(arXiv)

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.

(arXiv)

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 $5$.

(arXiv)

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.

To appear in

(arXiv)

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

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.

(Contributions in Algebra and Algebraic Geometry, Edited by S. G. Dani and S. K. Jain).

(arXiv), (AMS website)

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.

(from journal), (arXiv).

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.

(arXiv)

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.

(arXiv), (journal website)

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.

(arXiv), (from journal), (slide presentation of parts)

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.

(free download from SLC website), (arXiv)

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.

(arXiv)

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.

Electronic booklet of Proceedings of International Congress of Women Mathematicians (ICWM 2014). 2014 ICWM Poster Abstract Number: 20140020.

(arXiv)

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.

(arXiv).

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 {

with C. P. Anilkumar.

Free downlaod from SLC website, code and data files.

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.

with T. Geetha.

(arXiv), (from journal)

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 $n$.

with Pooja Singla and Steven Spallone.

(arXiv), (from journal)

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.

with Jayadev S. Athreya and Anish Ghosh.

(open access from journal)

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.

with Wesley Calvert and Kunal Dutta.

(arXiv), (from de Gruyter)

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.

with Jayadev S. Athreya and Anish Ghosh.

(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.

with Umesh V. Dubey and Pooja Singla.

(arXiv), (from journal)

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.

with Kunal Dutta.

(arXiv), (from journal)

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.

with Kunal Dutta.

(arXiv), (from journal)

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.

(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.

(arXiv), (from journal)

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.

with M. K. Vemuri.

(arXiv), (from journal), (MathSciNet)

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.

(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).

with Thomas J. Haines and Robert E. Kottwitz.

(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.

with M. K. Vemuri.

(arXiv), (from journal), (MathSciNet)

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.

with Anne-Marie Aubert, Uri Onn and Alexander Stasinski.

(arXiv), (from journal), (MathSciNet)

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.

(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.

with Nir Avni, Uri Onn and Leonid Vaserstein.

(arXiv), (from journal), (MathSciNet)

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.

with J. S. Athreya and Anish Ghosh.

(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.

with M. K. Vemuri.

(arXiv), (from journal), (MathSciNet)

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.

(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.

with Uri Onn and Leonid Vaserstein.

(arXiv), (from journal), (MathSciNet)

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.

(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.

(pdf), (from journal), (MathSciNet), (erratum), (published version of erratum)

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.

(arXiv), (SpringerLink), (IAS site), (MathSciNet)

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.

(from journal), (MathSciNet)

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.

The University of Chicago PhD thesis, June 2001.

(pdf).

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.