On root posets for noncrystallographic root systems

TL;DR

This paper investigates the properties of root posets in noncrystallographic root systems, demonstrating their uniqueness in certain types and exploring realizations and conjectures related to these structures.

Contribution

It characterizes root posets for noncrystallographic types, proves non-existence in one case, and provides a realization for type H3, along with conjectures for type H4.

Findings

Unique determination of root posets for dihedral types and H3

Non-existence of such posets for H4

Realization of H3 root poset as restricted roots of D6

Abstract

We discuss properties of root posets for finite crystallographic root systems, and show that these properties uniquely determine root posets for the noncrystallographic dihedral types and type $H_3$, while proving that there does not exist a poset satisfying all of the properties in type $H_4$. We do this by exhaustive computer searches for posets having these properties. We further give a realization of the poset of type $H_3$ as restricted roots of type $D_6$, and conjecture a Hilbert polynomial for the $q,t$-Catalan numbers for type $H_4$.