Algebras of block triangular matrices turn out to be quite important in the study of PI algebras and their extremal properties. Let \(G\) be a finite Abelian group and let \(F\) be an algebraically closed field of characteristic 0. The main result of the paper under review is a complete description of all \(G\)-gradings on the algebras of block triangular matrices over \(F\).
Let \(A=UT(d_1,\dots,d_m)\) be the algebra of upper block triangular matrices over \(F\); here \(d_1,\dots,d_m\) are the sizes of the blocks on the main diagonal. Suppose \(A\) is \(G\)-graded. The authors prove that there is a decomposition \(d_i=tp_i\), \(i=1,\dots,m\), a subgroup \(H\) of \(G\) and an \(n\)-tuple \((g_1,\dots,g_n)\) of elements of \(G\) where \(n=p_1+\cdots+p_m\) such that \(A\) is isomorphic to \(M_t(F)\otimes UT(p_1,\dots,p_m)\) as \(G\)-graded algebras. Here \(M_t(F)\) is \(H\)-graded with a fine \(H\)-grading (that is, each homogeneous component is of dimension \(\leq 1\)). Also the grading on \(UT(p_1,\dots,p_m)\) is the elementary one defined by the \(n\)-tuple \((g_1,\dots,g_n)\).