Lexicographic shellability of sects

TL;DR

This work addresses the topology of the Bruhat order on $(p,q)$-clans in a type $AIII$ symmetric variety by recasting sects as rook placements in a $p\times q$ rectangle and extending Can's EL-labeling to these sectors. The authors define a precise partial permutation model $\phi_\gamma$ and a minimal covering clan construction $mcc_\tau(\gamma)$ to produce a standard EL-labeling, proving that each sect $\mathcal{C}_{p,q}^\lambda$ is EL-shellable. Consequently, the Bruhat order on matrix Schubert varieties (a key special case) is EL-shellable, with broader implications for Cohen–Macaulayness and Möbius-function interpretations in these posets. The results provide evidence toward the shellability of the full clan poset $\mathcal{C}_{p,q}$ and illustrate a robust combinatorial bridge between rook placements and geometric orbit closures in spherical varieties. Overall, the paper delivers a concrete, combinatorial EL-shelling framework for sects and derives significant consequences for related geometric objects.

Abstract

In this paper, we show that the Bruhat order on any sect of a symmetric variety of type $AIII$ is lexicographically shellable. Our proof proceeds from a description of these posets as rook placements in a partition shape which fits in a $p \times q$ rectangle. This allows us to extend an EL-labeling of the rook monoid given by Can to an arbitrary sect. As a special case, our result implies that the Bruhat order on matrix Schubert varieties is lexicographically shellable.

Figures (4)

• Figure 1: The bijection of Proposition \ref{['prop:bij']} applied to a $(3,3)$-clan with base clan ${-}{+}{-}{-}{+}{+}$. Clans of this sect give rook placements of shape $\lambda=(3,3,1)$.
• Figure 2: Rook placement corresponding to a clan and its hidden rooks colored in gray. To the right of the hidden rooks, we indicate the symbols from the simple, innermost $1{+}{-}1$ pattern that gives rise to it. The hidden rook associated to the pattern $2{+}{-}2=\hat{c}_{i_2}\hat{c}_{j_2}\hat{c}_{i_5}\hat{c}_{j_5}$ is obtained from $\hat{\gamma}_1$, after changing the 1212 pattern to 1221 and deleting the symbols $3{+}{-}3=\hat{c}_{i_3}\hat{c}_{j_3}\hat{c}_{i_4}\hat{c}_{j_4}$.
• Figure 3: EL-labeling of $\mathcal{S}_3$ with the unique increasing maximal chain highlighted in red.
• Figure 4: EL-shellability of $\mathcal{C}_{3,3}^{(3,2,1)}$ with the unique maximal increasing chain highlighted in red.

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.3: see bingham21
• Example 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7: bingham21