On Bruhat intervals of small lengths for Weyl groups

TL;DR

This paper develops a new inductive invariant for Bruhat intervals in Weyl groups, enabling faster comparison of intervals and extending the understanding of their structure beyond small lengths.

Contribution

It introduces an invariant for Bruhat intervals that simplifies their comparison, allowing analysis of higher-dimensional intervals where previous geometric methods were impractical.

Findings

Invariant allows faster isomorphism checks

Method applicable to any interval length

Extends classification to higher dimensions

Abstract

The number of Bruhat intervals in Coxeter groups is finite, and for the first few lengths, the intervals were described up to an isomorphism by A. Hultman using the correspondence between Bruhat intervals and cell decompositions of a 2d sphere and straightforward computations. The main purpose of this paper consists of a description of the intervals in higher dimensions, as the Hultman's geometric method is hard to apply due to rapidly growing with length number of nonisomorphic intervals. We construct an invariant on subintervals in the Bruhat graphs, using their specific properties. This gives us a method of comparing two Bruhat interval, that is faster than the general algorithm for checking if two graphs are isomorphic. This construction is inductive, and thus, can be easily applied for any interval length and Weyl group.