Asymptotics for the standard and the Capelli identities.
For a T-ideal \(Id(A)\) of the polynomial identities of an algebra \(A\) and a T-ideal \(\langle f\rangle_T\) generated by a polynomial identity \(f\) over a field \(F\) of characteristic 0, let \(c_n(A)\), \(c_n(f)\), respectively, be the \(n\)-th codimensions. In the paper under review the authors study the exact asymptotics of the codimension sequences of the T-ideals generated by the standard polynomial \(St_m\) of degree \(m\) and the Capelli polynomial \(C_m\) in \(m\) skew-symmetric variables, and relate them with the codimensions of algebras of matrices. The authors establish that \(\langle St_{2k}\rangle_T=Id(M_k(F)\oplus B)\), \(\langle St_{2k+1}\rangle_T=Id(M_{k\times 2k}(F)\oplus M_{2k\times k}(F)\oplus D)\), \(\langle C_{k^2+1}\rangle_T=Id(M_k(F)\oplus E)\), where \(M_k(F)\) is the \(k\times k\) matrix algebra with entries from \(F\), \(M_{k\times 2k}(F)\) is the subalgebra of \(M_{2k}(F)\) consisting of the matrices with the last \(k\) rows equal to 0 (and similarly for \(M_{2k\times k}(F)\)), and the codimensions of the algebras \(B,D,E\) grow much slower than those of the other algebras in the direct sums. This implies the asymptotics of the codimension sequences \(c_n(St_{2k})\simeq c_n(C_{k^2+1})\simeq c_n(M_k(F))\) and \(c_n(St_{2k+1})\simeq c_n(M_{k\times 2k}(F)\oplus M_{2k\times k}(F))\), and the exact asymptotic behaviour of \(c_n(M_k(F))\) is known and that of the other algebras can be derived. It is known that the exponent of the Capelli identity, \(\lim_{n\to\infty}\root n\of{c_n(C_{m+1})}\), is an integer between \(m-3\) and \(m\), depending of the presentation of \(m\) and \(m-1\) as sums of 1, 2, 3 or 4 squares. In the present paper the authors determine also the asymptotics of \(c_n(C_{m+1})\) when the exponent of \(C_{m+1}\) is equal to \(m\) and \(m-1\). The former happens if and only if \(m\) is a square. The latter means that \(m-1\) is a square or a sum of two squares. Then \(\langle C_{m+1}\rangle_T\) coincides with the T-ideal of the direct sum of several algebras of the form \(M_{k\times 2k}(F)\), \(M_{2k\times k}(F)\) or upper triangular block matrices.
scientific article; zbMATH DE number 2053060