# Bruhat order and graph structures of lower intervals in Coxeter groups

## TL;DR

This paper investigates the structure of lower Bruhat intervals in Coxeter groups, revealing they are unions of two-sided cosets and introducing quotient structures to analyze their properties.

## Contribution

It introduces the concepts of quotient lower intervals and quotient Bruhat graphs, providing new tools to understand Coxeter group interval structures.

## Key findings

- Lower Bruhat intervals are unions of two-sided cosets
- Quotient lower intervals and quotient Bruhat graphs are introduced
- Structural descriptions in terms of graphs and posets are provided

## Abstract

We show that any lower Bruhat interval in a Coxeter group is a disjoint union of certain two-sided cosets as a consequence of Lifting Property and Subword Property. Furthermore, we describe these details in terms of Bruhat graphs, graded posets, and two-sided quotients altogether. For this purpose, we introduce some new ideas, quotient lower intervals and quotient Bruhat graphs.