Pieri resolutions for classical groups

We generalize the constructions of Eisenbud, Fl{\o}ystad, and Weyman for equivariant minimal free resolutions over the general linear group, and we construct equivariant resolutions over the orthogonal and symplectic groups. We also conjecture and provide some partial results for the existence of an equivariant analogue of Boij-S\"oderberg decompositions for Betti tables, which were proven to exist in the non-equivariant setting by Eisenbud and Schreyer. Many examples are given.Comment: 40 pages, no figures; v2: corrections to sections 2.2, 3.1, 3.3, and some typos; v3: important corrections to sections 2.2, 2.3 and Prop. 4.9 added, plus other minor corrections; v4: added assumptions to Theorem 3.6 and updated its proof; v5: Older versions misrepresented Peter Olver's results. See "New in this version" at the end of the introduction for more detail

Computing Intersection Multiplicity via Triangular Decomposition

Fulton’s algorithm is used to calculate the intersection multiplicity of two plane curves about a rational point. This work extends Fulton’s algorithm first to algebraic points (encoded by triangular sets) and then, with some generic assumptions, to l many hypersurfaces. Out of necessity, we give a standard-basis free method (i.e. practically efficient method) for calculating tangent cones at points on curves

Borel generators

We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to ordinary generators, this enables one to do manual computations much more easily. Moreover, this perspective allows us to find new connections to combinatorics involving Catalan numbers and their generalizations. We conclude with a surprising result relating the Betti numbers of certain principal Borel ideals to the number of pointed pseudo-triangulations of particular planar point sets.Comment: 23 pages, 2 figures; very minor changes in v2. To appear in J. Algebr