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arXiv · 2508.00464

Cocharacters of generalized polynomial identities

Sebastiano Argenti·Giovanni Busalacchi
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Abstract

In this paper we extend the cocharacter theory to generalized identities of $W$-algebras. We prove that the Hilbert series of the relatively free $W$-algebra admits an expansion in terms of Schur functions whose coefficients coincide with generalized cocharacter multiplicities. Moreover, we prove analogues of the Hook and Strip theorems for $W$-algebras and we derive growth bounds for generalized codimension and colenght sequences. Finally, we establish that every variety $\mathcal{V}$ of $W$-algebras is generated by the Grassmann envelope of a finitely generated $W$-superalgebra, and if $\mathcal{V}$ satisfies a generalized Capelli set, then it is generated by a finitely generated $W$-algebra.

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Sebastiano Argenti, Giovanni Busalacchi. 2025-08-01. Cocharacters of generalized polynomial identities. https://arxiv.org/abs/2508.00464

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