Plethysm and cohomology representations of external and symmetric products

TL;DR

This paper develops refined formulas for the characters of cohomology representations of external products, generalizing previous symmetric and alternating power results, with applications to spaces with symmetries and an abstract plethysm calculus.

Contribution

It introduces generalized generating series formulas for cohomology characters of external products, extending prior work to Schur functors and equivariant contexts.

Findings

Refined generating series formulas for cohomology characters.

Generalization to Schur functors and symmetric sequences.

Application to spaces with symmetries and equivariant settings.

Abstract

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules on spaces such as (possibly singular) complex quasi-projective varieties. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant K\"{u}nneth formula to a more general generating series identity for abstract characters of tensor powers $\mathcal{V}^{\otimes n}$ of an element $\mathcal{V}$ in a suitable symmetric monoidal category $A$. This abstract approach applies directly also in the equivariant context for spaces with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms), as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in $A$.