Theory and algorithms for w-stable ideals

Abstract

Strongly stable ideals are a class of monomial ideals which correspond to generic initial ideals in characteristic zero. Such ideals can be described completely by their Borel generators, a subset of the minimal monomial generators of the ideal. In [1], Francisco, Mermin, and Schweig develop formulas for the Hilbert series and Betti numbers of strongly stable ideals in terms of their Borel generators. In this thesis, a specialization of strongly stable ideals is presented which further restricts the subset of relevant generators. A choice of weight vector w ∈ Nn>0 restricts the set of strongly stable ideals to a subset designated as w-stable ideals. This restriction allows one to further compress the Borel generators to a subset termed the weighted Borel generators of the ideal. As in the non-weighted case, formulas for the Hilbert series and Betti numbers of strongly stable ideals can be expressed in terms of their weighted Borel generators. In computational support of this class of ideals, the new Macaulay2 package wStableIdeals.m2 has been developed and segments of its code support computations within the thesis. In a strengthening of combinatorial connections, strongly stable partitions are defined and shown to be in bijection with totally symmetric partitions.