Graduate Meaning of Slot-Naming Index Notation (tensor conversion)

[heptacle]

I'm studying the component representation of tensor algebra alone.
There is a exercise question but I cannot solve it, cannot deduce answer from the text. (text is concise, I think it assumes a bit of familiarity with the knowledge)

(a) Convert the following expressions and equations into geometric, index-free notation:
A^αB_γβ ;
A_αB_γ^β ;
S_α^βγ=S^γβ_α ;
A^αB_β=A^αB_βg^α_β

In this problem, I can't see any difference between first two expressions except for the index position, and my only solution for the expression of index position is using metric tensor g, which I think is included in slot-naming notation. Is "index-free" notation can express the difference?
Other expressions are similarly confusing for me.(b) Convert T(_,S(R(C,_),_),_) into slot-naming index notation.

I think this notation would be not so universal notation. These problem are from http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1202.1.K.pdf (Ex 2.7)
and the help of anyone who is familiar with the notation would be appreciated.

 

[nbryan5]

(a) The first two expressions can be expressed in geometric, index-free notation as A⊗B, where ⊗ denotes the tensor product. The third expression can be expressed as S = Sᵠᵢᵣ, where Sᵠᵢᵣ denotes the permutation of the indices. Finally, the fourth expression can be written as A⊗B = A⊗Bg, where g denotes the metric tensor. (b) T(_,S(R(C,_),_),_) can be expressed in slot-naming index notation as Tᵢⱼₖₗₘ, where Tᵢⱼₖₗₘ denotes the components of the tensor T with respect to the slots defined by the indices i, j, k, l and m.