Brewing Fubini-Bruhat Orders

TL;DR

The paper develops a Bruhat-type framework for subvarieties of the spanning line configuration space $X_{n,k}$ indexed by Fubini words, introducing three interrelated orders—decaf, medium roast, and espresso—that generalize classical Bruhat relations. It builds the Pawlowski–Rhoades (PR) structure with a Bruhat-like decomposition $\mathcal{F}_{k\times n}(\mathbb{C})=\bigsqcup_w U P_w T$, PR cells $C_w$, and PR varieties $\overline{C}_w$, and connects these to the cohomology ring $R_{n,k}$ and $S_n$-action; the Alpha Test based on Gale order provides combinatorial criteria for comparing words, and two sets of defining equations for $\overline{C}_w$ are given via flag minors and rank conditions. A detailed analysis of covering relations yields the decaf order as the transitive closure of Transposition and Pushback rules, and a discussion highlights that medium roast and espresso are not ranked in general. The essential-set framework Ess$^{*}(w)$ extends Fulton’s ideas to Fubini words, giving minimal rank-conditions that define $\overline{C}_w$ and a practical criterion for $v\le w$ in the medium roast order. Overall, the work connects combinatorial descriptions (Gale order, alpha vectors, convexification) to geometric objects (PR cells/varieties) and provides tools for succinct equation sets and rank-based comparisons in this generalized Bruhat setting.

Abstract

The Bruhat order on permutations arises out of the study of Schubert varieties in Grassmannians and flag varieties, which have been important for over 100 years. The purpose of this paper is to study variations on this theme related to subvarieties of the spanning line configurations $X_{n,k}$ as defined by Pawlowski and Rhoades. These subvarieties are indexed by Fubini words, or equivalently by ordered set partitions. Three natural partial orders arise in this context; we refer to them as the decaf, medium roast, and espresso orders. The decaf order is a generalization of the weak order on permutations defined by covering relations using simple transpositions. The medium roast order is a generalization of the (strong) Bruhat order defined by the closure relationship on the subvarieties. The espresso order is the transitive closure of a relation based on intersecting subvarieties. Many properties of Schubert varieties and Bruhat order extend to one or more of the three Fubini-Bruhat orders. We examine some of the many possibilities in this work.

Figures (1)

• Figure 1: Diagram of $44253136541$ with cells in the essential set boxed.

Theorems & Definitions (29)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 1.7
• Theorem 1.8
• Corollary 1.9
• Definition 2.1