Eulerian polynomial identities on matrix rings
type: Item
instance of: scholarly article
title: Eulerian polynomial identities on matrix rings
zbMATH Open document ID: 0802.16015
author: Zsolt Tuza
author: Gábor Révész
author: Jenő Szigeti
published in: Journal of Algebra
publication date: 1994-10-24T00:00:00Z
review text: It is well known that the Amitsur-Levitzki theorem can be proved using some graph-theoretical constructions, see, e.g. \textit{R. G. Swan} [Proc. Am. Math. Soc. 14, 367-373 (1963; Zbl 0118.018)]. The paper under review is a further development in this direction. Assume \(G\) is an Eulerian directed graph having \(k\) vertices and \(N\) edges, and let \(P(G)\) be the set of all covering directed paths of \(G\). Then the ring \(M_ n(R)\) of the \(n\) by \(n\) matrices over an associative commutative unital ring \(R\) satisfies the polynomial (multilinear) identity \(\sum \text{sgn}(p) x_{p(1)} \dots x_{p(N)} = 0\), \(p \in P(G)\) for \(N \geq 2kn\). It is shown that the standard and the double Capelli identities can be obtained in a similar fashion using rather simple Eulerian graphs.
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 05C45
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 554264
zbMATH Keywords: multilinear identity
zbMATH Keywords: Amitsur-Levitzki theorem
zbMATH Keywords: Eulerian directed graph
zbMATH Keywords: Capelli identities
MaRDI profile type: Publication
full work available at URL: jabr.1993.1207
OpenAlex ID: W1964394099
DOI: 10.1006/JABR.1993.1207
description: scientific article; zbMATH DE number 554264