For a given (finite, undirected) graph G, we construct two algebras whose dimensions are both equal to the number of the spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, and the other is the quotient modulo certain powers of linear forms. These ideals are interesting examples of a more general class of monotone monomial ideals and their deformations.

We construct monomial bases for both quotients. We show that the Hilbert series of a monotone monomial ideal is always bounded by that of its deformations. This project (joint with Alex Postnikov, see [PS]) was motivated by [PSS] and [SS], where certain algebras of curvature forms on flag varieties were studied.

[PS]

A.Postnikov and B.Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals.

[PSS]

A.Postnikov, B.Shapiro, and M.Shapiro, Algebras of curvature forms on homogeneous manifolds. Differential topology, infinite-dimensional Lie algebras, and applications, 227-235, Amer. Math. Soc. Transl. Ser. 2, 194, 1999.

[SS]

B.Shapiro and M.Shapiro, On ring generated by Chern 2-forms on SL_n/B, C.R.Acad.Sci. Paris Ser. I Math. 326 (1998), no. 1, 75-80.