Combinatorial Cremona automorphisms and Coxeter arrangement matroids

We explore birational geometry of matroids by investigating automorphisms of their coarse Bergman fans. Combinatorial Cremona maps provide such automorphisms of Bergman fans which are not induced by matroid automorphisms. We investigate the structure of matroids allowing combinatorial Cremona maps and prove a realizability criterion in the presence of two different Cremonas. We also prove that for all matroids associated to Coxeter arrangements the group of coarse automorphisms of the Bergman fan is generated by the matroid automorphisms and at most one combinatorial Cremona map.