Simplifying and Unifying Bruhat Order for BGB, PGB, KGB, and KGP

TL;DR

This paper unifies and simplifies the combinatorial descriptions of Bruhat order across various contexts, enabling easier analysis and understanding of these structures through analogous and straightforward relations.

Contribution

It introduces a unified, simplified combinatorial framework for Bruhat order applicable to BGB, PGB, KGB, and KGP, reducing complexity and enhancing study of these orders.

Findings

Bruhat order has property Z for spherical subgroups.

Simplified combinatorial descriptions of Bruhat order are provided.

Applications include concise proofs and parametrizations related to Bruhat order.

Abstract

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 descriptions. Such analogies are valuable as they permit the study of PGB and KGB by reducing to BGB rather than by introducing additional machinery. A concise definition for reduced expressions and a simple proof of the exchange condition for PGB are provided as applications of this philosophy. A geometric argument for spherical subgroups, which includes all of the cases considered, shows that Bruhat order has property Z and therefore satisfies the subexpression property. Thus, Bruhat order can be described using only simple relations, and it is the simple relations which we simplify combinatorially. A parametrization of KGP is a simple consequence of understanding the Bruhat order of KGB restricted to a P-orbit.