Abstract

The contact vectors of a lattice \(L\) are vectors \(l\) which are minimal in the \(l^2\)-norm in their parity class. It is shown that, in the space of all symmetric matrices, the set of all contact vectors of the lattice \(L\) defines the subspace \(M(L)\) containing the Gram matrix \(A\) of the lattice \(L\). The notion of extremal set of contact vectors is introduced as a set for which the space \(M(L)\) is one-dimensional. In this case, the lattice \(L\) is rigid. Each dual cell of the lattice \(L\) is associated with a set of contact vectors contained in it. A dual cell is extremal if its set of contact vectors is extremal. As an illustration, we prove the rigidity of the root lattice \(D_n\) for \(n\ge 4\) and the lattice \(E_6^*\) dual to the root lattice \(E_6\).

中文翻译：

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点格的接触向量

摘要

晶格\(L\)的接触向量是向量\(l\) ，其在奇偶校验类的\(l^2\)范数中最小。结果表明，在所有对称矩阵的空间中，格的所有接触向量的集合\(L\)定义了包含格的Gram矩阵\(A\)的子空间\(M(L)\) \(L\)。引入接触向量极值集的概念作为空间\(M(L)\)是一维的集合。在这种情况下，晶格\(L\)是刚性的。晶格的每个双单元\(L\)与其中包含的一组接触向量相关联。如果对偶单元的接触向量集是极值，那么它就是极值。作为说明，我们证明了根格子\(D_n\)对于\(n\ge 4\)的刚性以及与根格子\(E_6\)对偶的格子\(E_6^*\)。