Asymptotics for Capelli Polynomials with Involution

Abstract

Let $F\langle X, \ast \rangle$ be the free associative algebra with involution $\ast$ over a field $F$ of characteristic zero. We study the asymptotic behavior of the sequence of $\ast$-codimensions of the T-$\ast$-ideal $\Gamma_{M+1,L+1}^\ast$ of $F\langle X, \ast \rangle$ generated by the $\ast$-Capelli polynomials $Cap^\ast_{M+1} [Y,X]$ and $Cap^\ast_{L+1} [Z,X]$ 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 $\ast$-simple algebra is isomorphic to one of the following algebras: \begin{itemize} \item [$\cdot$]$(M_{k}(F),t)$ the algebra of $k \times k$ matrices with the transpose involution; \item [$\cdot$]$(M_{2m}(F),s)$ the algebra of $2m \times 2m$ matrices with the symplectic involution; \item [$\cdot$]$(M_{h}(F)\oplus M_{h}(F)^{op}, exc)$ the direct sum of the algebra of $h \times h$ matrices and the opposite algebra with the exchange involution. \end{itemize} We prove that the $\ast$-codimensions of a finite dimensional $\ast$-simple algebra are asymptotically equal to the $\ast$-codimensions of $\Gamma_{M+1,L+1}^\ast$, for some fixed natural numbers $M$ and $L$. In particular: $$ c^{\ast}_n(\Gamma^{\ast}_{\frac{k(k+1)}{2} +1,\frac{k(k-1)}{2} +1})\simeq c^{\ast}_n((M_k(F),t)); $$ $$ c^{\ast}_n(\Gamma^{\ast}_{m(2m-1)+1,m(2m+1)+1})\simeq c^{\ast}_n((M_{2m}(F),s)); $$ and $$ c^{\ast}_n(\Gamma^{\ast}_{h^2+1,h^2+1})\simeq c^{\ast}_n((M_{h}(F)\oplus M_{h}(F)^{op},exc)). $$