Research Projects

This project focuses on the study of restriction coefficients in the representation theory of finite groups—specifically, the positivity of the coefficients that arise when restricting irreducible representations from a group to its subgroup.More precisely, for an irreducible representation U of H, a subgroup of G, and an irreducible representation V of G, we say that U appears positively in V if the restriction of V from G to H contains U. In this case we also say that restriction coefficient of V with respect to U is positive. The positivity (i.e., non-zero nature) of these coefficients encodes important structural information about the interaction between G and ??, and has deep connections with character theory, branching rules, and modular representation theory. One of the very important conjecture in this direction is Mackay-conjecture (now a theorem due to Cabanes and Spath) and its refinements. In the big umbrella of restriction coefficients, we consider familiar (and important) subgroups and try to solve the problem of positivity of the restriction coefficients. The first one is a question of Tiep and Zalesski where the subgroup H is cyclic subgroups of G. Problem 1. Let g be an element of G and (ρ,V) be an irreducible representation of G. Let m(g) denote the order of the element g in G/ZG. Then classify pairs (g,V) such that the degree of the minimal polynomial of ρ(g) is strictly less than m(g) and strictly greater than 1. In the first instance, answer this question when m(g) is a prime power. They have also mentioned that the priority to this question to the groups which are close to simple. The second problem is the conjecture of Giannelli and Navarro concerning Sylow-p-subgrpups. Conjecture 1. Let (ρ,V) be a representation of G with degree divisible by a prime p. If the restriction of V to a sylow-p-subgroup P of G contains a linear character, then it contains at least p different linear characters of P. We solved the problem 1 for Symmetric and Alternating groups. The conjecture 1 was verified for symmetric and alternating groups. Encouraged by these results, we are very close to provide a complete solution to the problem 1 and Conjecture 1 for the double cover of symmetric and alternating groups. The problem 1 is , in general, difficult when working with modular representations. The best result we have in this direction is due to Thompson who showed the decomposition of the restriction of irreducible representations of Sn to the cyclic subgroup H= ⟨(1 2 ... n)⟩when n is a power of prime q (less than 2p). With the ideas developed while working in characteristic 0, we are hoping to extend Thompson’s result to all n in the near future. Once this is done, we will try to provide a solution to Problem 1 for the symmetric groups in characteristic p. We heavily use SAGE and GAP to make conjectures.