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

Classification and homological invariants of compact quantum groups of combinatorial type

Compact quantum groups can be found by solving certain combinatorics problems, as first shown by Banica and Speicher. Any system of partitions of finite sets which is closed under reflection and two kinds of concatenation gives rise to a quantum subgroup of the free orthogonal quantum group. Later Freslon, Tarrago and Weber extended this construction to colored partitions. Only recently, Mančinska and Roberson generalized this from finite sets to finite graphs. The present thesis contributes to the classification programs for quantum groups induced by two-colored partitions in Chapter 1 and those induced by uncolored graphs in Chapter 2. While these constructions produce numerous quantum groups, little is known about which of those are actually new and not isomorphic to others. In an effort to elucidate this, Chapter 3 shows that any such quantum group interpolating the unitary group and the free unitary quantum group can be written as a quotient of a wreath graph product of one of the two. Another way of making distinctions between such quantum groups of combinatorial type is to study quantum group invariants, such as cohomology. Chapter 4 computes the first order with trivial coefficients for the discrete duals of all of Tarrago and Weber’s quantum groups. For a handful of those Chapter 5 computes the L²-Betti numbers following Bichon, Kyed and Raum’s method. Chapter 6 proposes a common categorial framework covering all the aforementioned constructions for the first time.Durch das Lösen gewisser Kombinatorikrätsel lassen sich kompakte Quantengruppen finden, wie von Banica und Speicher gezeigt. Jede Sammlung von Partitionen endlicher Mengen, die unter Spiegelung und zwei Arten Konkatenierung abgeschlossen ist, ergibt eine Unterquantengruppe der freien orthogonalen Quantengruppe. Freslon, Tarrago und Weber erweiterten dies auf “gefärbte Partionen”. Erst kürzlich ersetzten Mancinska und Roberson die endlichen Mengen durch endliche Graphen. Die Dissertation trägt zu zwei entsprechenden Klassifikationsvorhaben bei: zweifarbige Partitionen in Kapitel 1, ungefärbte Graphen in Kapitel 2. Zwar ergeben sich viele Quantengruppen. Doch ist nur wenig darüber bekannt, welche davon tatsächlich neu sind. Um dieser Frage nachzugehen, wird in Kapitel 3 bewiesen, dass jede solche Quantengruppe zwischen der unitären Gruppe und der freien unitären Quantengruppe Quotient eines Kranzgraphprodukts einer dieser beiden ist. Eine andere Möglichkeit, solche Quantengruppen kombinatorischen Typs von einander zu unterscheiden bieten Invarianten wie Kohomologie. Von letzterer, mit trivialen Koeffizienten, wird in Kapitel 4 die erste Ordnung berechnet, und zwar für die diskreten Dualen aller von Tarrago und Webers Quantengruppen. Für eine handvoll davon werden in Kapitel 5 noch nach der Methode von Bichon, Kyed und Raum die L2-Betti-Zahlen bestimmt. Kapitel 6 enthält den Vorschlag eines gemeinsamen Rahmens für erstmals alle zuvor genannten Konstruktionen von Quantengruppen.Deutsche Forschungsgemeinschaft, SFB-TRR 195, IRTG scholarshi