Asymptotics for Capelli Polynomials with Involution

Abstract

Let $F⟨X,\ast ⟩$ 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 $F⟨X,\ast ⟩$ generated by the $\ast$-Capelli polynomials $Ca{p}_{M+1}^{\ast }[Y,X]$ and $Ca{p}_{L+1}^{\ast }[Z,X]$ 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: $$c_n^{\ast }({\Gamma }_{\frac{k(k+1)}{2}+1,\frac{k(k-1)}{2}+1}^{\ast })\simeq c_n^{\ast }((M_k(F),t));$$ $$c_n^{\ast }({\Gamma }_{m(2m-1)+1,m(2m+1)+1}^{\ast })\simeq c_n^{\ast }((M_{2m}(F),s));$$ and $$c_n^{\ast }({\Gamma }_{h^2+1,h^2+1}^{\ast })\simeq c_n^{\ast }((M_h(F)\oplus M_h(F{)}^{op},exc)).$$