Related papers: Homotopy type of intervals of the second higher Br…
In a companion manuscript, we introduce a stratification of intersections of a top dimensional real Bruhat cells with another arbitrary cell. This intersection is naturally identified with a subset of the lower triangular group: these…
We study the appearance of notable interval structures -- lattices, modular lattices, distributive lattices, and boolean lattices -- in both the Bruhat and weak orders of Coxeter groups. We collect and expand upon known results for…
We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular we study the restriction of these orders to noncrossing partitions and show…
In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that,…
The number of Bruhat intervals in Coxeter groups is finite, and for the first few lengths, the intervals were described up to an isomorphism by A. Hultman using the correspondence between Bruhat intervals and cell decompositions of a 2d…
We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type $\widetilde{A}_2$ we show that each interval coincides with a generalized permutohedron minus a star-shaped polygon, and we prove a subtler…
We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded…
We show that the order complex of intervals of a poset, ordered by inclusion, is a Tchebyshev triangulation of the order complex of the original poset. Besides studying the properties of this transformation, we show that the dual of the…
We study the poset topology of lattices arising from orientations of 1-skeleta of directionally simple polytopes, with Bruhat interval polytopes $Q_{e,w}$ as our main example. We show that the order complex $\Delta ((u,v)_w)$ of an interval…
Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these…
A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order.…
We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals,…
The higher Bruhat orders are partial orders that generalize the weak order on the symmetric group $S_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in $S_n$, where covering…
The Bruhat order on a Coxeter group is often described by examining subexpressions of a reduced expression. We prove that an analogous description applies to the Bruhat order on double cosets. This establishes the compatibility of the…
We propose versions of higher Bruhat orders for types $B$ and $C$. This is based on a theory of higher Bruhat orders of type~A and their geometric interpretations (due to Manin--Shekhtman, Voevodskii--Kapranov, and Ziegler), and on our…
Let u and v be permutations on n letters, with u <= v in Bruhat order. A Bruhat interval polytope Q_{u,v} is the convex hull of all permutation vectors z = (z(1), z(2),...,z(n)) with u <= z <= v. Note that when u=e and v=w_0 are the…
Arrangements of pseudohyperplanes are widely studied in computational geometry. A rich subclass of pseudohyerplane arrangements, which has gained more attention in recent years, is the so-called signotopes. Introduced by Manin and…
In this paper we study those generic intervals in the Bruhat order of the symmetric group that are isomorphic to the principal order ideal of a permutation w, and consider when the minimum and maximum elements of those intervals are related…
The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations…
Applying a classical theorem of Smith, we show that the poset property of being Gorenstein$^*$ over $\mathbb{Z}_2$ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every…