induced representation. All the cuspidal
representations for GL_{4}(O_{2}) are constructed.
Not all these representations are strongly cuspidal.

The Journal of Analysis, volume 17,
pages 73-86, 2009. (arXiv), (MathSciNet)

We develop a simple algebraic approach to the study of the Weil
representation associated to a finite abelian group. As a result,
we obtain a simple proof of a generalisation of a well-known
formula for the absolute value of its character. We also obtain a
new result about its decomposition into irreducible
representations. As an example, the decomposition of the Weil
representation of Sp_{2g}(Z/NZ) is described for odd N.

In this paper similarity classes of three by three matrices over
a local principal ideal commutative ring are analyzed. When the
residue field is finite, a generating function for the number of
similarity classes for all finite quotients of the ring is computed
explicitly.

Discrete and Continuous Dynamical Systems -
Series S, volume 2, no. 2, pages 337-348, June 2009. (arXiv), (from journal),
(MathSciNet)

We announce ultrametric analogues of the results of
Kleinbock-Margulis for shrinking target properties of semisimple
group actions on symmetric spaces. The main applications are
S-arithmetic Diophantine approximation results and logarithm laws
for buildings, generalizing the work of Hersonsky-Paulin on
trees.

Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid
are studied in the spirit of the treatment of the spherical
harmonics by Stein and Weiss. As a special case, a simple
self-contained proof of Laplace's integral for a Legendre function
is obtained.

Mathematics Newsletter of the Ramanujan Mathematical
Society, volume 16, pages 73-78, 2007. (pdf),
(djvu)

An element x of a finite group G is said to be p-regular if its
order is not divisible by p. Brauer gave several proofs of the fact
that the number of isomorphism classes of irreducible
representations of G over an algebraically closed field of
characteristic p is the same as the number of conjugacy classes in
G that consist of p-regular elements. One such proof is presented
here.

Let A be a local commutative principal ideal ring. We study the
double coset space of GL(n,A) with respect to the subgroup of upper
triangular matrices. Geometrically, these cosets describe the
relative position of two full flags of free primitive submodules of
A^{n}. If k is the length of the ring, we determine for
which of the pairs (n,k) the double coset space depend on the ring
in question. For n=3, we give a complete parametrisation of the
double coset space and provide estimates on the rate of growth of
the number of double cosets.

Bulletin of the Kerala Mathematics Association, Special
issue on Harmonic Analysis and Quantum Groups, December 2005. (arXiv),
(MathSciNet)

This article gives conceptual statements and proofs relating
parabolic induction and Jacquet functors on split reductive groups
over a non-Archimedean local field to the associated Iwahori-Hecke
algebra as tensoring from and restricting to parabolic subalgebras.
The main tool is Bernstein's presentation of the Iwahori-Hecke
algebra.

Let G be a split semisimple group over a finite field
F_{q}, F the field F_{q}(t) of
rational functions in t with coefficients in F_{q}
and A the adèles of F. We describe the irreducible
automorphic representations of G(A) which have non-zero
vectors invariant under Iwahori subgroups at two places and under
maximal compact subgroups at all other places in terms of the
irreducible square-integrable representations of an Iwahori-Hecke
algebra associated to G and the Satake isomorphism.

Let G be a split reductive group over a finite field
F_{q}. Let F=F_{q}(t) and let
A denote the adèles of F. We show that every double coset in
G(F)\G(A)/K has a representative in a maximal split torus of
G. Here K is the set of integral adelic points of G. When G ranges
over general linear groups this is equivalent to the assertion that
any algebraic vector bundle over the projective line is isomorphic
to a direct sum of line bundles.

Let G be a split adjoint group defined over
F_{q}, let
F_{q}(t), and let A be the
adèles of F. We describe the local constituents at two
points of automorphic representations of G in the discrete
part of L^{2}(G(F)\G(A))
which have vectors invariant under Iwahori subgroups at these two
points and are unramified at all other points.

Let G be a split semisimple group over a finite field
F_{q}, let F = F_{q} (t), and let
A denote the adèles of F. For all the irreducible
representations of G(A) occurring in the discrete part of
L^{2}(G(F)\G(A)) which have vectors invariant under
Iwahori subgroups at two places of F and maximal compact subgroups
at all other places, we describe the local constituents at those
two places in terms of the irreducible square integrable
representations of an Iwahori Hecke algebra. We include proofs of
certain well known results about the classification of principal
G-bundles on the projective line which we use in our
calculations.