On the colength sequence of algebras with graded involution

TL;DR

This paper investigates the structure of algebras with graded involution, providing explicit decompositions of their cocharacters and classifying varieties with bounded colength sequences.

Contribution

It offers an explicit description of cocharacter decompositions for $(G,*)$-algebras and classifies varieties with bounded colengths.

Findings

Explicit decomposition of $ abla$-cocharacters for key $(G,*)$-algebras

Classification of varieties with bounded colength sequences

Insights into the structure of algebras with graded involution

Abstract

In recent years, many results have been established regarding classifications of varieties whose colength sequences are bounded by a fixed constant. In this work, we explore this theme in the setting of algebras endowed with a graded involution, called $(G,*)$-algebras. We give an explicit description of the decomposition of the $\langle n \rangle$-cocharacter for some important $(G,*)$-algebras $A$, for every $\langle n \rangle=(n_1, \ldots, n_{2t})$. For each algebra $A$, the $n$th colength is defined as the number of irreducible components that appear in these decompositions. Our aim is to classify varieties whose $n$th colengths are bounded by a fixed constant.