Non-rigidity Degree of a Lattice and Rigid Lattices

Baranovskii, E. P.; Grishukhin, Viatcheslav

doi:10.1006/eujc.2001.0510

Cited by 19 publications

(30 citation statements)

References 7 publications

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“…The Leech lattice gives even a strongest possible example, in the sense that it is rigid (see Section 2 for details). Our proof in Section 3 immediately applies to E 8 , giving a new proof of E 8 's rigidity, first observed by Baranovskii and Grishukhin [BG01]. …”

Section: Introductionmentioning

confidence: 57%

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Schuermann¹,

Vallentin²

2005

Internat. Math. Res. Notices

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We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E 8 . For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E 8 . The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A * 8 . Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense.

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Section: Introductionmentioning

confidence: 57%

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Schuermann¹,

Vallentin²

2005

Internat. Math. Res. Notices

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“…In [1], the notion of the non-rigidity degree of a form f and the corresponding lattice L( f ) is introduced. It is denoted by nrd f and is equal to the dimension of the L-type domain containing f .…”

Section: Peter Engel and Viacheslav Grishukhinmentioning

confidence: 99%

An Example of a Non-simplicialL-type Domain

Engel

Grishukhin²

2001

European Journal of Combinatorics

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We present an example of a non-simplicial five-dimensional L-type domain of forms on five variables. Its closure has 10 non-simplicial facets each having five extreme rays. This domain is the L-type domain of the form 42.240 of is the dimension of P n , i.e., the dimension of the space of coefficients ofIn The set of all L-type domains is partitioned into classes of unimodularly equivalent domains. For n ≤ 3, there is only one class, i.e., only one general L-type. For n = 4, there are three general L-types.For n ≤ 4, all L-type domains are simplicial, i.e., the closure of a k-dimensional domain has k facets and k extreme rays. In [6], Voronoi paid special attention to this fact. However, for n = 5, there are non-simplicial L-type domains. In 1972, Barnes and Ternery [2] first noticed this fact. In 1976, Ryshkov and Baranovskii [5] mentioned (see the end of Section 13) that non-simplicial L-type domains are typical. For some non-simplicial domains, the authors present their numbers in Table III of [5] as well as the numbers of extreme rays of these domains.In this paper we explicitly describe an example of a five-dimensional special non-simplicial L-type domain for 5-ary quadratic functions. This special L-type domain is a facet of the closure of several general L-type domains which are therefore also non-simplicial.We take a representative form of this domain from Table 2 of [4]. This form is denoted in Table 2 by the symbol 42.240. This symbol (the same applies to the form symbols mentioned below) means that the Voronoi polytope of this form has 42 facets and 240 vertices. We denote this form by f 0 .Let {b 1 , b 2 , e 1 , e 2 , e 3 } be the basis corresponding to f 0 . Then the coefficients of the form f 0 are as follows: The coefficients of f 0 take a more symmetric form if we apply the unimodular transforma-

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“…Now, we show that D n lies in the intersection of the hyperplanes (9). It is proved in [BG01] that the equations of the hyperplanes in the intersection of which an L-type domain lies are given by some linear forms on norms of minimal vectors of cosets of 2L in L. Some of such linear forms are obtained by equating norms of minimal vectors of a non-simple coset. There are L-type domains for which linear forms of last type are sufficient for to describe the space, where this L-type domain lies.…”

Section: The Domain D Nmentioning

confidence: 99%

“…In [BG01], a notion of non-rigidity degree of a form f and the corresponding lattice L(f ) is introduced. It is denoted by nrdf and is equal to dimension of the L-domain containing f .…”

Section: Introductionmentioning

confidence: 99%

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Nonrigidity Degrees of Root Lattices and their Duals

Deza

Grishukhin

2004

Geometriae Dedicata

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Non-rigidity degree of a lattice L, nrdL, is dimension of the L-type domain to which L belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of D * n , and the case of E * 7 are decided. We describe explicitly the L-type domain D(D * n ), n ≥ 4. For n odd, it is a nonsimplicial polyhedral open cone of dimension n. For n even, it is one-dimensional, i.e. for even n, D * n is an edge form.Mathematics Subject Classification. Primary 11H06, 11H55; Secondary 52B22, 05B45.

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Non-rigidity Degree of a Lattice and Rigid Lattices

Cited by 19 publications

References 7 publications

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An Example of a Non-simplicialL-type Domain

Nonrigidity Degrees of Root Lattices and their Duals

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