A graph theoretic approach to graded identities for matrices

Abstract

We consider the algebra M k (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 S k via the regular representation and imbedding S k in M k (C) in the usual way. This induces a natural G−grading on M k (C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g 1 , g 2 , . . . , g k ) of distinct elements g i ∈ G. We study the graded polynomial identities for M k (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 M k (C).

Graded polynomial identities for matrices with the transpose involution

“…But if they have the same graph then the ending vertices of the path for m and the path for α(m) must be the same: it is either e or the only vertex other that e at which there are an odd number of edges either beginning or ending there. The group value of this final vertex is then the common value for condition (1).…”

“…Remarks: (1) Notice that these generators are independent of the group; they refer to the e-component only.…”

“…The third application is an asymptotic formula for the codimension of graded * -identities for M k (C). (For a discussion of the notion of codimension and its history see the introduction to our paper [1].) To state the result we let P G n , for each positive integer n, be the vector space of strongly multilinear polynomials of degree n: P G n = span{x ǫ 1 σ(1),g 1 x ǫ 2 σ(2),g 2 • • • x ǫn σ(n),gn | σ ∈ S n , g 1 , g 2 , .…”

A graph-theoretic approach for comparing dimensions of components in simply-graded algebras

“…Simply-graded algebras are vastly investigated, e.g. see [1][2][3][4][5][6][7][8]. Let (1) be a Gsimple grading.…”

“…By removing the vertices {v s } s∈F as well as their corresponding edges we get a new graph Γ = (V , E ) with n − j − 1 vertices, which satisfies the theorem by the induction assumption. That is,(7) |T (Γ )| ≥ |E |.…”