Fano Plane Research Articles

Graded contractions of the [formula omitted]-grading on [formula omitted]

Graded contractions of the Z23-grading on the complex exceptional Lie algebra g2 are classified up to equivalence and up to strong equivalence. The non-toral fine Z23-grading is highly symmetric, with all the homogeneous components Cartan subalgebras. This makes possible a combinatorial treatment based on certain nice subsets of the set of 21 edges of the Fano plane. There are 24 such nice sets up to collineation. Each of these is the support of an admissible graded contraction, one of which is present in every equivalence class of graded contractions. Each nice set gives rise to a single Lie algebra, except for three of the cases in which families depending on one or two parameters are found. In particular, a large family of 14-dimensional Lie algebras arise, most of which are solvable. The properties of each of these Lie algebras are studied.

How to Design a Stable Serial Knockout Competition

We investigate a new tournament format that consists of a series of individual knockout tournaments; we call this new format a serial knockout competition (SKC). This format has recently been adopted by the Professional Darts Corporation. Depending on the seedings of the players used for each of the knockout tournaments, players can meet in the various rounds (e.g., first round, second round … semifinal, final) of the knockout tournaments. Following a fairness principle of treating all players equal, we identify an attractive property of an SKCl each pair of players should potentially meet equally often in each of the rounds of the SKC. If the seedings are such that this property is indeed present, we call the resulting SKC stable. In this note, we formalize this notion, and we address the following question. Do there exist seedings for each of the knockout tournaments such that the resulting SKC is stable? We show using a connection to the Fano plane that the answer is yes for eight players, and we prove that the resulting SKC is unique up to permutations of the players. We further prove that stable SKCs exist for any numbers of players that are a power of two, and we provide stable schedules for competitions on 16 and 32 players. Funding: The research of F. Spieksma was partly funded by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 024.002.003].

Harmonic ovals

We show that a central oval in a Moufang Fano plane is a Moufang oval if and only if the oval is harmonic. This shows, for the first time, that harmonic ovals that are not conics exist. In addition, we show that, if a strongly harmonic oval in a Moufang Fano plane exists, then the plane is Pappian and the oval is a conic and that if a harmonic oval in a Moufang plane that is not a Fano plane exists, then the plane is Pappian and the oval is a conic. We also show local versions of these results, characterising conics by the property of: being harmonic at external points on a fixed secant line, being harmonic at external points on a fixed external line, being harmonic at points on a fixed tangent line and being harmonic at all secant lines on a fixed point of the oval. Finally, we give some related characterisation of conics in terms of degenerations of Pascal’s theorem, including the theorem that an oval in a Moufang plane with the four point Pascal property is a conic in a Pappian plane.

Atoms of the lattices of residuated mappings

Given a lattice L, we denote by Res(L) the lattice of all residuated maps on L. The main objective of the paper is to study the atoms of Res(L) where L is a complete lattice. Note that the description of dual atoms of Res(L) easily follows from earlier results of Shmuely (1974). We first consider lattices L for which all atoms of Res(L) are mappings with 2-element range and give a sufficient condition for this. Extending this result, we characterize these atoms of Res(L) which are weakly regular residuated maps in the sense of Blyth and Janowitz (Residuation Theory, 1972). In the rest of the paper we investigate the atoms of Res(M) where M is the lattice of a finite projective plane, in particular, we describe the atoms of Res(F), where F is the lattice of the Fano plane.

Geometric properties of special orthogonal representations associated to exceptional Lie superalgebras

From an octonion algebra [Formula: see text] over a field [Formula: see text] of characteristic not two or three, we show that the fundamental representation [Formula: see text] of the derivation algebra [Formula: see text] and the spinor representation [Formula: see text] of [Formula: see text] are special orthogonal representations. They have particular geometric properties coming from their similarities with binary cubics and we show that the covariants of these representations and their Mathews identities are related to the Fano plane and the affine space [Formula: see text]. This also permits to give constructions of exceptional Lie superalgebras.

Incidence geometry of the Fano plane and Freudenthal’s ansatz for the construction of octonions and split octonions

Incidence geometry of the Fano plane and Freudenthal’s ansatz for the construction of octonions and split octonions

On The Fano Planes in (7n-1)-dimensional Projective Spaces

In this study, we obtain Fano planes whose points are (n-1)-dimensional subspaces and lines are (3n-1)-dimensional subspaces of P in (7n-1)-dimensional projective space P ,using (n;k)-SCID. We give examplesof Fano planes whose the lines of Fano configuration are planes and the points are points ofPG(6,2)and thelines of Fano configuration are 5-spaces and the points are lines ofPG(13,2).

Simplicial complexes from finite projective planes and colored configurations

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as T1⊔T2, such that each Ti represents the lines of a copy of the Fano plane PG(2,F2). We generalize this observation by constructing, for each prime power q, a simplicial complex Xq with q2+q+1 vertices and 2(q2+q+1) facets consisting of two copies of PG(2,Fq). Our construction works for any colored k-configuration, defined as a k-configuration whose associated bipartite graph G is connected and has a k-edge coloring χ:E(G)→[k], such that for all v∈V(G), a,b,c∈[k], following edges of colors a,b,c,a,b,c from v brings us back to v. We give one-to-one correspondences between (1) Sidon sets of order 2 and size k+1 in groups with order n, (2) linear codes with radius 1 and index n in the lattice Ak, and (3) colored (k+1)-configurations with n points and n lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.

Irruption of the New: Truth, Events, History, Parallels, Fidelity

A historical meditation on non-Euclidean geometry, three Jesuits, a radical egalitarian mathematical philosopher, and the atom bomb, structured by word-count with attention to divisors of 441 and the Fano plane.

On alpha -points of q-analogs of the Fano plane

Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. A point P is called an alpha -point of D if the derived design of D in P is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non-alpha -point. For the binary case q = 2, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-alpha -points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of alpha -points implies the existence of a partition of the symplectic generalized quadrangle W(q) into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes q and all even values of q.

Clean tangled clutters, simplices, and projective geometries

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0−1 points, called the setcore of C.In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor.Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

Factors in randomly perturbed hypergraphs

AbstractWe determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a k‐graph H with minimum vertex degree to ensure an F‐factor with high probability, for any F that belongs to a certain class of k‐graphs, which includes, for example, all k‐partite k‐graphs, and the Fano plane. In particular, taking F to be a single edge, this settles a problem of Krivelevich, Kwan, and Sudakov. We also address the case in which the host graph H is not dense, indicating that starting from certain such H is essentially the same as starting from an empty graph (namely, the purely random model).

A Class of Three-Qubit Contextual Configurations Located in Fano Pentads

Given the symplectic polar space of type W(5,2), let us call a set of five Fano planes sharing pairwise a single point a Fano pentad. Once 63 points of W(5,2) are appropriately labeled by 63 non-trivial three-qubit observables, any such Fano pentad gives rise to a quantum contextual set known as a Mermin pentagram. Here, it is shown that a Fano pentad also hosts another, closely related, contextual set, which features 25 observables and 30 three-element contexts. Out of 25 observables, ten are such that each of them is on six contexts, while each of the remaining 15 observables belongs to two contexts only. Making use of the recent classification of Mermin pentagrams (Saniga et al., Symmetry 12 (2020) 534), it was found that 12,096 such contextual sets comprise 47 distinct types, falling into eight families according to the number (3,5,7,…,17) of negative contexts.

Half-Arc-Transitive Graphs and the Fano Plane

A subgroup G of the automorphism group of a graph \(\Gamma\) acts half-arc-transitively on \(\Gamma\) if the natural actions of G on the vertex-set and edge-set of \(\Gamma\) are both transitive, but the natural action of G on the arc-set of \(\Gamma\) is not transitive. When \(G ={\text{Aut}}(\Gamma )\) the graph \(\Gamma\) is said to be half-arc-transitive. Given a bipartite cubic graph with a certain degree of symmetry two covering constructions that provide infinitely many tetravalent graphs admitting half-arc-transitive groups of automorphisms are introduced. Symmetry properties of constructed graphs are investigated. In the second part of the paper the two constructions are applied to the Heawood graph, the well-known incidence graph of the Fano plane. It is proved that the members of the infinite family resulting from one of the two constructions are all half-arc-transitive, and that the infinite family resulting from the second construction contains a mysterious family of arc-transitive graphs that emerged within the classification of tightly attached half-arc-transitive graphs of valence 4 back in 1998 and 2008.

Constructing partial geometries from overlarge sets of Steiner systems

In Woldar (Geometric groups of second order and related combinatorial structures, J. Combin. Designs 28(4) (2020), 307–326), the second author gave a computer-free construction of three overlarge sets of Steiner systems: two of type $$\mathcal {S}(2,3,7)$$ (the Fano plane) and one of type $$\mathcal {S}(3,4,8)$$ (affine design). Presently, we use these constructions to derive four computer-free models of partial geometries. Two of these geometries are of type $$\mathrm{pg}(8,9,4)$$ , one of which is classical. The remaining two, which are of type $$\mathrm{pg}(5,7,3)$$ , are due to Mathon, who also proved that these are the only partial geometries with (5, 7, 3).

Rainbow Erdös--Rothschild Problem for the Fano Plane

The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It is known that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ ve...

M-theory cosmology, octonions, error correcting codes

We study M-theory compactified on twisted 7-tori with G2-holonomy. The effective 4d supergravity has 7 chiral multiplets, each with a unit logarithmic Kähler potential. We propose octonion, Fano plane based superpotentials, codifying the error correcting Hamming (4, 7) code. The corresponding 7-moduli models have Minkowski vacua with one flat direction. We also propose superpotentials based on octonions/error correcting codes for Minkowski vacua models with two flat directions. We update phenomenological α-attractor models of inflation with 3α = 7, 6, 5, 4, 3, 1, based on inflation along these flat directions. These inflationary models reproduce the benchmark targets for detecting B-modes, predicting 7 different values of r=12alpha /{N}_e^2 in the range 10−2 ≳ r ≳ 10−3, to be explored by future cosmological observations.

Turán and Ramsey numbers in linear triple systems

In this paper we study Turán and Ramsey numbers in linear triple systems, defined as 3-uniform hypergraphs in which any two triples intersect in at most one vertex.A famous result of Ruzsa and Szemerédi is that for any fixed c>0 and large enough n the following Turán-type theorem holds. If a linear triple system on n vertices has at least cn2 edges then it contains a triangle: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called s-configurations. The main tool is a generalization of the induced matching lemma from aba-patterns to more general ones.We slightly generalize s-configurations to extendeds-configurations. For these we cannot prove the corresponding Turán-type theorem, but we prove that they have the weaker, Ramsey property: they can be found in any t-coloring of the blocks of any sufficiently large Steiner triple system. Using this, we show that all unavoidable configurations with at most 5 blocks, except possibly the ones containing the sail C15 (configuration with blocks 123, 345, 561 and 147), are t-Ramsey for any t≥1. The most interesting one among them is the wicket, D4, formed by three rows and two columns of a 3 × 3 point matrix. In fact, the wicket is 1-Ramsey in a very strong sense: all Steiner triple systems except the Fano plane must contain a wicket.

A Note on Embedding the 4-arcs of Fano Plane with Quadric and Cubic Veroneseans to Projective Spaces

The 3-spaces generate intersect pairwise in at least a line (each pair of 4-sets share two points). But since the union of two such 4-sets is a 6-set, this union generates the whole space, so they pairwise intersect in a line. Dually, the lines that we consider have empty intersection. Other examples of such sets are spreads of lines, or spreads of any kind. In this work, we present SCID properties spaces generated by 4-arcs of Fano plane............................................................................

On flag-transitive 2-(v,k,2) designs

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n≥3. Alongside this analysis we give a construction for a flag-transitive 2-(v,k−1,k−2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line. Taking the design of points and lines of the projective space PG(n−1,3) as input to this construction yields a G-flag-transitive 2-(v,3,2) design where G has socle PSL(n,3) and v=(3n−1)/2. Apart from these designs, our classification yields exactly one other example, namely the complement of the Fano plane.