Noncrossing partitions and Bruhat order

2011

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik--Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik--Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations

1995

We explore the connection between polygon posets, which is a class of ranked posets with an edgelabeling which satisfies certain 'polygon properties', and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true. The class of polygon posets is seen to include the class of generalized quotients defined by Bjorner and Wachs, while itself being included in the class of alternative generalized quotients also considered by these authors. By studying polygon posets we are then able to answer an open question about common properties of these two classes.

Transactions of the American Mathematical Society, 2011

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection π c ↓ mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of π c ↓ essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of π c ↓. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Journal of Algebraic Combinatorics, 2012

In our recent paper , we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.

2011

We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations

2008

First, we investigate a generalization of the area statistic on Dyck paths for all crystallographic refection groups. In particular, we explore Dyck paths of type B together with an area statistic and a major index. Then, we construct bijections between non-nesting partitions and reverse non-crossing partitions for types A and B. These bijections simultaneously send the area statistic and the major index on non-nesting partitions to the length function and to the sum of the major index and the inverse major index on reverse non-crossing partitions. Finally, we construct bijections between non-nesting partitions and Coxeter sortable elements for types A and B having exactly the same properties.

Journal of Algebra, 2006

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies that any interval is Cohen-Macauley. We also give a Galois connection between intervals in this poset and a boolean poset. This allows us to compute the Möbius function for any interval.

Journal of Combinatorial Theory, Series A, 2003

We define a new family of partial orders generalizing the weak order on Coxeter groups called T-orders, where T is a set of reflections determining the covers in this order. We show that the Grassmann and Lagrange orders on the Coxeter groups of type A n and B n introduced by Bergeron and Sottile are in fact T-orders. These partial orders were used to compute certain products in the cohomology ring of the flag manifolds associated to the complex Chevalley groups of these types. We exhibit T-orders generalizing these orders to partial orders for the Coxeter groups of type D n ; E 6 ; and E 7 : r

arXiv: Combinatorics, 2005

Let $W$ be a finite coxeter group, let $W_I$, $W_J$ be standard parabolic subgroups, let $u$, $v$ be minimal double cosets representatives of double cosets in $W_I W / W_J$ and let $u'$, $v'$ be the maximal representatives of those same two cosets. We show that $u < v$ if and only if $u' < v'$ for the Bruhat order.

arXiv (Cornell University), 2011

In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group W afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of W. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of W related to the descent algebra of W. A consequence of our conjecture is that both the regular character of W and the character of the Orlik-Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of W , one for each conjugacy class of elements of W. The refined conjecture has been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjecture computationally for a given finite Coxeter group. We use these tools to verify the conjecture for all finite Coxeter groups of rank three and four, thus providing previously unknown decompositions of the regular characters and the Orlik-Solomon characters of these groups.