Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

The facial weak order in finite Coxeter groups

International audience We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. 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. Bjo ̈rner 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 its classes.

Tamari Lattices for Parabolic Quotients of the Symmetric Group

We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group $\mathfrak{S}_{n}$. We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.

Equivariant perverse sheaves on Coxeter arrangements and buildings

When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ of $W$-equivariant perverse sheaves on $E_{\mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtman's recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations. Comment: 28 pages, 6 figures. v5 processed for publication in Epiga

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H 3, i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter–Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H 3 is given and presented as a `pancake structure'.

Maximal length elements of excess zero in finite Coxeter groups

In this paper we prove that for W a finite Coxeter group and C a conjugacy class of W, there is always an element of C of maximal length in C which has excess zero. An element {w\in W} has excess zero if there exist elements {\sigma,\tau\in W} such that {\sigma^{2}=\tau^{2}=1,w=\sigma\tau} and {\ell(w)=\ell(\sigma)+\ell(\tau)} , {\ell} being the length function on W.

Generalized Non-Crossing Partitions and Buildings

For any finite Coxeter group $W$ of rank $n$ we show that the order complex of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give a new proof of the fact that the non-crossing partition lattice in type $A_n$ is supersolvable for all $n$. Moreover, we show that in case $B_n$, this is only the case if $n<4$. We also obtain a lower bound on the radius of the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the radius is $\binom{n}{2}$. A Corrigendum for this paper was added on May 17, 2018.

-GRAPHS AND GYOJA’S -GRAPH ALGEBRA

Let $(W,S)$ be a finite Coxeter group. Kazhdan and Lusztig introduced the concept of $W$-graphs, and Gyoja proved that every irreducible representation of the Iwahori–Hecke algebra $H(W,S)$ can be realized as a $W$-graph. Gyoja defined an auxiliary algebra for this purpose which—to the best of the author’s knowledge—was never explicitly mentioned again in the literature after Gyoja’s proof (although the underlying ideas were reused). The purpose of this paper is to resurrect this $W$-graph algebra, and to study its structure and its modules. A new explicit description of it as a quotient of a certain path algebra is given. A general conjecture is proposed which would imply strong restrictions on the structure of $W$-graphs. This conjecture is then proven for Coxeter groups of type $I_{2}(m)$, $B_{3}$ and $A_{1}$–$A_{4}$.