Geometric Invariant Theory

This website is a companion to the draft “Computing GIT Quotients of Semisimple Groups” by Patricio Gallardo, Jesus Martinez Garcia, Han-Bom Moon, and David Swinarski.

In that paper, we prove results and provide algorithms to describe any reductive quotient of a projective variety \( X \) by any reductive algebraic group \( G \). To do so, we do several simplifications that reduce the problem to the equivalent one of finding all \(T\)-unstable/non-stable/strictly polystable points in \(X=\mathbb P^n\). We refer to the paper for details.

In this page we provide an implementation of our algorithms in SageMath for simple groups of types A-D. We also provide the outputs of the programme for the list of cases described in the statistics section of the paper. Many of these are either known examples or examples of geometric interest whose moduli spaces have not yet been studied. We expect that the outputs below, and others not yet considered by us but easy to run, will result in papers by other researchers. In the paper, the interested reader may find references to some of these problems when we knew about them.

We welcome any comments on paper, code and examples.

SageMath code

These .sage files containing SageMath code have been posted here with an extra .txt extension here to make them easily viewable in a web browser. Remove the .txt before using them.

  1. GIT.sage
  2. SimpleGroup.sage

Using SageMath

SageMath is a free open-source mathematics software system whose language is based on Python 3.x and is licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more.

You can download SageMath and install it on your computer. It runs on iOS, Linux and Windows, although easiness of installation varies depending on the platform.

If you don’t want to install SageMath, there are a couple of alternatives online you may use. The online service Cocalc will allow you to run SageMath code and work with multiple persistent worksheets in Sage, IPython, LaTeX, and much, much more. The SageMathCell¬†project is an easy-to-use web interface to a free open-source mathematics software system¬†SageMath. However, we do not think it will be easy to run our code in it.

Demonstration: How to run our code

We demonstrate how to run our code to analyze the GIT problem for plane cubics. In this session we are running SageMath 10.0 in a Jupyter notebook.

Reading the output

Each problem output will state the Dynkin type (root system) of a group \(G\) as Xn where X is A,B,C,D,E,F,G and n is a positive integer. For instance, A3 corresponds to \(\mathrm{SL}_3 \).

The output will also state the representation the group acts on by listing the highest weight(s) of the representation. For instance, A3(3,0,0,0) means the group is \(\mathrm{SL}_3\) acting on a \(4\)-dimensional weight system whose highest weight is \(3\omega_1\).

The output also presents a list of non-stable, unstable and strictly polystable loci. Essentially, this is a list of one-parameter subgroups in \(G\). Note these are presented just by their weights up to multiplication by scalar. Note that groups of type \(A\) and groups of type \(B-E\) follow different basis and thus the weights of a one-parameter subgroup of type \(A\) add up to \(0\) while the rest do not. We hope this does not cause confusion.

For each one-parameter subgroup, a state is listed. This state will contain all the weights of the representation which are non-stable, unstable or strictly polystable with respect to the one-parameter subgroup. The program (and the results in the paper) guarantee that any \(T\)-non-stable (or unstable, strictly polystable, respectively) point in \(X\) must belong to one of these states. An example for cubic surfaces (whose output is listed below) is worked out in the paper. In it, one can read how to interpret this output to find all stable and strictly polystable points.

Output files for the examples in Table 1

Plane projective curves

Surfaces in \( \mathbb{P}^{3}\)

Threefolds in \( \mathbb{P}^{4}\)

More cubics

Pencils of quadrics

Nets of quadrics

Pencils of cubics

The representation \( \Gamma_{\omega_3}\) for different root systems

Byun-Lee quotients

Mukai models of \( {\overline{\mathrm{M}}}_g\)

  • Genus 7 (to be added)
  • Genus 8 (to be added)
  • Genus 9 (Stability and semistability analyses only)