Bruhat-Chevalley order on the rook monoid

Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations pi such that the elements in the closed lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations pi such that [id,pi] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincare polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincare polynomial of some particularly interesting intervals in the finite Weyl grou…

The Rook Monoid is Lexicographically Shellable

We prove that the Bruhat-Chevalley-Renner order on the rook monoid is EL-shellable. We determine the homeomorphism type of the associated order complex.

Lexicographic shellability of the Bruhat-Chevalley order on fixed-point-free involutions

The main purpose of this paper is to prove that the Bruhat-Chevalley ordering of the symmetric group when restricted to the fixed-point-free involutions forms an $EL$-shellable poset whose order complex triangulates a ball. Another purpose of this article is to prove that the Deodhar-Srinivasan poset is a proper, graded subposet of the Bruhat-Chevalley poset on fixed-point-free involutions.

Combinatorics of $B$-orbits and Bruhat--Chevalley order on involutions

Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $σ$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $Ω_σ\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of ro…

扇形的字典序壳性

Lexicographic shellability of sects

本文证明对称簇$AIII$的扇形Bruhat序具有字典序壳性，通过车放置描述并扩展EL-标记，特例涵盖矩阵Schubert簇。

On the Borel Submonoid of a Symplectic Monoid

In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid $MSp_n$. After showing that this order is completely determined by the Bruhat-Chevalley-Renner order on the linear algebraic monoid of $n\times n$ matrices $M_n$, we focus on the Borel submonoid of $MSp_n$. By using this submonoid, we introduce a new set of type B set partitions. We determine their count by using the ``folding'' and ``unfolding'' operators that we introduce. We show that the Borel submonoid of a rationally smooth reductive monoid with zero is rationally smooth. Finally, we analyze the …