Special idempotents and projections

TL;DR

This paper introduces special idempotents and projections in finite graded posets, analyzing their properties and applications in Eulerian posets and Coxeter groups, revealing their structural significance and generalizations.

Contribution

It defines a new class of idempotent functions called special idempotents and projections, and explores their properties and roles in poset and Coxeter group structures.

Findings

Special idempotents are interval retracts.

In Eulerian posets, projections' images are graded subposets.

All projections on certain Coxeter group quotients are special projections.

Abstract

We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. The idempotents of this monoid are called special idempotents. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings.