Algebras with additional structures and multiplicities bounded by a constant

TL;DR

The paper investigates when the $\langle n \rangle$-cocharacters of $G$-graded and $(G,\*)$-algebras have uniformly bounded multiplicities. It develops a representation-theoretic framework using $G$-polynomials, $T_G$-ideals, and highest weight vectors to connect multiplicities with specific $G$-identities and exclusion of certain graded algebras from the variety. For $G$-graded algebras, it extends MRZ-type results by showing multiplicities are bounded iff $UT_2^{g}$ are excluded from the variety and certain degree-$n$ identities hold for all $g\in G$. For $(G,\*)$-algebras, it characterizes the stronger condition of multiplicities bounded by $1$ via a concrete list of $(G,*)$-identities, with proofs leveraging highest weight decompositions and multitableaux, and provides illustrative examples in Grassmann-type settings. The results unify and extend classical cocharacter bounds in the graded and graded-involution contexts, offering a toolkit for identifying when bounded multiplicities arise from structural identities.

Abstract

$G$ be a finite group and $A$ a $G$-graded algebra over a field $F$ of characteristic zero. We characterize the varieties of $G$-graded algebras such that the multiplicities $m_{\langle λ\rangle}$ appering in the $\langle n \rangle $-cocharacters of $A$ are bounded by a constant, in terms of $G$-identities. If $A$ is endowed with a graded involution $\ast$, i.e. if $A$ is a $(G,\ast)$-algebra, we characterize the varieties of $(G,*)$-algebras whose multiplicities in the sequence of $\langle n\rangle$-cocharacters of $A$ are bounded by $1$ by showing a list of $(G,\ast)$-polynomial identities satisfied by such varieties.

Theorems & Definitions (46)

• Example 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Remark 3.1
• Theorem 3.2