Asymptotics for the standard and the Capelli identities

Let {c(n) (St(k))} and {c(n) (C-k)} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degree k and by the k-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a field F of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold: c(n)(St(2k)) similar or equal to c(n)(C-k(+1)2) similar or equal to c(n)(M-k(F)), c(n)(St(2k+1)) similar or equal to c(n)(M-k x 2k(F) circle plus M-2k x k(F)), where M-k(F) is the algebra of k x k matrices and M-k x l(F) is the algebra of (k + l) x (k + l) matrices having the last l rows and the last k columns equal to zero. The precise asymptotics of c(n)(M-k(F)) are known and those Of M-k x 2k(F) and M-2k x k(F) can be easily deduced. For Capelli polynomials we show that also upper block triangular matrix algebras come into play.