The Structure of the Grothendieck Rings of Wreath Product Deligne Categories and their Generalisations

Given a tensor category C over an algebraically closed field of characteristic zero, we may form the wreath product category W-n(C). It was shown in [10] that the Grothendieck rings of these wreath product categories stabilise in some sense as n -> infinity. The resulting "limit" ring, G(infinity)(Z)(C), is isomorphic to the Grothendieck ring of the wreath product Deligne category S-t(C) as defined by [9] (although it is also related to FIG-modules). This ring only depends on the Grothendieck ring G(C). Given a ring R that is free as a Z-module, we construct a ring G(infinity)(Z)(R) that specialises to G(infinity)(Z)(C) when R = G(C). We give a description of G(infinity)(Z)(R) using generators very similar to the basic hooks of [5]. We also show that G(infinity)(Z)(R) is a lambda-ring wherever R is and that G(infinity)(Z)(R) is (unconditionally) a Hopf algebra. Finally, we show that G(infinity)(Z)(R) is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in (W circle times(Z) R)(x), where W is the ring of Big Witt Vectors.