Asymptotics for Capelli Polynomials with Involution
F. S. Benanti, A. Valenti

TL;DR
This paper investigates the asymptotic behavior of the sequence of *-codimensions associated with certain *-Capelli polynomials in free associative algebras with involution, revealing their equivalence to those of finite-dimensional *-simple algebras.
Contribution
It establishes that the *-codimensions of finite-dimensional *-simple algebras are asymptotically equivalent to those generated by specific *-Capelli polynomials, providing explicit formulas for different algebra types.
Findings
Asymptotic equivalence of *-codimensions for matrix algebras with transpose involution
Asymptotic behavior for symplectic involution cases
Results for direct sums with exchange involution
Abstract
Let be the free associative algebra with involution over a field 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 symmetric variables and skew variables, respectively. It is well known that, if is an algebraic closed field of characteristic zero, every finite dimensional -simple algebra is isomorphic to one of the following algebras: \begin{itemize} \item [] the algebra of matrices with the transpose involution; \item [] the algebra of matrices with the symplectic involution; \item []$(M_{h}(F)\oplus…
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