Coxeter interchange graphs

TL;DR

The paper extends Brualdi and Li’s tournament interchange graphs to Coxeter types $B_n$, $C_n$, and $D_n$ by introducing Coxeter interchange graphs ${\rm IntGr}(\Phi,{\bf s})$ built from deterministic Coxeter tournaments on the complete $\Phi$-graph ${\mathcal K}_{\Phi}$. It proves regularity with degree $d(\Phi,{\bf s})=\frac{\|{\bf s}_{\Phi}\|^2-\|{\bf s}\|^2}{2}$ and provides a geometric interpretation via distances in the Coxeter permutahedron $\Pi_{\Phi}$, along with a Landau-type characterization of the score set ${\rm Score}(\Phi)$. The authors establish the set ${\rm Tour}(\Phi,{\bf s})$ and show two proofs of the main generator-counting formula; one algebraic via norm-difference and one combinatorial via embeddings to $A$-type tournaments. The work also reduces Coxeter scoring problems to classical cases to derive sufficiency, and lays groundwork for random-walk analyses on these graphs (cf. BKMP23).

Abstract

Brualdi and Li introduced tournament interchange graphs. In such a graph, each vertex represents a tournament. Traversing an edge corresponds to reversing a cyclically directed triangle. Such a triangle is neutral, in that its reversal does not affect the score sequence. An interchange graph encodes the combinatorics of the set of tournaments with a given score sequence, or equivalently, of a given fiber of the classical permutahedron from discrete geometry. Coxeter tournaments were introduced by the first author and Sanchez, in relation to the Coxeter permutahedra in Ardila, Castillo, Eur and Postnikov. Coxeter tournaments have collaborative and solitaire games, in addition to the usual competitive games in classical tournaments. We introduce Coxeter interchange graphs. These graphs are more intricate, as there are multiple neutral structures at play, which interact with one another. Our main result shows that the Coxeter interchange graphs are regular, and we describe the degree geometrically, in terms of distances in the Coxeter permutahedra. We also characterize the set of score sequences of Coxeter tournaments, generalizing a classical result of Landau.

Figures (15)

• Figure 1: For a root system of type $\Phi$, the permutahedron $\Pi(\Phi)$ is the convex hull of the orbit of the Weyl vector ${\bf s}(\Phi)$ under the Weyl group. In an interchange graph, vertices represent tournaments with score sequence ${\bf s}$, and edges correspond to the reorientation of one of the small neutral structures in \ref{['F_genD', 'F_genB', 'F_genC']}. \ref{['T_MainCount']} shows that the degree is $(\| {\bf s}(\Phi)\|^2-\| {\bf s}\|^2)/2$. The graphs above arise in type $C_3$ for ${\bf s}=(2,0,0)$, $(-1,0,1)$ and $(0,0,0)$, where ${\bf s}(C_3)=(1,2,3)$. The degrees (5, 6 and 7) increase as ${\bf s}$ approaches the center $(0,0,0)$ of $\Pi(C_3)$ in \ref{['F_permC']}.
• Figure 2: A Coxeter permutahedron $\Pi_\Phi$ of type $\Phi=C_3$.
• Figure 3: Cyclic and balanced triangles $\Delta_c$ and $\Delta_b$.
• Figure 4: Neutral pairs $\Omega_1$, $\Omega_2$ and $\Omega_3$.
• Figure 5: Neutral clovers $\Theta_1$ and $\Theta_2$.

Theorems & Definitions (22)

• Definition 1
• Definition 3
• Theorem 4
• Definition 5
• Theorem 6
• proof : Proof of \ref{['T_MainLan']} (only if part)
• Lemma 7
• proof
• Lemma 8
• proof