On the Asymptotics of Capelli Polynomials

Abstract

We present old and new results about Capelli polynomials, ℤ2 -graded Capelli polynomials, Capelli polynomials with involution and their asymptotics. Let Capm=∑σ∈Sm(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 Giambruno and Zaicev, Israel J Math 135:125–145, 2003) 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 (Benanti, Algebr Represent Theory 18:221–233, 2015) this result was extended to superalgebras proving that the ℤ2 -graded codimensions of the T2-ideal generated by the ℤ2 -graded Capelli polynomials CapM+10 and CapL+11 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 CapM+1+ and CapL+1−, for some fixed natural numbers M and L.