Multialternating Jordan polynomials and codimension growth of matrix algebras

Let R be the Jordan algebra of k × k matrices over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables of order k2 and we prove that f is not a polynomial identity of R. We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f, we are able to prove that the exponential rate of growth of the sequence of Jordan codimensions of R in precisely k2.