A graph-theoretic approach for comparing dimensions of components in simply-graded algebras

TL;DR

This paper introduces a graph-theoretic method to analyze simple group-gradings of finite-dimensional complex algebras, establishing bounds on the structure and showing the maximality of the trivial component's dimension.

Contribution

It provides a novel graph-theoretic framework to compare components in simply-graded algebras, revealing structural bounds and properties.

Findings

Proves a lower bound on digraph edge combinations related to algebra grading

Shows the dimension of the trivial component is maximal in simple group-gradings

Establishes a connection between graph properties and algebraic structure

Abstract

Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that $|E\circ E^{\text{op}}\cup E^{\text{op}}\circ E|\geq |E|$ for any digraph $\Gamma =(V,E)$ without parallel edges, and deduce that for any simple group-grading, the dimension of the trivial component is maximal.