Center symmetry of incidence matrices
type: Item
instance of: scholarly article
zbMATH Open document ID: 0966.15018
publication date: 2001-03-19T00:00:00Z
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 16N40
zbMATH DE Number: 1579113
zbMATH Keywords: center symmetry
zbMATH Keywords: Nagata-Higman theorem
zbMATH Keywords: index of nilpotence
zbMATH Keywords: incidence matrix
MaRDI profile type: Publication
author: Woo Lee
title: Center symmetry of incidence matrices
published in: Communications of the Korean Mathematical Society
review text: Let \(A\) be an algebra over a field \(F\) of characteristic \(0\) and let \(n>0\) be an integer. The Nagata-Higman theorem states that, if \(a^{n}=0\) for all \(a\in A,\) then \(A\) is nilpotent [see \textit{M. Nagata}, J. Math. Soc. Japan 4, 296-301 (1952; Zbl 0049.02402)]. It has been proved that the best possible value \(\mathcal{N} (n)\) of the index of nilpotence of \(A\) lies between \(n(n+1)/2\) and \(n^{2}\) but the precise value is not known in general. Let \(p_{n}(m)\) be the number of partitions of \(m\) into \(n\) parts. The author of the present paper observes that \(\mathcal{N}(n)\) is equal to the least integer \(m\) such that the rank of a certain \(p_{n}(m)\cdot m!\times m!\) incidence matrix has rank \(m!\), and finds some elementary properties of this matrix.
reviewed by: John D. Dixon
description: scientific article