Stable characters for symmetric groups and wreath products

Abstract

Given a Hopf algebra R, the Grothendieck group of C = R-mod inherits the structure of a ring. We define a ring [mathematical equation]), which is "the [mathematical equation] limit" of the Grothendieck rings of modules for the wreath products [mathematical equation]; it is the Grothendieck group of a certain wreath product Deligne category. The construction yields a basis of [mathematical equation] corresponding to irreducible objects. The structure constants of this basis are stable tensor product multiplicities for the wreath products [mathematical equation]. We generalise [mathematical equation], allowing an arbitrary ring to be substituted for the Grothendieck ring of C. Aside from being a Hopf algebra, [mathematical equation] is the algebra of distributions on a certain affine group scheme. In the special case where C is the category of vector spaces (over C, say), [mathematical equation] is the ring of symmetric functions. The basis obtained by our construction is the family of stable Specht polynomials, which is closely related to the problem of calculating restriction multiplicities from [mathematical equation]. We categorify the stable Specht polynomials by producing a resolution of irreducible representations of S[subscript n] by modules restricted from [mathematical equation].