Classification of 8-Dimensional Trilinear Alternating Forms over GF(2)

Let V be an n-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms we introduce a new invariant called radical polynomial and investigate its behaviour, in particular in the case of the 2-element field. We show that it is compatible with direct products of forms and how it is related to its values on dimension n ? 1. Moreover, it turns out that it is full up to dimension 7. On the other hand, on higher dimensions it is no more full and it is necessary to generalize it to obtain (using computer) a classification of forms on dimension 8 over the 2-element field. This classification is provided, together with the sizes of stabilizers of the corresponding forms.     

Singular lines of trilinear forms

We prove that an alternating e-form on a vector space over a quasi-algebraically closed field always has a singular (e-1)-dimensional subspace, provided that the dimension of the space is strictly greater than e. Here an (e-1)-dimensional subspace is called singular if pairing it with the e-form yields zero. By the theorem of Chevalley and Warning our result applies in particular to finite base fields. Our proof is most interesting in the case where e=3 and the space has odd dimension n; then it involves a beautiful equivariant map from alternating trilinear forms to polynomials of degree . We also give a sharp upper bound on the dimension of subspaces all of whose two-dimensional subspaces are singular for a non-degenerate trilinear form. In certain binomial dimensions the trilinear forms attaining this upper bound turn out to form a single orbit under the general linear group, and we classify their singular lines.     

Space of invariant bilinear forms

Let \({\mathbb {F}}\) be a field, V a vector space of dimension n over \({\mathbb {F}}\). Then the set of bilinear forms on V forms a vector space of dimension \(n^2\) over \({\mathbb {F}}\). For char \({\mathbb {F}}\ne 2\), if T is an invertible linear map from V onto V then the set of T-invariant bilinear forms, forms a subspace of this space of forms. In this paper, we compute the dimension of T-invariant bilinear forms over \({\mathbb {F}}\). Also we investigate similar type of questions for the infinitesimally T-invariant bilinear forms (T-skew symmetric forms). Moreover, we discuss the existence of nondegenerate invariant (resp. infinitesimally invariant) bilinear forms.     

Autotopisms and Isotopisms of Trilinear Alternating Forms

For a trilinear alternating form f on a vector space V, a generalization of the group of automorphisms group of autotopisms Atp(f), is introduced. An autotopism of f is a triple (α, β, γ) of automorphisms of V satisfying f(u, v, w) = f(α(u), β(v), γ(w)) for all u, v, w ∈ V. Basic results concerning this group are presented, and it is shown that the subgroup of Atp(f) containing autotopisms with identity in one coordinate is Abelian and that a mapping in this group has no fixed points if and only if its order is not a power of two.

Moreover, the notion of equivalence of two trilinear alternating forms is generalized in a similar way, and a partial result is given.

Examples of forms with both trivial (Atp(f) = Aut(f)) and nontrivial groups of autotopisms are presented.     

Trilinear products and comtrans algebra representations

Comtrans algebras are modules over a commutative ring R equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. The standard construction of a comtrans algebra uses the ternary commutator and translator of a trilinear product. If 6 is invertible in R, then each comtrans algebra arises in this standard way from the so-called bogus product.Consider a vector space E of dimension n over a field R. While the dimension of the space of all trilinear products on E is n⁴, the dimension of the space of all comtrans algebras on E is less, namely . The paper determines which trilinear products may be represented as linear combinations of the commutator and translator of a comtrans algebra. For R not of characteristic 3, the necessary and sufficient condition for such a representation is the strong alternativityxxy+xyx+yxx=0 of the trilinear product xyz. For R also not of characteristic 2, it is shown that the representation may be given by the bogus product. A suitable representation for the characteristic 2 case is also obtained.     

Arithmetic properties of similitude theta lifts from orthogonal to symplectic groups

Adapting the work of Kudla and Millson we obtain a lifting of cuspidal cohomology classes for the symmetric space associated to GO(V) for an indefinite rational quadratic space V of even dimension to holomorphic Siegel modular forms on GSp _n (A). For n =  2 we prove the Thom Lemma for hyperbolic 3-space, which together with results of Kudla and Millson imply an interpretation of the Fourier coefficients of the theta lift as period integrals of the cohomology class over certain cycles, and relates those over infinite geodesics to L-values of cusp forms for GL₂ over imaginary quadratic fields. This allows us to prove, for almost all primes p, the p-integrality of the lift for a particular choice of Schwartz function. We further calculate the Hecke eigenvalues (including for some “bad” places) for this choice in the case of V of signature (3,1).     

The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field

Generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over GF(q²) and for the number of equivalence (or congruence) classes of nondegenerate bilinear forms of rank n over GF(q).     

Finite Homogeneous 3-Graphs

By “3-graph” we mean a pair (V, E) such that E ? [V]³. We show that the only non-trivial finite 3-graphs homogeneous in the sense of Fraïssé are those associated with the projective planes over GF(2) and GF(3), and with the projective lines over GF(5) and GF(9). To exclude other possibilities we use the classification of doubly transitive finite permutation groups.     

Pfaffian structures and critical problems in finite symplectic spaces

Just as matroids abstract the algebraic properties of determinants in a vector space, Pfaffian structures abstract the algebraic properties of Pfaffians or skew-symmetric determinants in a symplectic space (that is, a vector space with an alternating bilinear form). This is done using an exchange-augmentation axiom which is a combinatorial version of a Laplace expansion or straightening identity for Pfaffians. Using Pfaffian structures, we study a symplectic analogue of the classical critical problem: given a setS of non-zero vectors in a non-singular symplectic spaceV of dimension2m, find its symplectic critical exponent, that is, the minimum of the set {m?dim(U):U∩S=0}, whereU ranges over all the (totally) isotropic subspaces disjoint fromS. In particular, we derive a formula for the number of isotropic subspaces of a given dimension disjoint from the setS by Möbius inversion over the order ideal of isotropic flats in the lattice of flats of the matroid onS given by linear dependence. This formula implies that the symplectic critical exponent ofS depends only on its matroid and Pfaffian structure; however, it may depend on the dimension of the symplectic spaceV.     

The multiplications in GF(q)[x]⧸(a(x)) considered as linear transformations

We study in this paper the ring V(a)=GF(q)[x?(a(x)), which also can be considered as a linear space over GF(q). In this context the multiplications are considered as linear mappings of V(a) into V(a). The ideals (invariant subspaces under multiplications) are used to formulate properties of subspaces of V(a).     

Orbits under actions of affine groups over GF (2)

Let G be a group (or vector space) and A a group of transformations of G. A then acts as a group of transformations of P(G), the set of subsets of G. It is meaningful to study the orbit structure of P(G) under the action of A. The question of the existence of elements of P(G) with trivial isotropy subgroup seems to be of interest in studying the action of A on G. In this paper actions of affine groups over GF (2) are considered. It is proved, by an inductive construction, that every vector space over GF (2) of dimension at least six contains a subset with trivial isotropy subgroup.     

Test vectors for trilinear forms when at least one representation is not supercuspidal

Given three irreducible, admissible, infinite dimensional complex representations of GL₂(F), with F a local non-Archimedean field, the space of trilinear functionals invariant by the group has dimension at most one. When it is one we provide an explicit vector on which the functional does not vanish assuming that not all three representations are supercuspidal.     

The moduli space of (3,3,3) trilinear forms

LetU, V andW be three dimensional vector spaces over ∉ (or an alebraically closed field with characteristic not equal to 2 or 3). We prove that the moduli space of trilinear forms onU ^* ⊗V ^* ⊗W ^* is isomorphic to ℙ² by applying Geometric Invariant Theory to the action ofPGL(U)×PGL(V)×PGL(W) on ℙU⊗V⊗W).     

Prehomogeneous vector spaces and ergodic theory II

We apply M. Ratner's theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.

     

Generators of Spn(V) over a quasi semilocal semihereditary ring

This paper is devoted to determine the minimal length of expressions of an isometry in a symplectic group Sp_n(V) by a product of transvections under the assumption that V is an n-ary nonsingular alternating space over a quasi semilocal semihereditary ring with 2 as a unit.     

An analytic version of the De Rham theorem

LetV be a real analytic paracompact variety; in §1 of this paper we prove that: $$H^q (V,R) \approx \frac{{closed analytic differentiable q - forms on V}}{{exact analytic differentiable q - forms on V}}$$ Then we prove that the closed (exact) analytic differentiableq-forms onV are dense, in the Whitney topology, in the set of closed (exact) differentiableq-forms onV. We also consider the problem of extending closed (exact) analytic differentiableq-forms, defined on a subvarietyX ofV, to closed (exact) analytic forms defined onV.     

On the trivectors of a 6-dimensional symplectic vector space

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp₆(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ?³V of V and this action defines five families of nonzero trivectors of V (four of whose are orbits for any choice of F). In this paper, we divide three of these five families into orbits for the action of Sp(V,f)⊆GL(V) on ?³V.     

The weight distribution of linear codes over GF(ql) having generator matrix over GF(q>)

We study the weight distribution of the linear codes over GF(q^l) which have generator matrices over GF(q) and their dual codes. As an application we find the weight distribution of the irreducible cyclic (23(2¹≈1), 111) codes over GF(2) for all lnot divisible by 11.     

A conversion algorithm for logarithms on GF(2n)

We present a method for expressing a root of one irreducible polynomial of degree n over GF(2) in terms of a basis of GF(2ⁿ) over GF(2) associated with another. This allows us, when both polynomials are primitive, to find logarithms relative to one polynomial from logarithms relative to the other.     

Directed graph representation of half-rate additive codes over GF(4)

We show that (n, 2 ⁿ ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 ⁿ ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.