Elementary and good group gradings of incidence algebras over partially-ordered sets with cross-cuts

摘要

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 .