Lamination and antilamination of Euclidean lattices
The author studies some invariants related to the Hermite-Korkine-Zolotareff reduction of Euclidean lattices (or of positive definite quadratic forms). A real positive definite quadratic form is said \textit{laminated} if its decomposition as a sum of squares \[ {}^t xAx= A_1(x_1+\alpha_{12} x_2+\dots\alpha_{1n}x_n)^2+A_2(x_2+\alpha_{23}x_3+\dots+\alpha_{2n}x_n)^2+\dots+A_nx_n^2, \] is obtained by successive minimizations of coefficients \(A_1, A_2,\dots, A_{n-1}\). The invariant \[ \gamma_-''(A)=\min_{\text{HKZ}(\Lambda)}{\sqrt{A_1/A_n}}, \] (as the historic Korkine-Zolotareff constant \(\gamma_+''(A)=\max_{\text{HKZ}(\Lambda)}{\sqrt{A_1/A_n}}\)) where \(\min\) and \(\max\) are for all the \(\text{HKZ}\) (Hermite-Korkine-Zolotareff) reductions of a lattice \(\Lambda\) of rank \(n\) of an euclidean space \(E\) with \(A\) a Gram \(\text{HKZ}\)-reduced matrix of \(\Lambda\). These invariants are introduced to give an upper bound of the Bergé-Martinet constant \(\gamma^\prime(A)=\sqrt{\gamma(A)\gamma(A^{-1})}\) where \(\gamma(A)\) is the Hermite invariant and to give (conjecturally) a lower bound of \(\gamma(A)\) [\textit{A.M. Bergé}, J. Number Theory 52, No. 2, 284--298 (1995; Zbl 0829.11036) and \textit{J. Martinet}, Perfect lattices in Euclidean spaces. Grundlehren der Mathematischen Wissenschaften. 327. Berlin: Springer (2003; Zbl 1017.11031)]. This article introduces the \textit{antilaminations}, obtained by successive maximizations of \(A_n\), \(A_{n-1},\dots,A_2\) and the corresponding invariants \(\gamma_\pm'''(A)\), examines the relations between these two invariants to compute the value of these constants for the laminated lattices defined by Conway and Sloane. The author defines and examines the behaviour of recurrence of \(\gamma_{n,\pm}''\) and \(\gamma_{n,\pm}'''\) and derives some inequalities, formulates the conjecture \(\gamma_{5,-}''=\sqrt{2}\) and derives some consequences (e.g. \(\gamma_8=2\)) and he shows the unexpected discontinuity of these invariants from the dimension \(3\). This article contains several computations on the Blichfeldt lattices [\textit{H. F. Blichfeldt}, Math. Z. 39, 1--15 (1934; Zbl 0009.24403, JFM 60.0924.04)] and on the root lattices.
scientific article; zbMATH DE number 5774797