Counting nearest faraway flats for Coxeter chambers

TL;DR

This paper provides a case-free enumeration formula for certain parabolic subgroups in finite Coxeter groups, linking geometric and combinatorial methods, and offers a new proof of Chapoton's reflection count formula.

Contribution

It introduces a case-free product formula for counting specific parabolic subgroups in Coxeter groups, combining geometric and combinatorial techniques.

Findings

Derived a case-free enumeration formula for parabolic subgroups.

Connected geometric interpretation with Crapo's beta invariant.

Provided the first case-free proof of Chapoton's reflection formula.

Abstract

In a finite Coxeter group $W$ and with two given conjugacy classes of parabolic subgroups $[X]$ and $[Y]$, we count those parabolic subgroups of $W$ in $[Y]$ that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of $W$ in $[X]$. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the "full support" property with a double counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group $W$.