arXiv subjects

K N Raghavan

Publications and source records attributed to K N Raghavan.

3 recordsLinked to original sources

arXiv2025

Kronecker Coefficients and Simultaneous Conjugacy Classes

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.

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arXiv2019

A study of Kostant-Kumar modules via Littelmann paths

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be of independent interest. We also discuss the existence of PRV components and generalised PRV components in KK modules. Specialising to the case of the special linear Lie algebra, we record a decomposition rule for KK modules in terms of Littlewood-Richardson tableaux. In this connection, we present a new procedure to determine the permutation that is the initial element of the minimal standard lift of a semi-standard Young tableau. The appendix, necessitated by the derivation of the tableau decomposition rule, deals with standard concatenations of Lakshmibai-Seshadri paths of arbitrary shapes, of which semi-standard Young tableaux form a very special case.

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arXiv2019

A relationship between Gelfand-Tsetlin bases and Chari-Loktev bases for irreducible finite dimensional representations of special linear Lie algebras

We consider two bases for an arbitrary finite dimensional irreducible representation of a complex special linear Lie algebra: the classical Gelfand-Tsetlin basis and the relatively new Chari-Loktev basis. Both are parametrized by the set of (integral Gelfand-Tsetlin) patterns with a fixed bounding sequence determined by the highest weight of the representation. We define the "row-wise dominance" partial order on this set of patterns, and prove that the transition matrix between the two bases is triangular with respect to this partial order. We write down explicit expressions for the diagonal elements of the transition matrix.

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