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

Given a tensor category $\mathcal{C}$ over an algebraically closed field of characteristic zero, we may form the wreath product category ${\mathcal{W}}_n(\mathcal{C})$. It was shown in \cite{Ryba} that the Grothendieck rings of these wreath product categories stabilise in some sense as $n \to \infty$\mathrm{.}Theresulting\mathrm{"}limit\mathrm{"}ring,$$\mathcal{G}$\infty^{$\mathbb{Z}$}($\mathcal{C}$)$, is isomorphic to the Grothendieck ring of the wreath product Deligne category $S_t(\mathcal{C})$ as defined by \cite{Mori}. This ring only depends on the Grothendieck ring $\mathcal{G}(\mathcal{C})$. Given a ring $R$ which is free as a $\mathbb{Z}$-module, we construct a ring ${\mathcal{G}}_{\mathrm{\infty }}^{\mathbb{Z}}(R)$ which specialises to ${\mathcal{G}}_{\mathrm{\infty }}^{\mathbb{Z}}(\mathcal{C})$ when $R = $\mathcal{G}$($\mathcal{C}$)$\mathrm{.}Wegiveadescriptionof$$\mathcal{G}$\infty^{$\mathbb{Z}$}®$ using generators very similar to the basic hooks of \cite{Nate}. We also show that ${\mathcal{G}}_{\mathrm{\infty }}^{\mathbb{Z}}(R)$ is a $\lambda$-ring wherever $R$ is, and that ${\mathcal{G}}_{\mathrm{\infty }}^{\mathbb{Z}}(R)$ is (unconditionally) a Hopf algebra. Finally we show that ${\mathcal{G}}_{\mathrm{\infty }}^{\mathbb{Z}}(R)$ is isomorphic to the Hopf algebra of distributions on the formal neighbourhood of the identity in $(W{\otimes }_{\mathbb{Z}}R{)}^{\times }$, where $W$ is the ring of Big Witt Vectors.