Impact of Weyl group optimisation on output simplification

Determine whether applying Weyl group optimisation—i.e., using Weyl group symmetries to aggregate equivalent one-parameter subgroups in the algorithms that compute maximal families describing the unstable, non-stable, and strictly polystable loci for Geometric Invariant Theory stability—produces any simplification of the output (reduces the number of families), or whether there is no simplification in the output, as conjectured in GMGMS Conjecture 7.4.

Background

CompGIT implements algorithms for determining non-stable, unstable, and strictly polystable loci in GIT problems, leveraging the Weyl group of a simple algebraic group to potentially simplify outputs and speed computations. The code uses SageMath’s WeylGroup to perform a "Weyl optimisation" intended to reduce redundant analysis across conjugate configurations.

The authors remark that it is believed (see GMGMS Conjecture 7.4) that Weyl optimisation does not actually simplify the final output—i.e., it does not reduce the number of distinct families that must be described—though it may accelerate the computations. Establishing whether this belief is universally true would clarify the net benefit of Weyl optimisation and guide future generalisations beyond simple groups.