Lower bound for ranks of invariant forms

TL;DR

This paper establishes a lower bound for the Waring and cactus ranks of invariant forms under algebraic group actions, improving existing bounds for ranks of determinants, Pfaffians, and symmetric matrix determinants.

Contribution

It introduces a new lower bound method for ranks of invariant forms, enhancing previous bounds for specific algebraic forms like determinants and Pfaffians.

Findings

Improved lower bounds for Waring and cactus ranks of determinants and Pfaffians.

Applicable to forms invariant under connected algebraic group actions.

Enhanced understanding of the complexity of specific algebraic forms.

Abstract

We give a lower bound for the Waring rank and cactus rank of forms that are invariant under an action of a connected algebraic group. We use this to improve the Ranestad--Schreyer--Shafiei lower bounds for the Waring ranks and cactus ranks of determinants of generic matrices, Pfaffians of generic skew-symmetric matrices, and determinants of generic symmetric matrices.