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On generating points of a lattice in the region

Published online by Cambridge University Press:  18 May 2009

D. M. E. Foster
D. M. E. Foster
Affiliation:
St. Salvator's College University of St. Andrews
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A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), where

for some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.

If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the form

and Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 6 , Issue 3 , January 1964 , pp. 141 - 155
Copyright
Copyright © Glasgow Mathematical Journal Trust 1964

References

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