A difficulty in understanding the indices in the matrix interpretation of the product of two representation.

Question

I have a difficulty the indices in the two lines above the line starting with "Hence, $X$ transforms according to ..." , could anyone explain for me how the indices are located?

Also I do not understand from where the dual sign representation on $S_{f}(g)`$ came, could anyone explain this for me please?

Thank you!

Answer 1

If $A=(a_{ij})$, $B=(b_{ij})$ and $X=(x_{ij})$, then the $ij$-component of $AXB$ is $$ \sum_{k,\ell}a_{ik}x_{kl}b_{\ell j} $$ The formula appearing above is $$ \sum_{k,\ell}a_{ik}x_{kl}b_{j \ell} $$ which is the $ij$-component of $AXB^t$. The line beginning "Hence, $X$ transforms $\ldots$" simply states this.