Classifying good gradings on structural matrix algebras

ABSTRACT

Let k be a field and let $\rho$ be a partial order on the set $\{1,\dots ,n\}$. If G is a group, we classify the good G-gradings on the associated structural matrix algebra $M(\rho ,k)$ as the orbits of the action of the automorphism group of $\rho$ on a certain set of functions. We explicitly count the number of isomorphism types of good gradings in some cases where G is finite and the graph associated with $\rho$ has a cyclic associated undirected graph.

KEYWORDS:

• structural matrix algebra
• group graded algebra
• poset
• directed graph
• orbits of a group action

AMS SUBJECT CLASSIFICATIONS:

• 16S50
• 16W50
• 06A06

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the second author was supported by a grant of Ministry of Research and Innovation, CNCS – UEFISCDI, [project number PN-III-P4-ID-PCE-2016-0065], within PNCDI III. The third author was supported by the National Research Foundation of South Africa [grant number 81141]. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and therefore the National Research Foundation does not accept any liability in regard thereto.