Polynomial identities for matrices symmetric with respect to the symplectic involution. (Q420664)

Let \(F\) be an arbitrary field and let \(H_{2m}(F,s)\) be the vector space of \(2m\times 2m\) matrices over \(F\) which are symmetric with respect to the symplectic involution, \(m>1\). In the paper under review the author studies the polynomial identities of \(H_{2m}(F,s)\). A theorem of Rowen gives that \(H_{2m}(F,s)\) satisfies the standard identity of degree \(4m-2\) and \(4m-2\) is the minimal degree of the known polynomial identities for \(H_{2m}(F,s)\). Now the author constructs a new polynomial identity of degree \(4m-3\) which holds for \(H_{2m}(F,s)\). As a consequence, combining with the theorem of Rowen, the author obtains two new identities of degree \(4m-2\) with sum equal to the standard identity. The paper closes with several interesting comments and open problems concerning the polynomial identities of minimal degree for \(H_{2m}(F,s)\).