62 research outputs found
Realizations of the associahedron and cyclohedron
We describe many different realizations with integer coordinates for the
associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the
Bott-Taubes polytope) and compare them to the permutahedron of type A_n and B_n
respectively.
The coordinates are obtained by an algorithm which uses an oriented Coxeter
graph of type A_n or B_n respectively as only input and which specialises to a
procedure presented by J.-L. Loday for a certain orientation of A_n. The
described realizations have cambrian fans of type A and B as normal fans. This
settles a conjecture of N. Reading for cambrian fans of these types.Comment: v2: 18 pages, 7 figures; updated version has revised introduction and
updated Section 4; v3: 21 pages, 2 new figures, added statement (b) in
Proposition 1.4. and 1.7 plus extended proof; added references [1], [29],
[30]; minor changes with respect to presentatio
On inversion sets and the weak order in Coxeter groups
In this article, we investigate the existence of joins in the weak order of
an infinite Coxeter group W. We give a geometric characterization of the
existence of a join for a subset X in W in terms of the inversion sets of its
elements and their position relative to the imaginary cone. Finally, we discuss
inversion sets of infinite reduced words and the notions of biconvex and
biclosed sets of positive roots.Comment: 22 pages; 10 figures; v2 some references were added; v2: final
version, to appear in European Journal of Combinatoric
Automata, reduced words, and Garside shadows in Coxeter groups
In this article, we introduce and investigate a class of finite deterministic
automata that all recognize the language of reduced words of a finitely
generated Coxeter system (W,S). The definition of these automata is
straightforward as it only requires the notion of weak order on (W,S) and the
related notion of Garside shadows in (W,S), an analog of the notion of a
Garside family. Then we discuss the relations between this class of automata
and the canonical automaton built from Brink and Howlett's small roots. We end
this article by providing partial positive answers to two conjectures: (1) the
automata associated to the smallest Garside shadow is minimal; (2) the
canonical automaton is minimal if and only if the support of all small roots is
spherical, i.e., the corresponding root system is finite.Comment: 21 pages, 7 figures; v2: 23 pages, 8 figures, Remark 3.15 added,
accepted in Journal of Algebra, computational sectio
Polytopal realizations of finite type -vector fans
This paper shows the polytopality of any finite type -vector fan,
acyclic or not. In fact, for any finite Dynkin type , we construct a
universal associahedron with the property
that any -vector fan of type is the normal fan of a
suitable projection of .Comment: 27 pages, 9 figures; Version 2: Minor changes in the introductio
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