Equivariance of isotropic principal stresses (${\sigma }_i({\lambda }_{\pi (1)},{\lambda }_{\pi (2)},{\lambda }_{\pi (3)})={\sigma }_{\pi (i)}(\lambda )$)

For an objective material isotropy law, permutation of the principal stretches permutes the labelled principal stresses. This follows by conjugating the left stretch tensor with signed permutation rotation matrices. Each principal stress is symmetric in the other two stretches, but need not be symmetric in all three. The law $\sigma =V^2$ provides the example ${\sigma }_i={\lambda }_i^2$. In an invariant representation $\sigma ={\beta }_{0}I+{\beta }_{1}V+{\beta }_2{V}^2$, the scalar coefficients are symmetric functions of the unordered stretches.