[Paper Review] A Geometric Interpretation of the Characteristic Polynomial of Reflection Arrangements

Mathias Drton, Caroline J. Klivans|ArXiv.org|Jun 11, 2009

Matrix Theory and Algorithms2 references3 citations

TL;DR

This paper establishes a geometric interpretation of the characteristic polynomial of reflection arrangements by showing that, for types $A_n$, $B_n$, and $D_n$, the coefficients are proportional to the spherical projection volumes of points projected onto faces of given dimension in the fundamental chamber. The key result extends prior work on order-restricted inference and provides strong evidence—via simulations—for the conjecture that this holds for all finite reflection groups, including the exceptional types.

ABSTRACT

We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of type $A_n$, $B_n$, and $D_n$, the coefficients of the characteristic polynomial of the reflection arrangement are proportional to the spherical volumes of the sets of points that are projected onto faces of a given dimension. We also provide strong evidence that the same connection holds for the exceptional, and thus all, reflection groups. These results naturally extend those of De Concini and Procesi, Stembridge, and Denham which establish the relationship for 0-dimensional projections. This work is also of interest for the field of order-restricted statistical inference, where projections of random points play an important role.

Motivation & Objective

• To establish a geometric link between the coefficients of the characteristic polynomial of reflection arrangements and the spherical projection volumes of points in the fundamental chamber.
• To generalize prior results by De Concini, Procesi, and Stembridge, which only addressed 0-dimensional projections, to higher-dimensional projections.
• To provide strong computational evidence that the coefficient-projection volume correspondence holds for all finite reflection groups, including the exceptional types.
• To support the conjecture that for any finite reflection group, the number of group elements with k-dimensional projection is equal to the absolute value of the coefficient of $t^k$ in the characteristic polynomial.

Proposed method

• The authors analyze projections of points in the fundamental chamber of a reflection group onto faces of the chamber, using orthogonal projection onto the polyhedral cone.
• They define projection volumes $\nu_k$ as the surface measure of the unit sphere where the projection is k-dimensional, and relate these to the coefficients of the characteristic polynomial.
• For types $A_n$, $B_n$, and $D_n$, they prove that the number of group elements $g$ such that $\pi_{\mathcal{C}}(gx)$ is k-dimensional equals the absolute value of the coefficient of $t^k$ in $\chi(t)$.
• They use a computational approach involving random sampling in the fundamental chamber and orbit traversal via Stembridge’s Maple package coxeter to verify the conjecture on exceptional groups.
• They precompute projection chambers using inequality representations derived from face definitions and dual roots, leveraging polymake for polyhedral computations.
• They decompose the fundamental chamber by orbit type, identifying equivalence classes of points with identical projection behavior, to support a potential computer-assisted proof.

Experimental results

Research questions

• RQ1 For reflection groups of type $A_n$, $B_n$, and $D_n$, is the coefficient of $t^k$ in the characteristic polynomial proportional to the spherical volume of points whose projection onto the fundamental chamber is k-dimensional?
• RQ2 Does this geometric interpretation of characteristic polynomial coefficients via projection volumes extend to all finite reflection groups, including the exceptional types?
• RQ3 Can the number of group elements with k-dimensional projection be exactly matched to the absolute value of the coefficient of $t^k$ in the characteristic polynomial for all finite reflection groups?
• RQ4 What is the relationship between the orbit type of a point in the fundamental chamber and the dimension of its projection under the group action?

Key findings

• For reflection groups of type $A_n$, $B_n$, and $D_n$, the coefficients of the characteristic polynomial $\chi(t)$ are proportional to the spherical projection volumes $\nu_k$ of points whose projection lies in k-dimensional faces of the fundamental chamber.
• The number of group elements $g$ such that $\pi_{\mathcal{C}}(gx)$ is k-dimensional equals the absolute value of the coefficient of $t^k$ in $\chi(t)$, as proven for $A_n$, $B_n$, and $D_n$.
• Simulations on $H_3$, $H_4$, $F_4$, $E_6$, and $E_7$ confirm that the conjecture holds for these exceptional groups, with no counterexamples found in 1000 or 50 random samples.
• The only exception is $E_8$, which was too large for current computational tools, but the authors strongly believe the conjecture holds for this group as well.
• The authors provide a computational framework for verifying the conjecture via orbit type decomposition of the fundamental chamber, though a case-free proof remains elusive.
• The study reveals that different orbits of points in the fundamental chamber can exhibit distinct projection behaviors, complicating a uniform interpretation but supporting the robustness of the conjecture.