Signatures of Stable Multiplicity Spaces in Restrictions of Representations of Symmetric Groups

Representation theory is a way of studying complex mathematical structures such as groups and algebras by mapping them to linear actions on vector spaces. Recently, Deligne proposed a new way to study the representation theory of finite groups by generalizing the collection of representations of a sequence of groups indexed by positive integer rank to an arbitrary complex rank, creating an abelian tensor category. In this project, we focused on the case of the symmetric groups Sn, the groups of permutations of n objects. Elements of the Deligne category Rep St can be constructed by taking a stable sequence of Sn representations for increasing n and interpolating the associated formulas to an arbitrary complex number t. In this project, we studied the case of restriction multiplicity spaces Vλ,ρ, counting the number of copies of an irreducible representation Vρ of Sn−k in the restriction Res Sn Sn−k Vλ of an irreducible representation of Sn. We found formulas for norms of orthogonal basis vectors in these spaces, and ultimately for signatures (the number of basis vectors with positive norm minus the number with negative norm), an invariant that multiplies over tensor products and has important combinatorial connections.