A graph theoretic approach to graded identities for matrices

We consider the algebra Mk(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by imbedding G in the symmetric group Sk via the regular representation and imbedding Sk in Mk(C) in the usual way. This induces a natural G-grading on Mk(C) which we call a crossed-product grading. This grading is the so-called elementary grading defined by any k-tuple (g1,g2,…,gk) of distinct elements gi∈G. We study the graded polynomial identities for Mk(C) equipped with a crossed-product grading. To each multilinear monomial in the free graded algebra we associate a directed labeled graph. This approach allows us to give new proofs of known results of Bahturin and Drensky on the generators of the T-ideal of identities and the Amitsur–Levitsky theorem.Our most substantial new result is the determination of the asymptotic formula for the G-graded codimension of Mk(C).