Degree p Invariants in Linear Tensor Products: Bishop & Goldberg, p. 86-87

[Rasalhague]

An Invariant I is of degree p if it is a linear invariant of the p-fold tensor product of the variable with itself, that is,

$$IA=J(A\otimes ...\otimes A)$$

- Bishop & Goldberg: Tensor Analysis on Manifolds, Dover 1980, p. 86

...

Problem 2.14.3. Show that (Tr A)^p, where A is a tensor of type (1,1), in an invariant of degree p. [The fact that it is an invariant follows from the fact that Tr A is invariant. The question is whether (Tr A)² is a linear function of the coefficients of $A\otimes A$ etc.]

- ~ p. 87

I think I've misunderstood their definition of an invariant. The pth power of the trace function seems to be homogeneous of degree p rather than linear:

$$I(\lambda A) = J((\lambda A)\otimes (\lambda A)) = (\text{Tr}\, \lambda A)^2$$

$$=\lambda\lambda A^i_kA^r_m\delta^k_i\delta^m_r=\lambda\lambda A^k_kA^m_m=\lambda^2 (\text{Tr}\, A)^2= \lambda^2 IA \neq \lambda IA,$$

where $\lambda$ is a scalar. (Summing over like indices.)

 

[fresh_42]

Nobody required $I$ to be linear. We have $I(\lambda A) = \lambda^2 A$. However, the full definition is unfortunately missing.