Semiadditive Alternating Powers and Twisted Power Operation

We study a class of representations of symmetric groups in higher semiadditive categories. For these representations in ${\mathrm{M}\mathrm{o}\mathrm{d}}_{E_n}^{\wedge }$, 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 $(\mathrm{\infty },n)$-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.