*-Graded Capelli polynomials and their asymptotics

Abstract

Let [Formula: see text] be the free *-superalgebra over a field [Formula: see text] of characteristic zero and let [Formula: see text] be the [Formula: see text]-ideal generated by the set of the *-graded Capelli polynomials [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] alternating on [Formula: see text] symmetric variables of homogeneous degree zero, on [Formula: see text] skew variables of homogeneous degree zero, on [Formula: see text] symmetric variables of homogeneous degree one and on [Formula: see text] skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of *-graded codimensions of [Formula: see text] In particular, we prove that the *-graded codimensions of the finite dimensional simple *-superalgebras are asymptotically equal to the *-graded codimensions of [Formula: see text], for some fixed natural numbers [Formula: see text] and [Formula: see text].