ON THE ASYMPTOTICS OF CAPELLI POLYNOMIALS
- Autori: Francesca Saviella Benanti; Angela Valenti
- Anno di pubblicazione: 2021
- Tipologia: Capitolo o Saggio
- OA Link: http://hdl.handle.net/10447/417002
Abstract. We present old and new results about Capelli polynomials, Z2-graded Capelli polynomials and Capelli polynomials with involution and their asymptotics. Let Capm = Pσ2Sm (sgnσ)tσ(1)x1tσ(2) · · · tσ(m−1)xm−1tσ(m) be the m-th Capelli polynomial of rank m. In the ordinary case (see ) it was proved the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial Capk2+1 and the codimensions of the matrix algebra Mk(F ). In  this result was extended to superalgebras proving that the Z2-graded codimensions of the T2-ideal generated by the Z2-graded Capelli polynomials Cap0 M+1 and Cap1 L+1 for some fixed M, L, are asymptotically equal to the Z2-graded codimensions of a simple finite dimensional superalgebra. Recently, the authors proved that the ∗-codimensions of a ∗-simple finite dimensional algebra are asymptotically equal to the ∗-codimensions of the T-∗-ideal generated by the ∗-Capelli polynomials Cap+ M+1 and Cap− L+1, for some fixed natural numbers M and L.