Group Gradings on Matrix Algebras

Abstract Let Φ be an algebraically closed field of characteristic zero, G a finite, not necessarily abelian, group. Given a G -grading on the full matrix algebra A = M n (Φ), we decompose A as the tensor product of graded subalgebras A = B ⊗ C, B ≅ M p (Φ) being a graded division algebra, while the grading of C ≅ M q (Φ) is determined by that of the vector space Φ n . Now the grading of A is recovered from those of A and B using a canonical “induction” procedure.