p-adic dimensions in symmetric tensor categories in characteristic p

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±(X), and in particular show that they don’t always coincide with each other, and can take any value in Z [subscript]p. We also discuss the connection of p-adic dimensions with the theory of λ-rings and Brauer characters. ©2018 Keywords: tensor categories; symmetric monoidal categories