Bruhat graphs and pattern avoidance

TL;DR

This paper characterizes permutations based on the topological properties of their Bruhat graphs, specifically those drawable in the plane or torus, by identifying finitely many pattern avoidances.

Contribution

It provides a finite pattern avoidance characterization for permutations with Bruhat graphs drawable on the plane or torus.

Findings

Permutations with planar Bruhat graphs are characterized by avoiding certain patterns.

Permutations with torus-drawable Bruhat graphs are characterized by avoiding a different finite set of patterns.

These properties are finitely characterized by pattern avoidance.

Abstract

We characterize permutations whose Bruhat graphs can be drawn in the plane and those whose Bruhat graphs can be drawn in the torus. In particular, we show these properties are characterized by avoiding finitely many permutations.