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 Kunneth 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 $A$.

中文翻译：

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外部和对称产品的体积和上同调表示

我们证明了适当系数的外积的（虚拟）上同调表示的特征的精细生成级数公式，例如，可构造或相干滑轮的（复数），或空间上的混合霍奇模（的复数），例如（可能是奇异的）复拟-投射变体。这些公式概括了我们先前对此类系数的对称和交替幂的结果，并且也适用于其他 Schur 函子。这些结果的证明通过一个等变的 Kunneth 公式简化为一个更一般的生成级数恒等式，用于一个元素 $\mathcal{V}$ 的张量幂 $\mathcal{V}^{\otimes n}$ 的抽象特征合适的对称幺半群类别 $A$。这种抽象方法也直接适用于具有额外对称性的空间（例如，有限群作用，