Elementary and Good Group Gradings of Incidence Algebras over Partially-Ordered Sets with Cross-Cuts

Abstract

Let G be a group, X a locally-finite partially-ordered set, and F a field. We provide an algorithmic method for finding all good G-gradings of the incidence algebra I(X,F) when X has a cross-cut of length one or two. In these cases, we show that the good gradings are determined by a “freeness” property. It is shown that the number of good gradings of the incidence algebra I(X,F), when X has a cross-cut of length one or two, depends only on the size of G and not on its structure, but this is no longer true when the shortest cross-cut of X has length greater than two. If X has a cross-cut of length one, then every good grading of I(X,F) is an elementary grading, but, when the shortest cross-cut of X has length greater than one, there may exist good gradings of I(X,F) that are not elementary gradings. Finally, we establish bounds on the number of good gradings of I(X,F) for any finite partially-ordered set X.