16 research outputs found
Conjugacy of Coxeter elements
For a Coxeter group (W,S), a permutation of the set S is called a Coxeter
word and the group element represented by the product is called a Coxeter
element. Moving the first letter to the end of the word is called a rotation
and two Coxeter elements are rotation equivalent if their words can be
transformed into each other through a sequence of rotations and legal
commutations.
We prove that Coxeter elements are conjugate if and only if they are rotation
equivalent. This was known for some special cases but not for Coxeter groups in
general
Posets from Admissible Coxeter Sequences
We study the equivalence relation on the set of acyclic orientations of an
undirected graph G generated by source-to-sink conversions. These conversions
arise in the contexts of admissible sequences in Coxeter theory, quiver
representations, and asynchronous graph dynamical systems. To each equivalence
class we associate a poset, characterize combinatorial properties of these
posets, and in turn, the admissible sequences. This allows us to construct an
explicit bijection from the equivalence classes over G to those over G' and G",
the graphs obtained from G by edge deletion and edge contraction of a fixed
cycle-edge, respectively. This bijection yields quick and elegant proofs of two
non-trivial results: (i) A complete combinatorial invariant of the equivalence
classes, and (ii) a solution to the conjugacy problem of Coxeter elements for
simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and
K. Eriksson using a much different approach.Comment: 16 pages, 4 figures. Several examples have been adde
Coxeter Groups and Asynchronous Cellular Automata
The dynamics group of an asynchronous cellular automaton (ACA) relates
properties of its long term dynamics to the structure of Coxeter groups. The
key mathematical feature connecting these diverse fields is involutions.
Group-theoretic results in the latter domain may lead to insight about the
dynamics in the former, and vice-versa. In this article, we highlight some
central themes and common structures, and discuss novel approaches to some open
and open-ended problems. We introduce the state automaton of an ACA, and show
how the root automaton of a Coxeter group is essentially part of the state
automaton of a related ACA.Comment: 10 pages, 4 figure
Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets
Though iterated maps and dynamical systems are not new to combinatorics, they have enjoyed a renewed prominence over the past decade through the elevation of the subfield that has become known as dynamical algebraic combinatorics. Some of the problems that have gained popularity can also be cast and analyzed as finite asynchronous cellular automata (CA). However, these two fields are fairly separate, and while there are some individuals who work in both, that is the exception rather than the norm. In this article, we will describe our ongoing work on toggling independent sets on graphs. This will be preceded by an overview of how this project arose from new combinatorial problems involving homomesy, toggling, and resonance. Though the techniques that we explore are directly applicable to ECA rule 1, many of them can be generalized to other cellular automata. Moreover, some of the ideas that we borrow from cellular automata can be adapted to problems in dynamical algebraic combinatorics. It is our hope that this article will inspire new problems in both fields and connections between them
Morphisms and order ideals of toric posets
Toric posets are cyclic analogues of finite posets. They can be viewed
combinatorially as equivalence classes of acyclic orientations generated by
converting sources into sinks, or geometrically as chambers of toric graphic
hyperplane arrangements. In this paper we study toric intervals, morphisms, and
order ideals, and we provide a connection to cyclic reducibility and conjugacy
in Coxeter groups.Comment: 28 pages, 8 figures. A 12-page "extended abstract" version appears as
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Noncrossing partitions, toggles, and homomesies
We introduce n(n-1)/2 natural involutions ("toggles") on the set S of non-crossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T-orbit is the same for all T-orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called "2-cliquish." More generally, the philosophy of this "toggle-action," proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics
Toric partial orders
We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders