Related papers: Inversion arrangements and the weak Bruhat order

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…

We extend the weak Bruhat order of a finite Coxeter group to the set of its coclasses, modulo parabolic standard subgroups. We use this order to describe associative algebra structures on the vector spaces spanned by the faces of…

For an element $w$ of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory $\mathcal{F}(w)$ of a module category over a preprojective algebra of Dynkin type. This paper aims at studying categorical properties of…

Hultman, Linusson, Shareshian, and Sj\"ostrand gave a pattern avoidance characterization of the permutations for which the number of chambers of its associated inversion arrangement is the same as the size of its lower interval in Bruhat…

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the $\mathfrak{sl}_{n+1}$ subregular W-algebra can be realised in terms of the…

In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof…

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary $W$-algebra. We show that for a minimal simple $W$-algebra…

We define a partial order $\mathcal{P}_n$ on permutations of any given size $n$, which is the image of a natural partial order on inversion sequences. We call this the ``middle order''. We demonstrate that the poset $\mathcal{P}_n$ refines…

We characterize permutations whose Bruhat graphs can be drawn in the plane and those whose Bruhat graphs can be drawn in the torus. In particular, we show these properties are characterized by avoiding finitely many permutations.

We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their…

We give a geometric proof of inverse Hamiltonian reduction for all finite W-algebras in type $A$, a certain embedding of the finite W-algebra corresponding to an arbitrary nilpotent in $\mathfrak{gl}_N$ into that corresponding to a larger…

We show that the order dimension of the weak order on a Coxeter group of type A, B or D is equal to the rank of the Coxeter group, and give bounds on the order dimensions for the other finite types. This result arises from a unified…

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the…

The classical permutohedron Perm is the convex hull of the points (w(1),...,w(n)) in R^n where w ranges over all permutations in the symmetric group. This polytope has many beautiful properties -- for example it provides a way to visualize…

Hyperplane arrangements dissect $\mathbb{R}^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem…

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of the weak order…

Let $s$ be West's deterministic stack-sorting map. A well-known result (West) is that any length $n$ permutation can be sorted with $n-1$ iterations of $s.$ In 2020, Defant introduced the notion of highly-sorted permutations -- permutations…

A permutation is so-called two stack sortable if it (i) avoids the (scattered) pattern 2-3-4-1, and (ii) contains a 3-2-4-1 pattern only as part of a 3-5-2-4-1 pattern. Here we show that the permutations on [n] satisfying condition (ii)…

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 investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x_i - x_j = 1, 1 \leq i<j…