Suppose $T=A⊗B⊗C$ where A, B, and C are (1,1) Tensors. What would the contraction $T^{iju}_{ubc}$ look like?

Question

I'm pretty sure $T^{ujk}_{ubc}=tr(A)B⊗C$ and $T^{iuk}_{ubc}=AB⊗C$ but I have no idea what $T^{iju}_{ubc}$ would be

Answer 1

I believe that your answers are correct. As for the third, I believe we end up with$(AC) \otimes B$, but I'm new to tensor calculations so I'm not confident.

Here's my work: $$ T^{iju}_{ubc} = e^{iju} (A \otimes B \otimes C) e_{ubc} = A^i_u B^j_bC^u_c = (A^i_u C^u_c) B^j_b = (AC)^i_c B^j_b = [(AC) \otimes B]^{ij}_{cb}. $$ Note that the indices don't work in quite the same way. For $T^{iju}_{ubc}$, the choice $i,c$ corresponds to an entry of $AC$. For $T^{iuk}_{ubc}$, the choice $i,c$ corresponds to an entry of $AB$.