ONE BY ONE EMBEDDING THE CROSSED HYPERCUBE INTO PANCAKE GRAPH

Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from vertices of G to the vertices of H. The dilation of embedding is the maximum distance between f(u), f(v) taken over edges (u, v) of G. The Pancake graph is one as viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The Pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computer. Some good attractive properties of this interconnection network include: vertex symmetry, small degree, a sub-logarithmic diameter, extendability, and high connectivity (robustness), easy routing and regularity of topology, fault tolerance, extensibility and embeddability of others topologies. In this paper, we give a construction of one by one embedding of dilation 5 of crossed hypercube into Pancake graph

Quantum permutations and quantum reflections

The permutation group $S_N$ has a quantum analogue $S_N^+$, which is infinite at $N\geq4$. We review the known facts regarding $S_N^+$, and its versions $S_F^+$, with $F$ being a finite quantum space. We discuss then the structure of the closed subgroups $G\subset S_N^+$ and $G\subset S_F^+$, with particular attention to the quantum reflection groups.Comment: 400 pages. arXiv admin note: text overlap with arXiv:1909.0815

Between primitive and 2-transitive : synchronization and its friends

The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe

On 4-Dimensional Point Groups and on Realization Spaces of Polytopes

This dissertation consists of two parts. We highlight the main results from each part. Part I. 4-Dimensional Point Groups. (based on a joint work with Günter Rote.) We propose the following classification for the finite groups of orthogonal transformations in 4-space, the so-called 4-dimensional point groups. Theorem A. The 4-dimensional point groups can be classified into * 25 polyhedral groups (Table 5.1), * 21 axial groups (7 pyramidal groups, 7 prismatic groups, and 7 hybrid groups, Table 6.3), * 22 one-parameter families of tubical groups (11 left tubical groups and 11 right tubical groups, Table 3.1), and * 25 infinite families of toroidal groups (2 three-parameter families, 19 two-parameter families, and 4 one-parameter families, Table 4.3.) In contrast to earlier classifications of these groups (notably by Du Val in 1962 and by Conway and Smith in 2003), see Section 1.7), we emphasize a geometric viewpoint, trying to visualize and understand actions of these groups. Besides, we correct some omissions, duplications, and mistakes in these classifications. The 25 polyhedral groups (Chapter 5) are related to the regular polytopes. The symmetries of the regular polytopes are well understood, because they are generated by reflections, and the classification of such groups as Coxeter groups is classic. We will deal with these groups only briefly, dwelling a little on just a few groups that come in enantiomorphic pairs (i.e., groups that are not equal to their own mirror.) The 21 axial groups (Chapter 6) are those that keep one axis fixed. Thus, they essentially operate in the three dimensions perpendicular to this axis (possibly combined with a flip of the axis), and they are easy to handle, based on the well-known classification of the three-dimensional point groups. The tubical groups (Chapter 3) are characterized as those that have (exactly) one Hopf bundle invariant. They come in left and right versions (which are mirrors of each other) depending on the Hopf bundle they keep invariant. They are so named because they arise with a decomposition of the 3-sphere into tube-like structures (discrete Hopf fibrations). The toroidal groups (Chapter 4) are characterized as having an invariant torus. This class of groups is where our main contribution in terms of the completeness of the classification lies. We propose a new, geometric, classification of these groups. Essentially, it boils down to classifying the isometry groups of the two-dimensional square flat torus. We emphasize that, regarding the completeness of the classification, in particular concerning the polyhedral and tubical groups, we rely on the classic approach (see Section 1.6). Only for the toroidal and axial groups, we supplant the classic approach by our geometric approach. We give a self-contained presentation of Hopf fibrations (Chapter 2). In many places in the literature, one particular Hopf map is introduced as “the Hopf map”, either in terms of four real coordinates or two complex coordinates, leading to “the Hopf fibration”. In some sense, this is justified, as all Hopf bundles are (mirror-)congruent. However, for our characterization, we require the full generality of Hopf bundles. As a tool for working with Hopf fibrations, we introduce a parameterization for great circles in S^3 , which might be useful elsewhere. Our main tool to understand tubical groups are polar orbit polytopes. (Chapter 1). In particular, we study the symmetries of a cell of the polar orbit polytope for different starting points. Part II. Realization Spaces of Polytopes (based on a joint work with Rainer Sinn and Günter M. Ziegler.) Robertson in 1988 suggested a model for the realization space of a d-dimensional polytope P, and an approach via the implicit function theorem to prove that the realization space is a smooth manifold of dimension NG(P) := d(f_0 + f_{d−1} ) - f{0,d-1} . We call NG(P) the natural guess for (the dimension of the realization space of) P. We build on Robertson's model and approach to study the realization spaces of higher-dimensional polytopes. We conclude combinatorial criteria (Sections 9.3.3 and 9.4.1) to decide if the realization space of the polytope in consideration is a smooth manifold of dimension given by the natural guess. As another application, we study the realization spaces of the second hypersimplices (Section 9.4.2). We apply these criteria on 4-polytopes with small number of vertices, and along the way, we analyze examples where Robertson’s approach fails, identifying the three smallest examples of 4-polytopes, for which the realization space is still a smooth manifold, but its dimension is not given by the natural guess (Section 9.5). Finally, we investigate the realization space of the 24-cell (Section 9.5.2). We construct families of realizations of the 24-cell, and using them we show that the realization space of the 24-cell has points where it is not a smooth manifold. This provides the first known example of a polytope whose realization space is not a smooth manifold. We conclude by showing that the dimension of the realization space of the 24-cell is at least 48 and at most 52.Diese Dissertation befasst sich mit zwei verschiedenen Themen, von denen jedes seinen eigenen Teil hat. Der erste Teil befasst sich mit 4-dimensionalen Punktgruppen. Wir schlagen eine neue Klassifizierung für diese Gruppen vor (siehe Theorem A), die im Gegensatz zu früheren Klassifizierungen eine geometrische Sichtweise betont und versucht, die Aktionen dieser Gruppen zu visualisieren und zu verstehen. Im Folgenden werden diese Gruppen kurz beschrieben. Die polyedrischen Gruppen (Kapitel 5) sind mit den regelmäßigen Polytopen verwandt. Die axialen Gruppen (Kapitel 6) sind diejenigen, die eine Achse festhalten. Die schlauchartigen Gruppen (Kapitel 3) werden als solche charakterisiert, die genau eine invariantes Hopf-Bündel haben. Sie entstehen bei einer Zerlegung der 3-Sphäre in schlauchartige Strukturen (diskrete Hopf-Faserungen). Die toroidalen Gruppen (Kapitel 4) sind dadurch gekennzeichnet, dass sie einen invarianten Torus haben. Wir schlagen eine neue, geometrische Klassifizierung dieser Gruppen vor. Im Wesentlichen läuft sie darauf hinaus, die Isometriegruppen des zweidimensionalen quadratischen flachen Torus zu klassifizieren. Nebenbei geben wir eine in sich geschlossene Darstellung der Hopf-Faserungen (Kapitel 2). Als Hilfsmittel für die Arbeit mit ihnen führen wir eine Parametrisierung für Großkreise in S 3 ein, die an anderer Stelle nützlich sein könnte. Der zweite Teil befasst sich mit Realisierungsräumen von Polytopen. Wir bauen auf Robertsons Modell und Ansatz auf, um die Realisierungsräume von Polytopen zu untersuchen. Wir stellen kombinatorische Kriterien auf (Abschnitte 9.3.3 und 9.4.1), um zu entscheiden, ob der Realisierungsraum des betrachteten Polytops eine glatte Mannigfaltigkeit der durch die “natürliche Vermutung” gegebenen Dimension ist. Als weitere Anwendung, untersuchen wir die Realisierungsräume der zweiten Hypersimplices (Abschnitt 9.4.2). Nebenbei identifizieren wir die kleinsten Beispiele von 4-Polytopen, für die dieser Ansatz versagt (Abschnitt 9.5). Schließlich untersuchen wir den Realisierungsraum der 24-Zelle (Abschnitt 9.5.2). Wir zeigen, dass es Punkte gibt, an denen sie keine glatte Mannigfaltigkeit ist. Zuletzt zeigen wir, dass seine Dimension mindestens 48 und höchstens 52 beträgt

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ in $\mathbb{R}^n$. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ is the projection of the positive Grassmannian $Gr^{\geq 0}_{k,n}$ into $Gr_{k,k+2}$ under a map $\tilde{Z}$ induced by a matrix $Z\in \text{Mat}_{n,k+2}^{>0}$. Introduced in the context of scattering amplitudes, it is not a polytope, and has dimension $2k$. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first noted by Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- images of positroid cells of $Gr^{\geq 0}_{k+1,n}$ under the moment map -- translate into sign conditions characterizing the T-dual Grasstopes -- images of positroid cells of $Gr^{\geq 0}_{k,n}$ under $\tilde{Z}$. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We prove the main conjecture of (LPW): a collection of positroid polytopes is a triangulation of $\Delta_{k+1, n}$ if and only if the collection of T-dual Grasstopes is a triangulation of $\mathcal{A}_{n,k,2}(Z)$ for all $Z$. Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of $\mathcal{A}_{n,k,2}(Z)$, and Lukowski--Parisi--Spradlin--Volovich's conjectures on $m=2$ cluster adjacency and on generalized triangles (images of $2k$-dimensional positroid cells which map injectively into $\mathcal{A}_{n,k,2}(Z)$). Finally, we introduce new cluster structures in the amplituhedron.Comment: 72 pages, many figures, comments welcome. v4: Minor edits v3: Strengthened results on triangulations and realizability of amplituhedron sign chambers. v2: Results added to Section 11.4, minor edit