Bounds on colength and maximal multiplicity sequences.
type: Item
instance of: scholarly article
title: Bounds on colength and maximal multiplicity sequences.
zbMATH Open document ID: 1214.16016
author: Allan Berele
published in: Journal of Algebra
publication date: 2011-03-04T00:00:00Z
review text: Let \(A\) be an associative algebra over a field of characteristic 0. In order to study the ideal \(T(A)\) of the polynomial identities satisfied by \(A\) it is sufficient to study the multilinear ones. The vector space \(P_n\) of the multilinear polynomials in \(x_1,\dots,x_n\) in the free associative algebra is a module over the symmetric group \(S_n\) acting by permuting the variables; in fact \(P_n\) is isomorphic to the regular module for \(S_n\). The multilinear identities for \(A\) in \(P_n\) form a submodule. It is well known that if \(A\) satisfies a nontrivial identity then \(T(A)\cap P_n\) becomes huge when \(n\to\infty\), so one is led to study the quotient \(P_n(A)=P_n/(P_n\cap T(A))\). There are several important numerical characteristics of the identities of \(A\) associated to the modules \(P_n(A)\). Probably the most widely studied is the codimension sequence of \(A\), \(c_n(A)=\dim P_n(A)\). The representations of \(S_n\) are described in terms of partitions and Young diagrams. Let \(P_n(A)=\oplus m_\lambda M(\lambda)\) be the decomposition of \(P_n(A)\) into irreducibles \(M(\lambda)\) with multiplicities \(m_\lambda\) where \(\lambda\) runs over the partitions of \(n\). The character \(\chi(A)=\chi(P_n(A))=\sum m_\lambda\chi^\lambda\) is called the \(n\)-th cocharacter of \(A\). Denote by \(\text{mlt}(A)\) the least number \(e\) such that all multiplicities \(m_\lambda\) are bounded above by a polynomial in \(n\) of degree \(e\). The \(n\)-th colength of \(A\) is defined as the sum \(\sum m_\lambda\), and \(\text{col}(A)\) is the least \(e\) such that the \(n\)-th colength of \(A\) is bounded by a polynomial of degree \(e\) in \(n\). Recall also that the exponent of \(A\), \(\exp(A)\), is the least \(e\) such that \(c_n(A)\) is bounded by \(f(n)e^n\) where \(f\) is a polynomial. In order to state the main results of the paper under review one needs the notion of a hook. Let \(a\) and \(b\) be integers, the hook \(H(a,b)\) is the set of all partit…
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 5860490
zbMATH Keywords: PI algebras
zbMATH Keywords: cocharacter sequences
zbMATH Keywords: T-ideals
zbMATH Keywords: prime product algebras
zbMATH Keywords: multilinear identities
zbMATH Keywords: codimension sequences
zbMATH Keywords: partitions
zbMATH Keywords: Young diagrams
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.10.007
OpenAlex ID: W2045096488
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DOI: 10.1016/J.JALGEBRA.2010.10.007
description: scientific article; zbMATH DE number 5860490