Geometry-Algebra-Singularities-Combinatorics Seminar |

Trees,
parking functions, syzygies, and deformations of monomial ideals. |

Abstract: 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. |