Good Matrix Gradings from Directed Graphs

Induced Good Gradings of Structural Matrix Rings

“…Instead we may use preordered directed graphs to define structural matrix rings as in Section 3.14 of [10]. However, we refer to directed edges as "arrows," which has the advantage of not overusing the term "edges" when referring to the underlying undirected graph (see [20]) or the edges of a Hasse Diagram, as in Section 2. It is also consistent with [9], which considers gradings of path algebras that are defined using directed graphs containing vertices and "arrows.…”

“…Then S = S (D, R) is G-graded if there is a direct sum S = g∈G S g , as a group under the addition of S. If D is preordered, S is a G-graded ring if S g S h ⊆ S g+h for all g, h ∈ G. A G-grading of S (D, R) is called good if the standard unit matrices are homogeneous. We recall some definitions and notation from [20]. A transitive triple in D is an ordered triple of vertices contained in…”

“…More recent work has resulted in a classification of gradings for many classical simple Lie algebras (see [1], [8], [12] and [13]). A method to induce good gradings on Lie superalgebras, Lie algebras, and matrix algebras using directed graphs was developed in [5], [10], [11], and [20]. An analogous method to induce good and elementary gradings on incidence rings was introduced in [14] and extended in [18].…”

“…As explained in Theorem 1.6, for a finite abelian group G the number of induced gradings divides |G| s , where s is the number of arrows that are not loops of the directed graph. The construction of induced good gradings was used in [20] to count induced good gradings of certain structural matrix rings. These results also give the number of induced good gradings of certain kinds of incidence rings (see [14], [15], [18], and [19]).…”

An Alternative Construction to the Transitive Closure of a Directed Graph

“…In this section we follow conventions set in [6]. Let M n (R) denote the set of square matrices over R, an arbitrarily chosen ring with unit.…”