The generalized lifting property of Bruhat intervlas

TL;DR

This paper characterizes finite, simply-laced Coxeter groups as exactly those satisfying the generalized lifting property for Bruhat intervals, extending previous results from symmetric groups.

Contribution

It proves that the generalized lifting property holds for a Coxeter group if and only if it is finite and simply-laced, providing a complete classification.

Findings

The property holds for finite, simply-laced Coxeter groups.

The property does not hold for infinite or non-simply-laced Coxeter groups.

Extends the understanding of Bruhat interval properties beyond symmetric groups.

Abstract

In [E. Tsukerman and L. Williams, {\em Bruhat Interval Polytopes}, Advances in Mathematics, 285 (2015), 766-810] it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced.