Computing GIT-fans with symmetry and the Mori chamber decomposition of $bar{M}_{0,6}$

We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry and group theory. We have implemented our algorithm in the Singular library gitfan.lib. Using our implementation, we compute the Mori chamber decomposition of the cone of movable divisors of $bar{M}_{0,6}$.