The Yuri Manin Ring and Its B_n-Analogue

The Manin ring is a family of quadratic algebras describing pointed stable curves of genus zero whose homology gives the solution of the Commutativity Equations. This solution was first observed by the physicist Losev. We show the Manin ring is the Stanley-Reisner ring of the standard triangulation of the n-cube modulo a system of parameters. Thus, the Hilbert series of the Manin ring is given by the Eulerian polynomial. One can also view the Manin ring as the Stanley-Reisner ring of the dual of the permutahedron modulo a system of parameters. Furthermore, we develop a B_n-analogue of the Manin ring. In this case the signed Manin ring is the Stanley-Reisner ring of the barycentric subdivision of the n-cube (equivalently, the dual of the signed permutahedron) modulo a system of parameters and its Hilbert series is the descent polynomial of augmented signed permutations.