Semiadditive Alternating Powers and Twisted Power Operation

TL;DR

This paper explores representations of symmetric groups in higher semiadditive categories, connecting character theory with monoidal categorifications and extending classical algebraic operations to new categorical contexts.

Contribution

It introduces a framework for representations in higher semiadditive categories, linking character theory with monoidal categorifications and extending classical algebraic operations.

Findings

Recovered the transchromatic character as monoidal characters

Developed explicit algorithms for character computation

Extended classical algebraic operations to higher categorical settings

Abstract

We study a class of representations of symmetric groups in higher semiadditive categories. For these representations in $\mathrm{Mod}^{\wedge}_{E_n}$, 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 $\lambda$-operations in $\mathrm{K}$-theory. We provide explicit computations in both the chromatic and higher categorical settings at low heights.