[Paper Review] An evaluation approach to computing invariants rings of permutation groups

Nicolas Borie, Nicolas M. Thiéry|arXiv (Cornell University)|Oct 17, 2011

Finite Group Theory Research12 references3 citations

TL;DR

This paper introduces a Gröbner basis-free evaluation-based method for computing secondary invariants of finite permutation groups by leveraging strategic point evaluations and symmetry reduction. The approach achieves significant complexity improvements—demonstrated by benchmarks showing it handles groups up to size 50 million where elimination-based methods fail—by confining computations to a smaller quotient space via graded evaluation techniques.

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.

Motivation & Objective

• Address the computational bottleneck in invariant theory caused by Gröbner basis methods, which scale poorly with group size and symmetry.
• Overcome the inefficiency of elimination-based techniques in quotient computations, especially for large permutation groups.
• Develop a practical and scalable algorithm for computing secondary invariants in the non-modular case using evaluation techniques.
• Enable computation of invariants for large permutation groups—previously infeasible with existing methods—by exploiting symmetry and quotient space reduction.
• Provide theoretical and empirical justification for the evaluation approach as a viable alternative to elimination-based methods in computational invariant theory.

Proposed method

• Use evaluation maps at carefully selected points to characterize secondary invariants via perturbation of the quotient ring and grading transfer.
• Replace Gröbner basis computations with evaluation-based linear algebra by mapping invariants to their values at a finite set of points.
• Leverage group symmetry to reduce the dimension of the ambient space by working in a quotient space defined by orbit representatives.
• Apply the Reynolds operator and Hilbert series via Molien’s formula to guide the selection of evaluation points and degree bounds.
• Construct secondary invariants by solving a system of linear equations derived from evaluations, using the graded structure of the invariant ring.
• Implement the algorithm in Sage, using efficient evaluation strategies and exploiting sparsity and symmetry to minimize computational cost.

Experimental results

Research questions

• RQ1 Can secondary invariants of finite permutation groups be computed without Gröbner bases by using evaluation techniques?
• RQ2 To what extent can symmetry reduction and evaluation-based methods improve algorithmic complexity compared to elimination-based approaches?
• RQ3 What is the minimal set of evaluation points required to uniquely determine secondary invariants, especially in graded components?
• RQ4 How does the evaluation-based approach scale on large permutation groups where traditional methods fail?
• RQ5 Can the method be generalized to matrix groups beyond permutation groups, and what perturbation strategies are needed to preserve grading and orbit structure?

Key findings

• The evaluation-based method successfully computes secondary invariants for permutation groups up to order |G| = 50,803,200, including a case with n=14 and 1,716 secondary invariants, which is infeasible for elimination-based methods.
• Benchmarks show the method's practical complexity is approximately cubic in the number of secondary invariants, i.e., O(n! / |G|)^3, suggesting favorable scaling for large groups.
• For the cyclic group C7, only 110 out of 720 possible evaluation points were sufficient to reconstruct the secondary invariants, indicating strong sparsity and efficiency in point selection.
• The method outperforms elimination-based implementations in Singular, especially for large groups, despite being slower on small groups due to overhead.
• Theoretical analysis shows a worst-case complexity gain of at least a factor of |G| compared to elimination-based methods, though precise bounds remain elusive due to the lack of fine-grained complexity estimates for Gröbner bases.
• The approach enables predictable and scalable computation on large symmetric and alternating groups, such as S10 and A10, which are otherwise out of reach with standard tools.