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Hans Havlicek

Publications and source records attributed to Hans Havlicek.

At least 73 records · Page 4Linked to original sources
arXiv2012

Tetrads of lines spanning PG(7,2)

Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of PG(7,2) fall into four orbits omega_1, omega_2, omega_3 omega_4; of respective lengths 12, 54, 108, 81: We show that the 135 points in omega_2 \cup omega_4 are the internal points of a hyperbolic quadric H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.

math.CO↗
arXiv2012

Weak Linear Mappings -- A Survey

There are various concepts of structure preserving mappings in geometry. It is the aim of the present paper to give a survey on geometrical characterizations of some of those mappings. We discuss the results for projective spaces in some detail and report on generalizations to other spaces.

math.AG↗
arXiv2012

Quadratic embeddings

The quadratic Veronese embedding $ρ$ maps the point set $P$ of $\PG{n,F)$ into the point set of $PG({n+2 \choose 2}-1, F$ ($F$ a commutative field) and has the following well-known property: If $M\subset P$, then the intersection of all quadrics containing $M$ is the inverse image of the linear closure of $M^ρ$. In other words, $ρ$ transforms the closure from quadratic into inear. In this paper we use this property to define "quadratic embeddings". We shall prove that if $ν$ is a quadratic embedding of $PG{n,F)$ into $PG(n',F')$ ($F$ a commutative field), then $ρ^{-1}ν$ is dimension-preserving. Moreover, up to some exceptional cases, there is an injective homomorphism of $F$ into $F'$. An additional regularity property for quadratic embeddings allows us to give a geometric characterization of the quadratic Veronese embedding.

math.AG↗
arXiv2012

The Veronese Surface in PG(5,3) and Witt's 5-$(12,6,1$ Design

A conic of the Veronese surface in PG(5,3) is a quadrangle. If one such quadrangle is replaced with its diagonal triangle, then one obtains a point model $K$ for Witt's 5-$(12,6,1)$ design, the blocks being the hyperplane sections containing more than three (actually six) points of $K$. As such a point model is projectively unique, the present construction yields an easy coordinate-free approach to some results obtained independently by H.S.M. Coxeter and G. Pellegrino, including a projective representation of the Mathieu group $M_{12}$ in PG(5,3).

math.CO↗
arXiv2012

Chow's Theorem for Linear Spaces

If $ϕ: L\to L'$ is a bijection from the set of lines of a linear space $(P,L)$ onto the set of lines of a linear space $(P',L')$ ($\dim P, \dim P'\geq 3$), such that intersecting lines go over to intersecting lines in both directions, then $ϕ$ is arising from a collineation of $(P,L)$ onto $(P',L')$ or a collineation of $(P,L)$ onto the dual linear space of $(P',L')$. However, the second possibility can only occur when $(P,L)$ and $(P',L')$ are 3-dimensional generalized projective spaces.

math.AG↗
arXiv2012

On the Matrices of Central Linear Mappings

We show that a central linear mapping of a projectively embedded Euclidean $n$-space onto a projectively embedded Euclidean $m$-space is decomposable into a central projection followed by a similarity if, and only if, the least singular value of a certain matrix has multiplicity $\ge 2m-n+1$. This matrix is arising, by a simple manipulation, from a matrix describing the given mapping in terms of homogeneous Cartesian coordinates.

math.AG↗
arXiv2012

A Characteristic Property of Elliptic Plücker Transformations

We discuss elliptic Plücker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for "classical" elliptic 3-spaces a bijection of its lines is already a Plücker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then "bijection" can be replaced by "injection".

math.AG↗
arXiv2012

Giuseppe Veronese and Ernst Witt -- Neighbours in PG(5,3)

Let $P$ be a point of the Veronese surface $\Vcal$ in \PG53. Then thereare four conics of $\Vcal$ through $P$. We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5-$(12,6,1)$ design. In fact, this construction is "dual" to a similar construction that has been established by the author. We give an explicit parametrization of the cap $\Kcal$; the domain is a dual affine plane which arises from \PG23 by removing one point. Thus, as a by--product, we obtain an easy approach to the extended ternary Golay code $G_{12}$. Finally, we discuss some other procedures that yield 12-sets of points from the Veronese surface.

math.CO↗
arXiv2012

Geometric structures on finite- and infinite-dimensional Grassmannians

In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils (lines of the Grassmann space) and with so-called Z-reguli. We analyse the interdependencies among these different structures.

math.AG↗
arXiv2011

Density conditions for quantum propositions

As has already been pointed out by Birkhoff and von Neumann, quantum logic can be formulated in terms of projective geometry. In three-dimensional Hilbert space, elementary logical propositions are associated with one-dimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis $\{{\vec u},{\vec v},{\vec w}\}$, successive application of the binary logical operation $(x,y)\mapsto (x\vee y)^\perp$ generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis $\{{\vec u},{\vec v},{\vec w}\}$ is orthogonal to the other ones.

math-ph↗
arXiv2011

Aspects of the Segre variety S_{1,1,1}(2)

We consider various aspects of the Segre variety S := S_{1,1,1}(2) in PG(7,2), whose stabilizer group G_S < GL(8, 2) has the structure N {\rtimes} Sym(3), where N := GL(2,2)\times GL(2,2)\times GL(2,2). In particular we prove that S determines a distinguished Z_3-subgroup Z < GL(8, 2) such that AZA^{-1} = Z, for all A in G_S, and in consequence S determines a G_S-invariant spread of 85 lines in PG(7,2). Furthermore we see that Segre varieties S_{1,1,1}(2) in PG(7,2) come along in triplets {S,S',S"} which share the same distinguished Z_3-subgroup Z < GL(8,2). We conclude by determining all fifteen G_S-invariant polynomial functions on PG(7,2) which have degree < 8, and their relation to the five G_S-orbits of points in PG(7,2).

math.CO↗
arXiv2011

On Invariant Notions of Segre Varieties in Binary Projective Spaces

Invariant notions of a class of Segre varieties $\Segrem(2)$ of PG(2^m - 1, 2) that are direct products of $m$ copies of PG(1, 2), $m$ being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains $\Segrem(2)$ and is invariant under its projective stabiliser group $\Stab{m}{2}$. By embedding PG(2^m - 1, 2) into \PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant under $\Stab{m}{2}$ as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as $m$ is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a $\Stab{m}{2}$-invariant geometric spread of lines of PG(2^m - 1, 2). This spread is also related with a $\Stab{m}{2}$-invariant non-singular Hermitian variety. The case $m=3$ is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under $\Stab{3}{2}$, while the points of PG(7, 2) form five orbits.

math.AG↗
arXiv2010

Geometry of free cyclic submodules over ternions

Given the algebra $T$ of ternions (upper triangular $2\times 2$ matrices) over a commutative field $F$ we consider as set of points of a projective line over $T$ the set of all free cyclic submodules of $T^2$. This set of points can be represented as a set of planes in the projective space over $F^6$. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that $T$ admits an $F$-linear antiautomorphism, the plane model of our projective line does not admit any duality.

math.RA↗
arXiv2009

Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group $\vG$, we first construct vector spaces over $\GF(p)$, $p$ a prime, by factorising $\vG$ over appropriate normal subgroups. Then, by expressing $\GF(p)$ in terms of the commutator subgroup of $\vG$, we construct alternating bilinear forms, which reflect whether or not two elements of $\vG$ commute. Restricting to $p=2$, we search for ``refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of $\vG$ is $\leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation'' of several distinct elements of $\vG$. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.

math-ph↗