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

TL;DR

This paper studies the Grothendieck rings of wreath product Deligne categories, showing their structure as Hopf algebras and their relation to Witt vectors, providing new algebraic descriptions and generalizations.

Contribution

It introduces a new construction of Grothendieck rings for wreath product categories based on a free ring R, and proves these rings are λ-rings and Hopf algebras, extending previous stabilization results.

Findings

The Grothendieck rings are isomorphic to distributions on formal neighborhoods in a Witt vector-based group.

These rings are λ-rings when R is a λ-ring.

They are explicitly described using generators similar to basic hooks.

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.