Trees, parking functions, syzygies, and deformations of monomial ideal

For a graph, we construct two algebras, whose dimensions are both equal to the number of spanning trees of the graph. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe a monomial basis of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of ideals associated with posets and their deformations. Hilbert series of such ideals are always bounded by the Hilbert series of their deformations. We prove several formulas for Hilbert series of these ideals and construct their minimal free resolutions in terms of the order complex of the poset.

This is a joint work with Boris Shapiro.