There are exactly 222 \(L\)-types of primitive five-dimensional lattices
type: Item
instance of: scholarly article
title: There are exactly 222 \(L\)-types of primitive five-dimensional lattices
zbMATH Open document ID: 1017.52006
author: Peter Engel
author: Viacheslav Grishukhin
published in: European Journal of Combinatorics
publication date: 2003-08-19T00:00:00Z
review text: Two positive definite quadratic forms on \(R^n\) are said to be of the same \(L\)-type if their Delaunay partitions (the duals of the Voronoĭ partitions) are affinely equivalent. This notion is due to Voronoĭ, who initiated the study of the problem of enumeration of \(L\)-types, with which this paper is also concerned. For \(n = 5\) dimensional space, \(221\) \(L\)-types were found in the paper by \textit{S. S. Ryšhkov} and \textit{E. P. Baranovskiĭ} [translated in Proceedings of Steklov Institute of Mathematics (1978, issue 4), AMS]; however, in [\textit{P. Engel}, Proc. Inst. Math. Natl. Acad. Sci. Ukr. Math. Appl. 21(2), 22-60 (1998; Zbl 0955.51011)], \(222\) distinct \(L\)-types were listed. This paper disposes of the nontrivial problem of determining which of the \(222\) had been omitted in the earlier enumeration.
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1795947
zbMATH Keywords: Voronoi partition
zbMATH Keywords: Delaunay partition
zbMATH Keywords: \(L\)-type
reviewed by: James F. Lawrence
MaRDI profile type: Publication
full work available at URL: eujc.2001.0551
OpenAlex ID: W2072505366
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DOI: 10.1006/EUJC.2001.0551
description: scientific article; zbMATH DE number 1795947