Coxeter interchange graphs

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.