Counting Generating Invariants Under Semisimple Group and Torus Actions

TL;DR

This paper investigates the growth of minimal generating sets for invariant rings under semisimple group actions, revealing super-polynomial growth rates and contrasting bounds for torus invariants.

Contribution

It provides the first analysis of the asymptotic size of minimal generating sets for invariants under semisimple groups, highlighting their rapid growth.

Findings

Minimal generating sets grow faster than any polynomial in n for semisimple groups.

Sub-exponential upper bounds are established for torus invariants on binary forms.

The results impact the understanding of computational complexity for invariant generation.

Abstract

Although degree bounds and algorithms for the generators of various invariant rings have been known for decades, little is known about the cardinality of minimal generating sets. Estimates of such would provide lower bounds for the runtime of algorithms that compute invariants. Fix a semisimple linear algebraic group, choose an irreducible representation of highest weight w, and consider the irreducible representations of highest weight nw. As n goes to infinity, the cardinality of a minimal set of generating invariants grows faster than any polynomial in n. On the other hand, combinatorial methods yield sub-exponential upper bounds for the growth of generating sets for torus invariants on the binary forms.