Asymptotics for Capelli polynomials with involution

Abstract

Let F 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 Γ∗ M+1,L+1 of F generated by the ∗-Capelli polynomials Cap∗ M+1[Y, X] and Cap∗ L+1[Z, X] 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: · (Mk(F ), t) the algebra of k × k matrices with the transpose involution; · (M2m(F ), s) the algebra of 2m × 2m matrices with the symplectic involution; · (Mh(F ) ⊕ Mh(F )op, exc) 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 Γ∗ M+1,L+1, for some fixed natural numbers M and L. In particular: c∗n(Γ∗k(k+1)2 +1, k(k2−1) +1) ≃ c∗ n((Mk(F ), t)); c∗n(Γ∗ m(2m−1)+1,m(2m+1)+1) ≃ c∗ n((M2m(F ), s)); and c∗n(Γ∗ h2+1,h2+1) ≃ c∗ n((Mh(F ) ⊕ Mh(F )op, exc)) Moreover the exact asymptotics of c∗n((Mk(F),t)) and c∗n((M2m(F),s)) are known and those of (Mh(F)⊕Mh(F)op,exc) can be easily deduced.