This page contains links to files created as support material for the article Free resolutions of orbit closures for the representations associated to gradings on Lie algebras of type E6, F4 and G2 and the dissertation Free resolutions of orbit closures for representation with finitely many orbits by Federico Galetto.

These files were prepared with and are intended for use with the software Macaulay2. Each file is labeled by a string "XijOn.m2", where Xi is the type of the Lie algebra, j is the index of a distinguished node on the Dynkin diagram of type Xi and n is the natural number indexing the orbit. Please consult the documents cited above for an explanation of the notation and how the ideals of the orbit closures were constructed.

The files for the normal orbits will load a polynomial ring A and the defining ideal for the orbit closure On, labeled In. The files for the non normal orbits will load in addition two modules: Nn, the normalization of the coordinate ring of the orbit closure, and Cn, the cokernel of the inclusion of the coordinate ring in its normalization. To use one of these files, download it by right-clicking on its name in your browser and save it in a directory on the path of Macaulay2 (your home directory is usually fine). Then start M2 and type: load "XijOn.m2", replacing XijOn.m2 by the name of the file you downloaded.

Warning: computing resolutions of the ideals and modules provided here can be time and resource consuming. When necessary we provide hints to speed up the computation. All time and memory estimates refer to a 2008 MacBook with a 2.4 GHz Intel Core 2 Duo processor and 8 GB of RAM running Macaulay2 version 1.4 on Mac OS X 10.6.

For questions, comments or bug reports feel free to contact the author at: galetto.federico [(AT)] gmail.com.

### (E

_{6}, α_{1})### (E

_{6}, α_{2})### (E

_{6}, α_{3})### (E

_{6}, α_{4})- E64O1.m2 (I1 resolves with the option DegreeLimit=>3 in ~5 minutes using ~1 GB of RAM)
- E64O2.m2
- E64O3.m2
- E64O4.m2
- E64O5.m2
- E64O6.m2
- E64O7.m2
- E64O8.m2
- E64O9.m2
- E64O10.m2
- E64O11.m2
- E64O12.m2 (the resolution of N12 comes preloaded with the symbol RN12)
- E64O13.m2 (N13 resolves with the option DegreeLimit=>4 in ~4 minutes using ~2 GB of RAM)
- E64O14.m2
- E64O15.m2
- E64O16.m2 (~1 MB, loads in ~3 minutes)

### (F

_{4}, α_{1})### (F

_{4}, α_{2})### (F

_{4}, α_{4})### (G

_{2}, α_{2})### (E

_{7}, α_{2})### (E

_{7}, α_{3})- E73O9.m2 (~8.5 MB, only contains presentations for N9 and C9, the resolution of N9 comes preloaded with the symbol RN9)
- E73O10.m2
- E73O11.m2
- E73O12.m2 (only contains presentations for N12 and C12, the resolution of N12 comes preloaded with the symbol RN12)
- E73O13.m2 (only contains presentations for N13 and C13)

Page created and maintained by Federico Galetto. Last updated 3/10/2014.