ADEPerfectIdeals

Macaulay2 code for producing examples of perfect ideals using representation theory.

Brief background

Consider a T-shaped graph T with left arm of length c-2, right arm of length d, and bottom arm of length t where c >= 2, d >= 0, and t >= 1 are integers. This graph is a Dynkin diagram exactly when 1/(c-1) + 1/(d+1) + 1/(t+1) > 1. We henceforth assume this to be the case. In this setting, let G be the associated simply connected complex Lie group, and P the maximal parabolic corresponding to the diagram obtained by deleting the left-most vertex from T. Inside of the homogeneous space G/P, there are two linked Schubert varieties of codimension c, in the sense that their union is cut out by a regular sequence of length c.

The purpose of this code is to produce the defining ideals of these Schubert varieties when restricted to certain cells. Each of these is a homogeneous ideal in a multigraded polynomial ring.

The interest in these examples is because they conjecturally give the generic examples of perfect ideals of grade c, deviation at most d, and type at most t. This is classically known for c = 2 and it is proven for c = 3 (over a field of characteristic zero) in this preprint.

Usage

Generic examples

To see the generic examples, i.e. the defining ideals of particular Schubert varieties, load the file Demo.m2. Find the example of interest, evaluate the line that looks like (c,d,t) = ...; load "preparation.m2" and then the line lambda = ..., sigmaraw = ... corresponding to the cell of interest. Finally, evaluate the line load "computation.m2" Depending on the values of c,d,t, this computes either the resolution F or some part of it. The part computed is written on the next line, e.g. if the next line is d3;d2; that means the differentials d3 and d2 were computed, but not the rest of the resolution.

Random specializations

There is also a file SpecializationDemo.m2 that shows how one can produce random homogeneous examples in a standard graded polynomial ring by choosing a homomorphism from the root lattice to the integers; you can try sequentially evaluating all the commands in that file.

Other Schubert varieties

The very particular Schubert varieties were the main motivation for writing this code, but this code is actually capable of computing the defining ideals of any Schubert variety in G/P provided that the user can provide the irreducible representation inside of which to embed G/P via the Plucker embedding. The file SchubertDemo.m2 shows how to do this.