TY - GEN

T1 - ASYMPTOTICS FOR CAPELLI POLYNOMIALS WITH INVOLUTION

AU - Benanti, Francesca Saviella

AU - Valenti, Angela

PY - 2019

Y1 - 2019

N2 - Let F be the free associative algebra with involution ∗ over a fieldF 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 variablesand L + 1 skew variables, respectively.It is well known that, if F is an algebraic closed field of characteristic zero, everyfinite 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 theopposite 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 Mand 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));andc∗n(Γ∗ h2+1,h2+1) ≃ c∗ n((Mh(F ) ⊕ Mh(F )op, exc))

AB - Let F be the free associative algebra with involution ∗ over a fieldF 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 variablesand L + 1 skew variables, respectively.It is well known that, if F is an algebraic closed field of characteristic zero, everyfinite 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 theopposite 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 Mand 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));andc∗n(Γ∗ h2+1,h2+1) ≃ c∗ n((Mh(F ) ⊕ Mh(F )op, exc))

UR - http://hdl.handle.net/10447/417213

M3 - Other contribution

ER -