Related papers: Eigenspace arrangements of reflection groups

This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and…

This paper continues the study of the poset of eigenspaces of elements of a unitary reflection group (for a fixed eigenvalue), which was commenced in [6] and [5]. The emphasis in this paper is on the representation theory of unitary…

The eigenspace theory of unitary reflection groups, initiated by Springer and Lehrer, suggests that the following object is worthy of study: the poset of eigenspaces of elements of a unitary reflection group, for a fixed eigenvalue, ordered…

Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable, a notion due to Stanley. Jambu and Terao showed that…

Let $W\subset GL(V)$ be a complex reflection group, and ${\mathscr A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X({\mathscr A}(W))$ of the reflection arrangement ${\mathscr A}(W)$ is a…

Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable. For X in L(A), it is known that the restriction A^X is…

For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice…

In this article we prove that the lattice of noncrossing partitions is EL-shellable when associated with the well-generated complex reflection group of type $G(d,d,n)$, for $d,n\geq 3$, or with the exceptional well-generated complex…

Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a…

Reflection groups, geometry of the discriminant and noncrossing partitions. When W is a well-generated complex reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of…

In a finite real reflection group, the reflection length of each element is equal to the codimension of its fixed space, and the two coincident functions determine a partial order structure called the absolute order. In complex reflection…

In this article we give a simple, almost uniform proof that the lattice of noncrossing partitions associated with a well-generated complex reflection group is lexicographically shellable. So far a uniform proof is available only for Coxeter…

Let $V$ be a finite dimensional complex vector space and $W\subset \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. A classical conjecture predicts that $V^{\reg}$ is a…

Ehrenborg and Jung recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a…

The complement of an arrangement A of a finite number of affine hyperplanes in complex n-space has the structure of a poset of spaces indexed by the intersection poset, L(A). The space corresponding to G in L(A) is homotopy equivalent to…

Suppose that W is a finite, unitary, reflection group acting on the complex vector space V. Let A = A(W) be the associated hyperplane arrangement of W. Terao has shown that each such reflection arrangement A is free. Let L(A) be the…

For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $\Pi^{\bullet}_{\vec{c}}$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with…

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 connected chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give…

Introduced by Reading, the shard intersection order of a finite Coxeter group $W$ is a lattice structure on the elements of $W$ that contains the poset of noncrossing partitions $NC(W)$ as a sublattice. Building on work of Bancroft in the…

We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…