On the Total Positivity of Contingency Metamatrices

TL;DR

The paper proves the total positivity of contingency metamatrices $M(W)$ for finite Coxeter groups, extending Kapranov–Schechtman’s conjecture beyond type $A$ to type $B_n$ and exceptional types. For type $B$, it develops a concrete description of double cosets via a faithful reflection representation, defines (generalized) signed contingency matrices, and shows a Vandermonde–diagonal decomposition that yields a polynomial in $(p+\tfrac12)$ and $(q+\tfrac12)$, establishing total positivity via Whitney’s theorem. For exceptional types, it leverages the bijection $W_I\backslash W/W_J \cong {}^I W^J$ to reduce $M_{pq}$ to counts of two-sided $W$-Eulerian numbers $N_{ij}$ and performs computational verifications (with symmetry) to confirm total positivity; explicit metamatrix data are provided for $I_2(m)$, $H_3$, $H_4$, $F_4$, $E_6$, $E_7$, and $E_8$. Overall, the work delivers a type-uniform perspective on positivity phenomena in Coxeter combinatorics, offering both a constructive type $B$ proof and comprehensive computational results for exceptional groups, along with potential interpretations via Peterson’s two-sided complexes and Whitney theory.

Abstract

M. Kapranov and V. Schechtman introduced the contingency metamatrix for a finite Coxeter group and conjectured that the contingency metamatrix is totally positive. For the Coxeter groups of type $A$, this conjecture has been proved by P. Etingof. In this article, we prove this conjecture for the Coxeter groups of type $B$ and exceptional types.

Theorems & Definitions (26)

• Conjecture 1.0.1: M. Kapranov, V. Schechtman
• Theorem 1.0.2
• Lemma 2.1.1
• Remark 2.1.2
• Definition 2.1.4: margin conditions
• Proposition 2.1.5
• proof
• Definition 2.1.6: signed contingency matrix
• Definition 2.1.7
• Proposition 2.1.8