Research Projects

Representation theory of groups is an important branch of mathematics which is well connected with geometry, number theory, algebraic combinatorics, physics, and even computer science. The building block representations of a group are given by irreducible representations. The multiplicity of an irreducible representation in a representation is a nonnegative integer counting the number of times the same irreducible representation occurs. A fundamental question in representation theory is to describe combinatorially or to find a closed positive formula computing the multiplicities of irreducible representations in a given representation. Now I elaborate two intrinsic ways that a representation of a group can occur and also along the way address the broad objectives of the proposal. 1. Given a group G and its subgroup H, a representation of G is also a representation of H, referred to as a restricted representation of H. These had appeared in classical works of Frobenius, Littlewood et al and ever since then have been extensively studied. The multiplicities of irreducible representations of H in a restricted representation of H are called restriction multiplicities. The restriction problem for the group G and subgroup H asks for explicit description of restriction multiplicities. The proposal concerns the restriction problem in two setups: a) for the general linear group and its subgroup of all permutation matrices; this was originated in a work of Littlewood and henceforth it has been of great importance in research due to various new/old reformulations and yet a very few explicit answers, b) for the symmetric group and its subgroup which is obtained by taking Kronecker product of permutation matrices. Both these restriction problems are deeply connected with symmetric functions which are central to the algebraic combinatorics. 2. The tensor product, a fundamental construction in the multilinear algebra, of two irreducible representations of a group G gives another representation of G which is often not irreducible. The multiplicities of irreducible representations in the tensor product of two irreducible representations of a fixed group are known as Kronecker multiplicities and notably these are of interest in physics too where they are referred as Clebsch--Gordon coefficients. The Kronecker problem for a group asks for explicit description of Kronecker multiplicities. Kronecker multiplicities for symmetric groups have also caught the attention of computer scientists working in geometric complexity theory. The main objective here is to study Kronecker multiplicities for a larger and important class of groups, namely complex reflection groups and these contain symmetric groups as special cases. A common theme to approach the above two broad objectives will be to establish various new Schur-Weyl dualities and discover new diagram algebras which will enable us to find explicit answers in both restriction and Kronecker problems.