Supervarieties of small graded colength.
type: Item
instance of: scholarly article
title: Supervarieties of small graded colength.
zbMATH Open document ID: 1271.16023
author: Yanyan Li
published in: Journal of Pure and Applied Algebra
publication date: 2013-04-18T00:00:00Z
review text: Let \(F\) be a field of characteristic zero. For an associative PI-algebra \(A\) over \(F\) the most important quantitative information on the polynomial identities of \(A\) can be derived from the cocharacter sequence \(\chi_n(A)=\sum_{\lambda\vdash n}m_\lambda(A)\chi_\lambda\), \(n=1,2,\dots\), where \(\chi_\lambda\) is the irreducible \(S_n\)-character associated with the partition \(\lambda\) of \(n\). Let \(c_n(A)\) be the \(n\)-th codimension of \(A\) (i.e., the degree of the \(S_n\)-character \(\chi_n(A)\)) and let \(l_n(A)=\sum_{\lambda\vdash n}m_\lambda(A)\) be the \(n\)-th colength of \(A\) (which counts the number of the irreducible characters appearing in \(\chi_n(A)\)). Results of A. R. Kemer from the late 1970's give that the codimension sequence \(c_n(A)\) is polynomially bounded if and only if the colength \(l_n(A)\) is bounded by a constant. Equivalently, the infinite dimensional Grassmann algebra \(G\) and the algebra \(UT_2(F)\) of \(2\times 2\) upper triangular matrices do not belong to the variety of algebras generated by \(A\). It is well known that the study of superpolynomial identities of associative superalgebras is both of independent interest and an essential ingredient of the modern theory of algebras with ordinary polynomial identities. Let \(A\) be a superalgebra over \(F\) which satisfies a superpolynomial identity. Then the \(n\)-th graded cocharacter of \(A\) is \(\chi^{gr}_n(A)=\sum_{|\lambda|+|\mu|=n}m_{\lambda,\mu}(A)\chi_{\lambda,\mu}\), where \(\chi_{\lambda,\mu}\) is the irreducible \(\mathbb Z_2\wr S_n\)-character associated with the pair of partitions \(\lambda\) of \(r\) and \(\mu\) of \(n-r\). As in the ordinary case one defines the \(n\)-th graded codimension \(c^{gr}_n(A)\) and the \(n\)-th graded colength \(l^{gr}_n(A)\). \textit{A. Giambruno, S. Mishchenko} and \textit{M. Zaicev} [Commun. Algebra 29, No. 9, 3787-3800 (2001; Zbl 1006.16024)] established that the sequence \(c^{gr}_n(A)\), \(n=1,2,\dots\), is polynomially…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 6155421
zbMATH Keywords: varieties of superalgebras
zbMATH Keywords: polynomial identities of superalgebras
zbMATH Keywords: codimension sequences
zbMATH Keywords: cocharacter sequences
zbMATH Keywords: graded colengths
zbMATH Keywords: superpolynomial identities
zbMATH Keywords: forbidden subalgebras
zbMATH Keywords: irreducible characters
zbMATH Keywords: superidentities
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.06.001
OpenAlex ID: W1971772083
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DOI: 10.1016/J.JPAA.2012.06.001
description: scientific article; zbMATH DE number 6155421