A characterization of the permanent function by the Binet-Cauchy theorem
type: Item
instance of: scholarly article
title: A characterization of the permanent function by the Binet-Cauchy theorem
zbMATH Open document ID: 0645.15005
DOI: 10.1016/0024-3795(88)90142-5
author: K. P. S. Bhaskara Rao
author: Konrad J. Heuvers
author: Larry Cummings
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: Let K be a field of characteristic 0. Denote by \(M_ n(K)\) the set of \(n\times n\) matrices with entries in K. One may interpret \(n\in K\) if necessary by writing n as \(\sum^{n}_{k=1}1\), e.g. let \(E\in M_ n(K)\) be the matrix with each entry equal to 1/n. Denote by \(G_ n\) the set of all sequences \(\omega =(\omega_ 1,\omega_ 2,...,\omega_ n)\) of integers such that \(1\leq \omega_ 1\leq \omega_ 2\leq...\leq \omega_ n\leq n\). The authors show that if \(f: M_ n(K)\to K\) is such that \[ f(AB)=\sum_{\omega \in G_ n}(1/\mu (\omega))f(A[1,...,n| \omega])f(B[\omega | 1,...,n]) \] with f(E)\(\neq 0\), then either \(f(A)=n!/n^ n\) or else \(f(A)=\phi\) (per A) where \(\phi\) is an isomorphism of K. Here for any \(C\in M_ n(K)\) and \(\alpha,\beta \in G_ n\), C[\(\alpha| \beta]\) is that member of \(M_ n(K)\) whose rows are indexed by \(\alpha\) and whose columns are indexed by \(\beta\) ; \(\mu (\omega)=\mu_ 1!\mu_ 2!...\mu_ n!\), where \(\mu_ i\) is the number of times i appears in \(\omega\), \(i=1,...,n\).
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 4051800
zbMATH Keywords: characterization
zbMATH Keywords: permanent function
zbMATH Keywords: Binet-Cauchy theorem
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90142-5
OpenAlex ID: W2079518226
cites work: Q5514010
cites work: Q4769056
cites work: On homomorphisms of the general linear group
cites work: Q3262498
cites work: Q5567950
cites work: Q4771383
cites work: Permanents
cites work: Multinomial matrices
description: scientific article; zbMATH DE number 4051800