Structural matrix algebras, generalized flags and gradings

TL;DR

This paper establishes a connection between structural matrix algebras and generalized flags, classifies their gradings via group actions, and offers a new method to compute automorphism groups of such algebras.

Contribution

It introduces a novel correspondence between structural matrix algebras and generalized flags, and classifies gradings through group actions, providing a new approach to automorphism group computation.

Findings

Structural matrix algebra is isomorphic to endomorphism algebra of a generalized flag.

Gradings on the algebra correspond to group actions on associated combinatorial objects.

New method for computing automorphism groups of structural matrix algebras.

Abstract

We show that a structural matrix algebra $A$ is isomorphic to the endomorphism algebra of an algebraic-combinatorial object called a generalized flag. If the flag is equipped with a group grading, an algebra grading is induced on $A$. We classify the gradings obtained in this way as the orbits of the action of a double semidirect product on a certain set. Under some conditions on the associated graph, all good gradings on $A$ are of this type. As a bi-product, we obtain a new approach to compute the automorphism group of a structural matrix algebra.