  Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of
$k$ by $k$ matrices over the complex numbers and view it as a crossed product
with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the
regular representation and embedding $S_k$ in $M_k(\mathbb{C})$ in the usual
way. This induces a natural $G$-grading on $M_k(\mathbb{C})$ which we call a
crossed-product grading. We study the graded $*$-identities for
$M_k(\mathbb{C})$ equipped with such a crossed-product grading and the
transpose involution. To each multilinear monomial in the free graded algebra
with involution we associate a directed labeled graph. Use of these graphs
allows us to produce a set of generators for the $(T,*)$-ideal of identities.
It also leads to new proofs of the results of Kostant and Rowen on the standard
identities satisfied by skew matrices. Finally we determine an asymptotic
formula for the $*$-graded codimension of $M_k(\mathbb{C})$.

Rings and Algebras