Asymptotics in the cocharacter sequence of \(A\otimes B\).
type: Item
instance of: scholarly article
title: Asymptotics in the cocharacter sequence of \(A\otimes B\).
zbMATH Open document ID: 1112.16024
published in: Advances in Applied Mathematics
publication date: 2007-01-26T00:00:00Z
review text: By the already classical theorem of Regev the codimension sequence \(c_n(A)\) of a PI-algebra \(A\) grows exponentially. As a consequence, the tensor product \(A\otimes B\) of two PI-algebras \(A\) and \(B\) is a PI-algebra again. The more recent theorem of Giambruno and Zaicev states that, over a field \(F\) of characteristic 0, the limit \(e(A)=\lim_{n\to\infty}\root n\of{c_n(A)}\) always exists and is an integer called the exponent of \(A\). In the paper under review the author obtains upper and lower bounds for \(e(A\otimes B)\) in terms of \(e(A)\) and \(e(B)\), namely \(e(A\otimes B)=e(A)\cdot e(B)\) or \(e(A\otimes B)=e(A)\cdot e(B)-1\). Both bounds can occur. For a fixed algebra \(A\), the equality \(e(A\otimes B)=e(A)\cdot e(B)\) holds for all \(B\) if and only if there exists a verbally prime algebra \(A'\) which satisfies all polynomial identities of \(A\) and \(e(A')=e(A)\). The minimum value of \(\{e(A\otimes B)\mid e(B)=b\}\) occurs when \(B\) is the \(b\times b\) upper triangular matrix algebra over \(F\). The second problem considered in the paper deals with the cocharacter sequence \(\chi_n(A\otimes B)=\sum_{\lambda\vdash n}m_\lambda(A\otimes B)\chi_\lambda\). It is known that the partitions \(\lambda\) participating with nonzero multiplicities \(m_\lambda\) are in a hook. The author studies the shape of the hook in terms of the corresponding hooks for \(A\) and \(B\). He obtains upper and lower bounds for the arm and the leg of this hook.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 5118776
zbMATH Keywords: polynomial identities
zbMATH Keywords: tensor products
zbMATH Keywords: codimension sequences
zbMATH Keywords: cocharacter sequences
zbMATH Keywords: exponents
zbMATH Keywords: PI-algebras
zbMATH Keywords: verbally prime algebras
zbMATH Keywords: upper triangular matrix algebras
zbMATH Keywords: hooks
author: Allan Berele
MaRDI profile type: Publication
full work available at URL: j.aam.2005.07.005
OpenAlex ID: W2060660436
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DOI: 10.1016/J.AAM.2005.07.005
description: scientific article; zbMATH DE number 5118776