Lattice Of Rank Research Articles

On the genera and the class number of unimodular lattices over the ring of integers in real quadratic fields

H. Pfeuffer [ J. Number Theory 11 (1979), 188–196] showed that totally positive quadratic forms with three or more variables and class number h = 1 exist only in a finite number of algebraic number fields. For even unimodular lattices, we already know that h = 1 when the rank n = 4 and fields F = Q(√2), Q(√5), or Q(√13) (see [P. J. Costello and J.S.Hsia, Adv. in Math. 64 (1987), 241–278; J. S. Hsia and D. C. Hung, Math. Ann. 283 (1989), 367–374; Y. Kitaoka, Nagoya Math. J. 66 (1977), 89–98; H. Maass, Math. Ann. 118 (1941), 65–84; I. Takada, Math. Japon. 30 (1985), 423–433]). In this paper we show that the class number of any even unimodular lattice of rank > 2 in a quadratic space having an orthonormal basis over a real quadratic field is more than one except in the three cases above.

Simultaneously good bases of a lattice and its reciprocal lattice

It is well-known that every lattice L has a basis consisting of relatively short vectors. To state a quantitative version of this fact, let 21(L) denote the i-th successive minimum of L, which is the smallest value such that there are i linearly independent vectors in L in the ball {x: [I x II _-< ~.i(L)}. In particular, let A, be the minimal constant such that every lattice of rank n possesses a basis which satisfies

Hecke operators on theta series attached to lattices of arbitrary rank

Hecke operators on theta series attached to lattices of arbitrary rank

On the structure of even unimodular extremal lattices of rank 40

On the structure of even unimodular extremal lattices of rank 40

Trees and discrete subgroups of Lie groups over local fields

Let K be a locally compact field and G a simple AT-group, G = G(K). A discrete subgroup T of G is called a lattice if G/F carries a finite G-invariant measure. It is a uniform (or cocompact) lattice if G/T is compact and nonuniform otherwise. When the jRf-rank of G is greater than one, Margulis [Ma, Z] proved that T is arithmetic, establishing the conjecture of Selberg and PiatetskiShapiro. This remarkable work left open the case of rank one groups. SLi{U) contains continuous families of lattices (the Teichmuller spaces) and in particular it contains nonarithmetic lattices. By the Mostow rigidity theorem (cf. [Ml]), SLi{R) is essentially the only real simple Lie group which allows this phenomenon. Some other real rank one groups are known to have nonarithmetic lattices: Gromov and Piatetski-Shapiro [GPS] showed that SO(n, 1) have such lattices (earlier it was shown by Makarov and Vinberg for small n), Mostow [M2] constructed nonarithmetic lattices in SU(2,1 ), and together with Deligne [DM] also in SU(3,1 ). For the other rank one real groups: SU(n, 1) (n > 4), Sp(n, 1) and F4 the problem is still open. A related problem is the congruence subgroup problem, which asked: Given T arithmetic, are all its finite index subgroups congruence subgroups? Serre [SI] conjectured that T has the congruence subgroup property (CSP) if and only if rank^(G) > 2. The affirmative part of this conjecture was proved to a large extent (but mainly for nonuniform lattices; see [RI, R2] for precise results and history). Less is known for arithmetic lattices in real rank one groups: It is easy to prove that none of the arithmetic lattices in PSL2(R) = SO(2,1)° has CSP. The same holds for PSL2(C) = SO(3,1) (see [SI] for the nonuniform case and [LI] for the uniform case). For SO(n, 1), general n, it is known only for some of the lattices (Millson [Mi]). Similarly, Kazdhan [Ka] showed that some lattices in SU(n, 1) do not have CSP. Again, nothing is known for Sp(n, 1) and ƒ% which are of rank one but have Kazdhan property (T) like groups of higher rank. In this note we will describe results on the structure of lattices in rank one groups over locally compact nonarchimedean fields, as well as some

On finite locally projective planar spaces

Let L be a finite geometric lattice of rank 4 (i.e., a planar space) such that any two planes of L meet in a line. There is a longstanding conjecture due to W. M. Kantor which states that every such lattice can be embedded into a projective space. If L is given as above, then for every point p ϵ L, L p is a projective plane of order n (independent of p). Recently, A. Beutelspacher has shown that if L has at least n 3 points then L can be embedded into a projective space. We give an alternative proof of his result, which applies to the more general class of finite locally projective planar spaces. Furthermore, our considerations lead to some more insight into the geometrical structure of a possible counterexample to Kantor's conjecture. For example, they can be used to show that the bound on n 3 is not tight.

Examples of even unimodular extremal lattices of rank 40 and their Siegel theta-series of degree 2

Using two newly discovered [40, 20] codes and a [40, 20] code, which is equivalent to a known one, we construct three non-isomorphic even unimodular extremal lattices of rank 40. Next we show these lattices are distinguished by their Siegel theta-series of degree 2.

Local rigidity of discrete groups acting on complex hyperbolic space

The superrigidity theorem of Margulis, see Zimmer [17], classifies finite dimensional representations of lattices in semi-simple Lie groups of real rank strictly larger than 1. It is a fundamental problem to obtain the classification of finite dimensional representations of lattices in rank I semi-simple Lie groups. That this problem will be considerably harder than the previous one is suggested by the existence of continuous families of inequivalent representations or, in other words, the existence of non-trivial deformations. In Johnson-Millson [6] and Kourouniotis [8] such deformations were constructed for certain representations of lattices in SO(n, 1) based on a construction of Thurston called bending. The deformation space of the representation of a lattice F=SO(n, 1) obtained by restricting an inclusion SO(n, 1)-,SO(n+ l, 1) to F is of particular interest. If n > 2 the space of infinitesimal deformations is Hi(F, R "+1) where F acts on R "+1 by the restriction of the standard action of SO(n, 1). The space of infinitesimal deformations is non-zero for the standard arithmetic examples and the main point of the papers cited above was to establish that some of these infinitesimal deformations are integrable (in [6] it is also shown that some are not). In this paper, we study the complex analogue of the above example. We let F be a cocompact torsion free lattice in SU(n, 1) and consider the deformation space of the representation of F obtained by restricting an inclusion SU(n, 1)--*SU(n+I, 1). If n > l the space of infinitesimal deformations is H~(F, R)@H~(F, C,+ 1). In the first summand F acts trivially and the infinitesimal deformations are tangent to the obvious deformations obtained by deforming F in U(n, 1) by a curve of homomorphisms into the center of U(n, 1) (observe that the above inclusion factors as SU(n, 1)~U(n , 1)~SU(n+I, 1)). In the second summand F acts by the restriction of the standard action of SU(n, 1). This summand is non-zero for the standard arithmetic examples,

Partial geometric lattices, I: Regularity conditions

This paper introduces partial geometric lattices, which generalize to arbitrary rank, partial geometries studied by Bose, wich are partial geometries of rank 3, and the partial geometries of dimension 3 studied by Laskar and Dunbar, which are a subclass of the partial geometric lattices of rank 4. The main result is that the number of spaces of a given rank i + k which contain a given i-space and are contained in a given ( i + s)-space depends only on the ranks i, i + k and i + s. This function is called φ( i, s, k). It, along with related functions, is calculated for rank 3 and 4. These combinatorial structures should prove useful in the study of hierarchical experimental designs both factorial and non-factorial, and in coding theory and combinatorial information theory.

A Classification of Modularly Complemented Geometric Lattices

A geometric lattice G is said to be modularly complemented if for every point in G , there exists a modular copoint not containing it. We prove that a connected modularly complemented geometric lattice of rank at least four is either a Dowling lattice or the lattice of flats of a projective geometry with some of its points deleted.

On even unimodular positive definite quadratic lattices of rank 32

Let Z be the ring of rational integers, and ~ the field of rational numbers. A finitely generated Z-module L in Q" with a positive definite metric is called a quadratic lattice. Since we treat only the quadratic lattices, we shall omit the adjectives "positive definite quadratic". A lattice L is integral if L satisfies (x ,y)sZ for any x, y e L where ( , ) is the bilinear form associated to the metric. The dual L e of L is defined by

On the configurations of even unimodular lattices of rank 48

On the configurations of even unimodular lattices of rank 48

A proof of the Bundle Theorem for certain semimodular locally projective lattices of rank 4

The Bundle Theorem is proved for geometric locally projective lattices of rank 4 which for every given line (rank 2 element) do not contain too many lines that are on a common plane (rank 3 element) with this line, but on no common point (rank 1 element). By a result of J. Kahn ( Math. Z. 175 (1980), 219–247), this implies that these lattices are projectively embeddable.

Reliability calculations for electrical transmission systems on the basis of mean failure indices : S. K. Banerjee and B. R. Reddi. IEEE Trans Reliab.R-32, 346 (October 1983)

We construct many new invariant solutions to the Strominger system with respect to a 2-parameter family of metric connections ∇ε,ρ in the anomaly cancellation equation. The ansatz ∇ε,ρ is a natural extension of the canonical 1-parameter family of Hermitian connections found by Gauduchon, as one recovers the Chern connection ∇c for (ε,ρ)=(0,12), and the Bismut connection ∇+ for (ε,ρ)=(12,0). In particular, explicit invariant solutions to the Strominger system with respect to the Chern connection, with non-flat instanton and positive α′ are obtained. Furthermore, we give invariant solutions to the heterotic equations of motion with respect to the Bismut connection. Our solutions live on three different compact non-Kähler homogeneous spaces, obtained as the quotient by a lattice of maximal rank of a nilpotent Lie group, the semisimple group SL(2,C) and a solvable Lie group. To our knowledge, these are the only known invariant solutions to the heterotic equations of motion, and we conjecture that there is no other such homogeneous space admitting an invariant solution to the heterotic equations of motion with respect to a connection in the ansatz ∇ε,ρ.

The characters of the wreath product group acting on the homology groups of the Dowling lattices

Let G be a finite group, n a positive integer, Q n ( G) the Dowling lattice of rank n based on G and W n the wreath product group G wr S n . It is easily seen that W n acts as a group of automorphisms of Q n ( G). This action lifts to a representation of W n on each homology group of Q n ( G). The character values of these representations are computed. Let σ be an element of W n . Consider σ as an n × n permutation matrix σ whose nonzero entries have been replaced by elements of G. If C is a cycle of σ, the weight of C is the product of the elements of G which lie in the cycle C. The type of C is the conjugacy class of G containing the weight of C. Let c l, u denote the number of l-cycles of σ of type u. The conjugacy class of σ in W n depends only on the numbers c l, u . For each i = 0, 1,…, n − 1 let ( Q n ( G)) i be the geometric lattice obtained from Q n ( G) by deleting ranks i + 1 through n − 1 (so ( Q n ( G)) n − 1 = Q n ( G)). Let β i denote the character of the representation of W n on the unique non-vanishing reduced homology group of Q n ( G)) i . For each σ ϵ W n , let B σ ( λ) be the polynomial B σ ( λ) = ∑ i = 0 n − 1 β i ( σ)) λ n − 1 − i . It is shown that B 0(λ)=(−1) n ∏ l,u C l,u!(l|C G(u)|) cl,u λ+1 F(l,u, l) − C l,u F(l,u,−λ) where F(l, u, λ) = ( −1 l¦G¦ ) ∑ t¦lb(t,u) μ(t) λ l t , and where b(t, u) is the number of solutions h ϵ G to h t = u. The formula (∗) allows for the explicit computation of the characters β i . Using this information, several facts about the characters are deduced. For example, it is shown that the trivial character appears exactly once in each β i and it is shown that β n − 1 can be realized in a simple way as a sum of induced characters.

The Lattices of Six-Dimensional Euclidean Space

The lattices of full rank of the six-dimensional Euclidean space are classified according to their automorphism groups (Bravais classification). We find 826 types of such lattices.

A q-analogous of the characterization of hypercubes as graphs

We give an arithmetical characterization of graphs which are realizable as graphs of the lattice ℒ of the subspaces of a graphic (or projective) space of finite dimension and of finite order q ≥ 1. In other words ℒ is any complemented modular lattice of finite rank and of finite order q. When q ≥ 2 and the rank is at least four, ℒ is the lattice of the subspaces of a finite dimensional vector space over the field GF(q). Two independent axioms are required involving the number of geodesies between any two vertices. This number must be a simple function of the distance.

Regular join endomorphisms on a complemented modular lattice of finite rank

The concept of regularity is defined for a class of join endomorphisms on a complemented modular lattice of finite rank. The properties of this class of endomorphisms are studied and it is shown that they are generalizations of endomorphisms on a finite dimensional vector space.

The lattices of six-dimensional Euclidean space

The lattices of full rank of the six-dimensional Euclidean space are classified according to their automorphism groups (Bravais classification). We find 826 types of such lattices.

On the class number of a unit lattice over a ring of real quadratic integers

Let K be a totally real algebraic number field. In a positive definite quadratic space over K a lattice En is called a unit lattice of rank n if En has an orthonormal basis {e1 …, en}. The class number one problem is to find n and K for which the class number of En is one. Dzewas ([1]), Nebelung ([3]), Pfeuffer ([6], [7]) and Peters ([5]) have settled this problem.