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

### Pavel Etingof

Massachusetts Institute of Technology, Cambridge, USA### Nate Harman

University of Chicago, USA### Victor Ostrik

University of Oregon, Eugene, USA

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## Abstract

To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers Dim$_+(X)$ and Dim$_-(X)$ whose reduction modulo $p$ is the categorical dimension dim$(X)$ of $X$, coinciding with the usual dimension when $X$ is a vector space. We study properties of 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.

## Cite this article

Pavel Etingof, Nate Harman, Victor Ostrik, $p$-adic dimensions in symmetric tensor categories in characteristic $p$. Quantum Topol. 9 (2018), no. 1, pp. 119–140

DOI 10.4171/QT/104