Asymptotics for Capelli polynomials with involution

Abstract

Let be the free associative algebra with involution * over a field F of characteristic zero. We study the asymptotic behavior of the sequence of *-codimensions of the T-*-ideal of generated by the *-Capelli polynomials and alternanting on M + 1 symmetric variables and L + 1 skew variables, respectively. It is well known that, if F is an algebraic closed field of characteristic zero, every finite dimensional *-simple algebra is isomorphic to one of the following algebras: the algebra of k × k matrices with the transpose involution; the algebra of matrices with the symplectic involution; the direct sum of the algebra of h × h matrices and the opposite algebra with the exchange involution. We prove that the *-codimensions of a finite dimensional *-simple algebra are asymptotically equal to the *-codimensions of for some fixed natural numbers M and L. In particular: and Moreover the exact asymptotics of and are known and those of can be easily deduced.