On the ideal of the shortest vectors in the Leech lattice and other lattices

Martin, William J. ; Steele, Corre L.

Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 707-726.

Résumé

Let $X\subset\mathbb R^m$ be a spherical code (i.e., a finite subset of the unit sphere) and consider the ideal of all polynomials in $m$ variables which vanish on $X$. Motivated by a study of cometric ($Q$-polynomial) association schemes and spherical designs, we wish to determine certain properties of this ideal. After presenting some background material and preliminary results, we consider the case where $X$ is the set of shortest vectors of one of the exceptional lattices $E_6$, $E_7$, $E_8$, $\Lambda _{24}$ (the Leech lattice) and determine for each: (i) the smallest degree of a non-trivial polynomial in the ideal and (ii) the smallest $k$ for which the ideal admits a generating set of polynomials all of degree $k$ or less. As it turns out, in all four cases mentioned above, these two values coincide, as they also do for the icosahedron, our introductory example. The paper concludes with a discussion of these two parameters, two open problems regarding their equality and a few remarks concerning connections to cometric association schemes.

Détail

Classification : 05E30, 05B30, 94B60
Keywords: spherical code, spherical design, association scheme, polynomial ideal, lattice