Related papers: Sorting orders, subword complexes, Bruhat order an…
It is shown that there is an order isomorphism $\phi'$ from the poset $V$ of $B\times B$-orbits on the wonderful compactification of a semi-simple adjoint group $G$ with Weyl group $W$ to an interval in reverse Chevalley-Bruhat order on a…
Let (W, S) be a Coxeter system. We investigate combinatorially certain partial orders, called extended Bruhat orders, on a (W x W)-set W(N,C), which depends on W, a subset N of S, and a component C of N. We determine the length of the…
We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence.…
By explicitly describing a cellular decomposition we determine the Borel invariant cycles that generate the Chow groups of the quotient of a reductive group by a Levi subgroup. For illustrations we consider the variety of polarizations…
First, we review the basic mathematical structures and results concerning the gauge orbit space stratification. This includes general properties of the gauge group action, fibre bundle structures induced by this action, basic properties of…
Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are known to be related to the enumeration of chains in a new partial order on S_\infty, which we call the universal k-Bruhat order. Here we…
The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of adjacent transpositions that each increase the number of…
We give a necessary and sufficient condition for that the support $\tau$-tilting poset of a finite dimensional algebra is isomorphic to the poset of symmetric group with weak order. Moreover we show that there are infinitely many finite…
A new class of partial order-types, class $\gbqo^+$ is defined and investigated here. A poset $P$ is in the class $W^+ $ iff the free poset algebra $F(P)$ is generated by a better quasi-order $G$ that is included in the free lattice $L(P)$.…
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…
Let H be a complex Hilbert space and denote by Bs(H) the set of all self-adjoint bounded linear operators on H. In this paper we describe the form of all bijective maps (no linearity or continuity is assumed) on Bs(H) which preserve 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 $n$).…
We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with…
The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is isomorphic to a poset on commutation classes of reduced words…
The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this…
This paper provides a unifying and simplifying approach to Bruhat order in which the usual Bruhat order, parabolic Bruhat order, and Bruhat order for symmetric pairs are shown to have combinatorially analogous and relatively simple…
It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (CABA's) taking a set to its power-set and, reciprocally, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a…
In this paper, we consider a natural generalization of the concept of order of an element in a group: an element $g \in G$ is said to have order $k$ in a subgroup $H$ of $G$ (\resp \wrt a coset $Hu$) if $k$ is the first strictly positive…
The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Pl\"ucker coordinates. The circular Bruhat order is a poset isomorphic to…
A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of…