An evaluation approach to computing invariants rings of permutation groups

TL;DR

This paper introduces a new, efficient method for computing invariant rings of permutation groups by leveraging evaluation points and symmetry, avoiding Gr"obner bases, and demonstrating improved performance through extensive benchmarks.

Contribution

It presents a Gr"obner basis free evaluation-based approach for calculating secondary invariants, optimizing computations for large permutation groups.

Findings

Effective reduction of computational complexity for large groups

Successful implementation in Sage with extensive benchmarking

Improved efficiency over traditional methods

Abstract

Using evaluation at appropriately chosen points, we propose a Gr\"obner basis free approach for calculating the secondary invariants of a finite permutation group. This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups. This is confirmed by extensive benchmarks using a Sage implementation.