$p$-adic dimensions in symmetric tensor categories in characteristic $p$

TL;DR

This paper introduces $p$-adic dimensions for objects in symmetric tensor categories over fields of characteristic $p$, revealing their properties, differences, and connections to $mbda$-rings and Brauer characters.

Contribution

It defines and studies $p$-adic dimensions in symmetric tensor categories, showing they can differ and take any $p$-adic integer value, extending the understanding of categorical dimensions.

Findings

$p$-adic dimensions can differ from each other.

They can take any value in $al{Z}_p$.

Connections to $mbda$-rings and Brauer characters are established.

Abstract

To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and $\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension $\text{dim}(X)$ of $X$, coinciding with the usual dimension when $X$ is a vector space. We study properties of $\text{Dim}_{\pm}(X)$, and in particular show that they don't always coincide with each other, and can take any value in $\mathbb{Z}_p$. We also discuss the connection of $p$-adic dimensions with the theory of $\lambda$-rings and Brauer characters.