The intermediate orders of a Coxeter group

TL;DR

This paper introduces intermediate partial orders on Coxeter groups based on reflection sets, exploring their properties, grading, and related structures like $k$-Bruhat graphs, with conjectures and open problems.

Contribution

It defines new intermediate orders on Coxeter groups, analyzes their properties, and introduces related concepts such as $k$-Bruhat graphs and $k$-absolute length.

Findings

Posets are graded by the length function.

Projections on right parabolic quotients are order preserving.

Introduces $k$-Bruhat graph, $k$-absolute length, and $k$-absolute order.

Abstract

We define a class of partial orders on a Coxeter group associated with sets of reflections. In special cases, these lie between the left weak order and the Bruhat order. We prove that these posets are graded by the length function and that the projections on the right parabolic quotients are always order preserving. We also introduce the notion of $k$-Bruhat graph, $k$-absolute length and $k$-absolute order, proposing some related conjectures and problems.