284 research outputs found

### Long fully commutative elements in affine Coxeter groups

An element of a Coxeter group $W$ is called fully commutative if any two of
its reduced decompositions can be related by a series of transpositions of
adjacent commuting generators. In the preprint "Fully commutative elements in
finite and affine Coxeter groups" (arXiv:1402.2166), R. Biagioli and the
authors proved among other things that, for each irreducible affine Coxeter
group, the sequence counting fully commutative elements with respect to length
is ultimately periodic. In the present work, we study this sequence in its
periodic part for each of these groups, and in particular we determine the
minimal period. We also observe that in type $A$ affine we get an instance of
the cyclic sieving phenomenon.Comment: 17 pages, 9 figure

### On some polynomials enumerating Fully Packed Loop configurations

We are interested in the enumeration of Fully Packed Loop configurations on a
grid with a given noncrossing matching. By the recently proved
Razumov--Stroganov conjecture, these quantities also appear as groundstate
components in the Completely Packed Loop model. When considering matchings with
p nested arches, these numbers are known to be polynomials in p. In this
article, we present several conjectures about these polynomials: in particular,
we describe all real roots, certain values of these polynomials, and conjecture
that the coefficients are positive. The conjectures, which are of a
combinatorial nature, are supported by strong numerical evidence and the proofs
of several special cases. We also give a version of the conjectures when an
extra parameter tau is added to the equations defining the groundstate of the
Completely Packed Loop model.Comment: 27 pages. Modifications reflecting the recent proof of the
Razumov--Stroganov conjecture; also added some references and a more detailed
conclusio

### Fully Packed Loops in a triangle: matchings, paths and puzzles

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the
study of ordinary Fully Packed Loop configurations (FPLs) on the square grid
where they were used to show that the number of FPLs with a given link pattern
that has m nested arches is a polynomial function in m. It soon turned out that
TFPLs possess a number of other nice properties. For instance, they can be seen
as a generalized model of Littlewood-Richardson coefficients. We start our
article by introducing oriented versions of TFPLs; their main advantage in
comparison with ordinary TFPLs is that they involve only local constraints.
Three main contributions are provided. Firstly, we show that the number of
ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs
and thus it suffices to consider the latter. Secondly, we decompose oriented
TFPLs into two matchings and use a classical bijection to obtain two families
of nonintersecting lattice paths (path tangles). This point of view turns out
to be extremely useful for giving easy proofs of previously known conditions on
the boundary of TFPLs necessary for them to exist. One example is the
inequality d(u)+d(v)<=d(w) where u,v,w are 01-words that encode the boundary
conditions of ordinary TFPLs and d(u) is the number of cells in the Ferrers
diagram associated with u. In the third part we consider TFPLs with d(w)-
d(u)-d(v)=0,1; in the first case their numbers are given by
Littlewood-Richardson coefficients, but also in the second case we provide
formulas that are in terms of Littlewood-Richardson coefficients. The proofs of
these formulas are of a purely combinatorial nature.Comment: 40 pages, 31 figure

### 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

### Combinatorics of fully commutative involutions in classical Coxeter groups

An element of a Coxeter group $W$ is fully commutative if any two of its
reduced decompositions are related by a series of transpositions of adjacent
commuting generators. In the present work, we focus on fully commutative
involutions, which are characterized in terms of Viennot's heaps. By encoding
the latter by Dyck-type lattice walks, we enumerate fully commutative
involutions according to their length, for all classical finite and affine
Coxeter groups. In the finite cases, we also find explicit expressions for
their generating functions with respect to the major index. Finally in affine
type $A$, we connect our results to Fan--Green's cell structure of the
corresponding Temperley--Lieb algebra.Comment: 25 page

### A Bijection between well-labelled positive paths and matchings

A well-labelled positive path of size n is a pair (p,\sigma) made of a word
p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters
of any prefix is non-negative, together with a permutation \sigma of
{1,2,...,n} such that p_i=-1 implies \sigma(i)<\sigma(i+1), while p_i=1 implies
\sigma(i)>\sigma(i+1). We establish a bijection between well-labelled positive
paths of size $n$ and matchings (i.e. fixed-point free involutions) on
{1,2,...,2n}. This proves that the number of well-labelled positive paths is
(2n-1)!!. By specialising our bijection, we also prove that the number of
permutations of size n such that each prefix has no more ascents than descents
is [(n-1)!!]^2 if n is even and n!!(n-2)!! otherwise. Our result also prove
combinatorially that the n-dimensional polytope consisting of all points
(x_1,...,x_n) in [-1,1]^n such that the sum of the first j coordinates is
non-negative for all j=1,2,...,n has volume (2n-1)!!/n!

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