Cocharacters of generalized polynomial identities

TL;DR

This paper extends cocharacter theory to generalized identities of W-algebras, providing new structural insights, growth bounds, and generation results for varieties of W-algebras.

Contribution

It introduces a generalized cocharacter theory for W-algebras, proves analogues of classical theorems, and characterizes varieties generated by finitely generated W-superalgebras.

Findings

Hilbert series expansion in Schur functions with multiplicities

Analogues of Hook and Strip theorems for W-algebras

Growth bounds for codimension and colength sequences

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.