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

Research paper by **Christopher Ryba**

Indexed on: **25 Oct '18**Published on: **25 Oct '18**Published in: **arXiv - Mathematics - Representation Theory**

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#### Abstract

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 $n \to
\infty$. The resulting "limit" 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 $\mathcal{G}_\infty^{\mathbb{Z}}(R)$
which specialises to $\mathcal{G}_\infty^{\mathbb{Z}}(\mathcal{C})$ when $R =
\mathcal{G}(\mathcal{C})$. We give a description of
$\mathcal{G}_\infty^{\mathbb{Z}}(R)$ using generators very similar to the basic
hooks of \cite{Nate}. We also show that $\mathcal{G}_\infty^{\mathbb{Z}}(R)$ is
a $\lambda$-ring wherever $R$ is, and that $\mathcal{G}_\infty^{\mathbb{Z}}(R)$
is (unconditionally) a Hopf algebra. Finally we show that
$\mathcal{G}_\infty^{\mathbb{Z}}(R)$ 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.