Induced Good Gradings of Structural Matrix Rings

TL;DR

This paper studies how to assign group-based gradings to structural matrix rings defined over directed graphs, identifying conditions for the existence of such gradings and counting their equivalence classes.

Contribution

It introduces the concept of good gradings over preordered directed graphs and characterizes when G-grading sets exist, extending previous results and providing enumeration methods.

Findings

G-grading sets exist for transitive graphs when G is a prime order group.

Counterexamples show G-grading sets may not exist for certain cyclic groups of even order.

The paper counts elementary gradings of matrix rings over arbitrary fields.

Abstract

Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group $G$, a $G$ -grading set is a set of arrows with the property that any assignment of these arrows to elements of $G$ uniquely determines an induced good grading. One of our main results is that a $G$-grading set exists for any transitive directed graph if $G$ is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have $G$-grading sets for any cyclic group $G$ of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.