On the number of gradings on matrix algebras

Abstract

We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group G, the sequence of the numbers E(G,m) of isomorphism classes of elementary G-gradings on the algebra Mm(F) of m×m matrices with entries in a field F characterizes G. A formula for the number of isomorphism classes of gradings by a finite abelian group on an algebra of upper block-triangular matrices over an algebraically closed field, with mild restrictions on its characteristic, is also provided. Finally, if G is a finite abelian group, F is an algebraically closed field and N(G,m) is the number of isomorphism classes of G-gradings on Mm(F) we prove that N(G,m)∼1|G|!m|G|−1∼E(G,m).