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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 $A$.