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 sho w the Manin ring is the Stanlw-Reisner ring of the standard triangulation o f the n-cube module a system of parameters. Thus, the Hilbert series of the Manin ring is given by the Eulerian polynomial. One can also view the Mani n ring as the Stanley-Reisner ring of the dual of the permutahedron module 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 th e barycentric subdivision of the n-cube (equivalently, the dual of the sign ed permutahedron) module a system of parameters and its Hilbert series is t he descent polynomial of augmented signed permutations. (C) 2001 Academic P ress.