“Relatively little is known about integrity bases for tensors of order higher than two, for any of the transformation groups of interest in continuum mechanics. An exception is the set of results presented by J. Betten [Tensorrechnung für Ingenieure (1987; Zbl 0625.15019), Chapter 11]. Apart from these, there appears to be no thorough systematic treatment, although some isolated results appear in the literature.”
This quotation and the following one are taken from an article by A. J. M. Spencer published in [Applications of tensor functions in solid mechanics (1987 ; Zbl 0657.73001)]:
“Presumably the reduction to a finite number and even a minimal set of polynomial invariants may in principle be effected by application of identities of the form \[ \left|\begin{matrix} \delta_{ip} &\delta_{iq} &\delta_{ir} &\delta_{is}\\ \delta_{jp} &\delta_{jq} &\delta_{jr} &\delta_{js}\\ \delta_{kp} &\delta_{kq} &\delta_{kr} &\delta_{ks}\\ \delta_{lp} &\delta_{lq} &\delta_{lr} &\delta_{ls}\end{matrix}\right|\;=\;0, \] but the details are certain to be algebraically complicated.”
Equations of this kind will be used to achieve results concerning integrity bases of tensors of rank four in \(n\)-dimensional Euclidean space for \(n=2,3,4\).
For the entire collection see [Zbl 0839.00015].