On the multiplicities of the central cocharacter of algebras with polynomial identities
Wesley Quaresma Cota, Thais Silva do Nascimento
TL;DR
The paper studies the multiplicities of central cocharacters for algebras with polynomial identities over characteristic zero, using cocharacters $\chi_n(A)$, central cocharacters $\chi_n^z(A)$ and proper central cocharacters $\chi_n^\delta(A)$ together with colength sequences. It develops a $GL_m$-representation framework and constructs explicit algebras with small colength to compute multiplicities and cocharacters. It then delivers a complete PI-equivalence classification of varieties with bounded colength (up to a constant) and a parallel classification for algebras with small central colength, identifying finite lists of base algebras plus nilpotent components. These results advance understanding of Amitsur-type growth phenomena, provide explicit central-cocharacter data for many examples, and furnish a toolkit for analyzing PI-equivalence via colength and central-colength constraints.
Abstract
For an associative algebra $A$ over a field of characteristic zero, let $P_n(A)$ and $P_n^z(A)$ denote the spaces of multilinear polynomials of degree $n$ modulo the polynomial identities and the central polynomials of $A$, respectively. We also write $Δ_n(A)$ for the space of multilinear central polynomials of degree $n$ modulo the polynomial identities of $A$. The corresponding sequences of colengths, central colengths and proper central colengths measure the number of irreducible components in the $S_n$-module decompositions of $P_n(A)$, $P_n^z(A)$ and $Δ_n(A)$, respectively. In this paper, we investigate several examples of PI-algebras and explicitly describe their cocharacter, central cocharacter and proper central cocharacter sequences. As a consequence, we obtain a complete classification, up to PI-equivalence, of all algebras whose sequences of colengths and central colengths are bounded by a constant.