Asymptotics for the standard and the Capelli identities.
type: Item
instance of: scholarly article
title: Asymptotics for the standard and the Capelli identities.
zbMATH Open document ID: 1048.16012
DOI: 10.1007/BF02776053
published in: Israel Journal of Mathematics
publication date: 2004-03-08T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 2053060
zbMATH Keywords: polynomial identities
zbMATH Keywords: T-ideals
zbMATH Keywords: varieties of algebras
zbMATH Keywords: codimension sequences
zbMATH Keywords: growth sequences
zbMATH Keywords: Capelli polynomials
zbMATH Keywords: exponents
author: Antonio Giambruno
author: Mikhail Zaicev
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
cites work: On the codimensions of the verbally prime P.I. algebras
cites work: Codimensions of products and of intersections of verbally prime T-ideals
cites work: Exponential growth for codimensions of some p. i. algebras
cites work: Q5340151
cites work: On codimension growth of finitely generated associative algebras
cites work: Exponential codimension growth of PI algebras: an exact estimate
cites work: Minimal varieties of algebras of exponential growth
cites work: Asymptotics of the codimension growth functions of the standard identity and the Capelli identities
cites work: Q4494228
cites work: Q4710870
cites work: The Polynomial Identities of the Grassman Algebra
cites work: The exponential growth of codimensions for Capelli identities
cites work: Existence of identities in \(A \otimes B\)
cites work: Codimensions and trace codimensions of matrices are asymptotically equal
cites work: Q3910639
cites work: Invariant Wedderburn factors
full work available at URL: https://doi.org/10.1007/bf02776053
OpenAlex ID: W2032054800
description: scientific article; zbMATH DE number 2053060