Semiadditive Alternating Powers and Twisted Power Operation

Abstract

We study a class of representations of symmetric groups in higher semiadditive categories. For these representations in ${\mathrm{Mod}}_{E_n}^{\wedge }$, the transchromatic character of Hopkins--Kuhn--Ravenel and Stapleton is recovered as a sequence of monoidal characters on suitable categorifications, giving an explicit algorithm for its computation, and relating it to the iterated monoidal character in $(\infty ,n)$-categories. These representations also give rise to notions of alternating powers and power operations in semiadditive categories, extending the classical alternating powers and $\hat{I}»$-operations in $\mathrm{K}$-theory. We provide explicit computations in both the chromatic and higher categorical settings at low heights.

Small mistakes and typos were fixed. 49 pages. Comments are welcome!