Mathematics – Combinatorics
Scientific paper
2003-01-11
Mathematics
Combinatorics
33 pages; v2: appendix on sandpiles added, references added, typos corrected; v3: references added
Scientific paper
For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. 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 the set of monomials that forms a linear basis in each 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 monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the sandpile model.
Postnikov Alexander
Shapiro Boris
No associations
LandOfFree
Trees, parking functions, syzygies, and deformations of monomial ideals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Trees, parking functions, syzygies, and deformations of monomial ideals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Trees, parking functions, syzygies, and deformations of monomial ideals will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-583917