Quadrirational Yang-Baxter maps and the elliptic Cremona system

This paper connects the quadrirational Yang-Baxter maps, which are two-dimensional integrable discrete systems of KdV type, and the elliptic Cremona system, which is a higher analogue of discrete Painlev'e equations associated with ${\tilde{E}}_8$ symmetry. This is a natural connection between integrable systems in different dimensions that is outside of the usual paradigm of reductions. Our approach is based on formulation of both systems in terms of birational Coxeter groups.