Related papers: Higher Bruhat Orders in Type B

It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order." We describe simplex equations (including the Yang-Baxter equation) as realizations of higher…

Partially ordered sets of type (k, n) are the sets such that a) cardinality of each set is n, b) dimension of each set is two, c) length of the maximal antichain in each set is k. Let \alpha_k(n) be the number of partially ordered sets of…

Let n_n(C) be the algebra of strictly upper-triangular n x n matrices over the field of complex numbers and X_2 the subset of matrices of nilpotent order 2. Let B_n(C) be the group of invertible upper-triangular matrices acting on n_n(C) by…

We define combinatorially a partial order on the set partitions and show that it is equivalent to the Bruhat-Chevalley-Renner order on the upper triangular matrices. By considering subposets consisting of set partitions with a fixed number…

Weak Bruhat interval modules of the $0$-Hecke algebra in type $A$ provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were generalized from type $A$…

To any saturated chain in the affine Weyl group whose translation parts are sufficiently regular, we associate a near path and a far path in the quantum Bruhat graph. Using this, working in the Bruhat order on the minimal-length…

We classify all equivalences between the indecomposable abelian categories which appear as blocks in BGG category O for reductive Lie algebras. Our classification implies that a block in category O only depends on the Bruhat order of the…

We explore lattice structures on integer binary relations (i.e. binary relations on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$) and on integer posets (i.e. partial orders on the set $\{1, 2, \dots, n\}$ for a fixed integer $n$).…

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair $(X,Y)$ of subsets of the symmetric group $\mathfrak{S}_n$, the…

We define the notion of a separable element in a finite Weyl group, generalizing the well-studied class of separable permutations. We prove that the upper and lower order ideals in weak Bruhat order generated by a separable element are…

Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset…

For every partially ordered sets I, having a finite cofinal subset, and every field K we build a unital, locally matricial and hence unit-regular K-algebra B(I) such that the lattice of all its ideals is order isomorphic to the lattice of…

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{{\bf u}}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure…

The combinatorially and the geometrically defined partial orders on the set of permutations coincide. We extend this result to $(0,1)$-matrices with fixed row and column sums. Namely, the Bruhat order induced by the geometry of a Cherkis…

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for…

We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its M\"obius function. We show that the weak order on Coxeter groups of type A, B, affine A, and the flag weak order…

For an integer $n\geq 1$, we display a poset $\Phi_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat \Phi_n$ is obtained from the $n$-cube by adding some edges. We show that…

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…

We derive a formula for computing the size of lower Bruhat intervals for elements in the dominant cone of an affine Weyl group of type $A$. This enumeration problem is reduced to counting lattice points in certain polyhedra. Our main tool…

We present a matrix-theoretic approach for studying and enumerating finite posets through their incidence representations, referred to as poset matrices. Naturally labelled posets are encoded as Boolean lower triangular matrices, allowing a…