6533b855fe1ef96bd12b072a

RESEARCH PRODUCT

Asymptotics for the standard and the Capelli identities

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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 degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold: $$\begin{gathered} c_n \left( {St_{2k} } \right) \simeq c_n \left( {C_{k^2 + 1} } \right) \simeq c_n \left( {M_k \left( F \right)} \right), \hfill \\ c_n \left( {St_{2k + 1} } \right) \simeq c_n \left( {M_{k \times 2k} \left( F \right) \oplus M_{2k \times k} \left( F \right)} \right), \hfill \\ \end{gathered} $$ whereM k (F) is the algebra ofk×k matrices andM k×l (F) is the algebra of (K+l)×(k+l) matrices having the lastl rows and the lastk columns equal to zero. The precise asymptotics ofc n (M k (F)) are known and those ofM k×2k (F) andM 2k×k (F) can be easily deduced. For Capelli polynomials we show that also upper block triangular matrix algebras come into play.