Kronecker Coefficients and Simultaneous Conjugacy Classes

with Jyotirmoy Ganguly, Digjoy Paul, K N Raghavan, and Velmurugan S

A Kronecker coefficient is the multiplicity of an irreducible representation of a finite group G in a tensor product of irreducible representations. We define Kronecker Hecke algebras and use them as a tool to study Kronecker coefficients in finite groups. We show that the number of simultaneous conjugacy classes in a finite group G is equal to the sum of squares of Kronecker coefficients, and the number of simultaneous conjugacy classes that are closed under elementwise inversion is the sum of Kronecker coefficients weighted by Frobenius-Schur indicators. We use these tools to investigate which finite groups have multiplicity-free tensor products. We introduce the class of doubly real groups, and show that they are precisely the real groups which have multiplicity-free tensor products. We show that non-Abelian groups of odd order, non-Abelian finite simple groups, and most finite general linear groups do not have multiplicity-free tensor products.

On the Existence of Elementwise Invariant Vectors in Representations of Symmetric Groups

with Amrutha P and Velmurugan S

We determine when a permutation with cycle type $\mu$ admits a non-zero invariant vector in the irreducible representation $V_\lambda$ of the symmetric group. We find that a majority of pairs $(\lambda,\mu)$ have this property, with only a few simple exceptions.

Representation zeta functions of arithmetic groups of type $A_2$ in positive characteristic

with Uri Onn and Pooja Singla

We prove two conjectures regarding the representation growth of groups of type $A_2$. The first, conjectured by Avni, Klopsch, Onn and Voll, regards the uniformity of representation zeta functions over local complete discrete valuation rings. The second is the Larsen--Lubotzky conjecture on the representation growth of irreducible lattices in groups of type $A_2$ in positive characteristic assuming Serre's conjecture on the congruence subgroup problem.

Enumeration of Anti-Invariant Subspaces and the q-Hermite Catalan Matrix

with Samrith Ram, Adv. Appl. Math. 154, 102654, 2024. (arXiv version)

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the $q$-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

Representation Theory: A Combinatorial Viewpoint

Cambridge University Press, 2015.