A Lower Bound for the Cardinality of Function Basis of Tensor Invariants

TL;DR

This paper establishes a lower bound on the size of function bases for tensor invariants under compact group actions and applies it to symmetric traceless tensors in mechanics.

Contribution

It proves a lower bound for the cardinality of function bases of tensor invariants and shows minimal integrity bases are irreducible in a specific tensor space.

Findings

Lower bound for function basis cardinality based on dimension differences

Minimal integrity bases are irreducible in the space of third order symmetric traceless tensors

Solves a problem in applied mechanics regarding tensor invariants

Abstract

In this article, we give a proof for that the cardinality of a function basis of the invariants for a finite dimensional real vector space by a compact group is lower bounded by the intuitive difference of the dimensions of the vector space and the group. An application is given to the space of third order three dimensional symmetric and traceless tensors, showing that each minimal integrity basis is an irreducible function basis, which solves a problem in applied mechanics.