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 \(E_6\).
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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 667–676 https://doi.org/10.4213/mzm13479.
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Grishukhin, V.P. Contact Vectors of Point Lattices. Math Notes 113, 642–649 (2023). https://doi.org/10.1134/S0001434623050048
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DOI: https://doi.org/10.1134/S0001434623050048