Matrix algebras with involution and central polynomials (Q1604382)

In the paper under review the author continues her investigation on polynomial identities and central polynomials for matrix algebras with involution in the spirit of her recent papers [Bull. Aust. Math. Soc. 60, No. 3, 469-477 (1999; Zbl 0942.16028); Commun. Algebra 28, No. 10, 4879-4887 (2000; Zbl 0961.16013)]. As in the considerations of \textit{E. Formanek} [J. Algebra 23, 129-132 (1972; Zbl 0242.15004)] and in a preprint by \textit{G. M. Bergman} written in the early 80's which never appeared in journal form, the author makes a correspondence between the polynomial \(g(t_1,\dots,t_{n+1})=\sum\alpha_pt_1^{p_1}\cdots t_{n+1}^{p_{n+1}}\) in \(n+1\) commuting variables, \(\alpha_p\in K\), \(K\) being a field of characteristic 0, and the polynomial \(v(g)(x,y_1,\dots,y_n)=\sum\alpha_px^{p_1}y_1x^{p_2}y_2\cdots y_nx^{p_{n+1}}\) in noncommuting variables. Then she studies polynomials of the form \(f=\sum v(g_i)(x,y_{i_1},\dots,y_{i_n})\), \(g_i\in K[t_1,\dots,t_{n+1}]\), which are central when evaluated over the skew-symmetric elements in the \(2n\times 2n\) matrix algebra with symplectic involution. The author constructs classes of new examples of such central polynomials of arbitrary high degree for \(n=2\) and \(n=3\). In particular, these examples contain polynomials of minimal possible degree.