SageMath code for computations related to the cohomology of the cohomology of complements of toric arrangements (in particular computations related to cohomology of moduli spaces of geometrically marked Del Pezzo surfaces of degrees 3 and 4).
The simplest way to use the program is the following:
- Load the files into Sage.
- Initiate a root system, e.g. "ra2=RootSystem(["A",2])".
- Compute the Weyl group, e.g. "wa2=ra2.ambient_space().weyl_group()".
- Generate representatives for the conjugacy classes of the Weyl group, e.g. "ca2=wa2.conjugacy_classes_representatives()".
- Compute the values of the equivariant Poincaré polynomial for all conjugacy classes, e.g. "polsa2=many_pols(ca2,ra2,0,len(ca2)-1)".
There is functionality for computing e.g. the value at a single conjugacy class or computing the Poincaré polynomial for the action extended by {+/-1} (and Z/3Z in "relative_toric.sage"). More information about this is provided in the comments (you might also want to consult the code for the method "many_pols" in "toric_generate_polynomial.sage").
The file "relative_toric.sage" is a version of the program which is convenient e.g. for computations with a root system inside a larger root system.
WARNING: some of the function names are reused in "relative_toric.sage" and there are compatability issues (you will get an error message if you have loaded the files in an incorrect order). However, as long as you don't run both versions in the same Sage-session no issues will arise (or make sure to reload the version you want to use).