Sorting orders, subword complexes, Bruhat order and total positivity

TL;DR

This paper reveals new connections between subword complexes, sorting orders, and totally nonnegative spaces through a poset map, providing new proofs and geometric interpretations related to Bruhat order.

Contribution

It constructs a poset map linking Boolean algebra to Bruhat order, offering new proofs and geometric insights into sorting orders and their relation to the weak order and Bruhat order.

Findings

Proper part of Bruhat order is homotopy equivalent to a sphere.

Intersection of all sorting orders is the weak order.

Union of sorting orders equals the Bruhat order.

Abstract

In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof of Bj\"orner and Wachs' result \cite{BW} that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra --- that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is the Bruhat order.