848 research outputs found
Parabolic double cosets in Coxeter groups
International audienceParabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group
A two-sided analogue of the Coxeter complex
For any Coxeter system of rank , we introduce an abstract boolean
complex (simplicial poset) of dimension that contains the Coxeter
complex as a relative subcomplex. Faces are indexed by triples , where
and are subsets of the set of simple generators, and is a
minimal length representative for the parabolic double coset . There
is exactly one maximal face for each element of the group . The complex is
shellable and thin, which implies the complex is a sphere for the finite
Coxeter groups. In this case, a natural refinement of the -polynomial is
given by the "two-sided" -Eulerian polynomial, i.e., the generating function
for the joint distribution of left and right descents in .Comment: 26 pages, several large tables and figure
Distance regularity in buildings and structure constants in Hecke algebras
In this paper we define generalised spheres in buildings using the simplicial
structure and Weyl distance in the building, and we derive an explicit formula
for the cardinality of these spheres. We prove a generalised notion of distance
regularity in buildings, and develop a combinatorial formula for the
cardinalities of intersections of generalised spheres. Motivated by the
classical study of algebras associated to distance regular graphs we
investigate the algebras and modules of Hecke operators arising from our
generalised distance regularity, and prove isomorphisms between these algebras
and more well known parabolic Hecke algebras. We conclude with applications of
our main results to non-negativity of structure constants in parabolic Hecke
algebras, commutativity of algebras of Hecke operators, double coset
combinatorics in groups with -pairs, and random walks on the simplices of
buildings.Comment: J. Algebra, to appea
Deformations of permutation representations of Coxeter groups
The permutation representation afforded by a Coxeter group W acting on the
cosets of a standard parabolic subgroup inherits many nice properties from W
such as a shellable Bruhat order and a flat deformation over Z[q] to a
representation of the corresponding Hecke algebra. In this paper we define a
larger class of ``quasiparabolic" subgroups (more generally, quasiparabolic
W-sets), and show that they also inherit these properties. Our motivating
example is the action of the symmetric group on fixed-point-free involutions by
conjugation.Comment: 44 page
Minimal length elements in some double cosets of Coxeter groups
We study the minimal length elements in some double cosets of Coxeter groups
and use them to study Lusztig's -stable pieces and the generalization of
-stable pieces introduced by Lu and Yakimov. We also use them to study the
minimal length elements in a conjugacy class of a finite Coxeter group and
prove a conjecture in \cite{GKP}.Comment: 35 page
The facial weak order and its lattice quotients
We investigate a poset structure that extends the weak order on a finite
Coxeter group to the set of all faces of the permutahedron of . We call
this order the facial weak order. We first provide two alternative
characterizations of this poset: a first one, geometric, that generalizes the
notion of inversion sets of roots, and a second one, combinatorial, that uses
comparisons of the minimal and maximal length representatives of the cosets.
These characterizations are then used to show that the facial weak order is in
fact a lattice, generalizing a well-known result of A. Bj\"orner for the
classical weak order. Finally, we show that any lattice congruence of the
classical weak order induces a lattice congruence of the facial weak order, and
we give a geometric interpretation of their classes. As application, we
describe the facial boolean lattice on the faces of the cube and the facial
Cambrian lattice on the faces of the corresponding generalized associahedron.Comment: 40 pages, 13 figure
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