I have to decompose some representations of $S_d \wr S_n$. I understand better and better how it works, I still have a case I don't know how to deal with.

For simplicity I take $d=2$ and $n=4$. $S^{(2)}$ is the trivial irrep and $S^{(1,1)}$ the sign irrep of $S_2$.

Let us consider the irrep of $S_2 \wr S_4$: $$ \bigl( U \boxtimes V \bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4} $$ where $U=S^{(1,1)}$ and $V={S^{(2)}}^{\widetilde{\otimes 3}}$

and the other irrep $$ \bigl( U \boxtimes (V\otimes S^{(2,1)}) \bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4} $$

I'd like to decompose the product of the two, I don't know how this works...

Thanks a lot!

PS:

- I hope my writting is correct and rigorous but I'm not absolutly sure... I hope this is understandable
- related question: Decomposition into irreducible of a representation of the wreath product $S_d \wr S_m$ (2)
- I think I know how to decompose the product of the first with itself $ \bigl( U \boxtimes V \bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4} \otimes \bigl( U \boxtimes V \bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4} $. I would write it like this:

$... = \bigl({S^{(1,1)}}^{\otimes 2}\boxtimes {S^{(2)}}^{\widetilde{\otimes 3}}\bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}\oplus \bigl({S^{(1,1)}}\boxtimes{S^{(1,1)}}\boxtimes {S^{(2)}}^{\widetilde{\otimes 2}}\bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{1}\times S_2\wr S_{2}}^{S_2\wr S_4}$

With ${S^{(1,1)}}^{\otimes 2}=S^{(2)}$, the left term is $\bigl({S^{(2)}}\boxtimes {S^{(2)}}^{\widetilde{\otimes 3}}\bigr)\bigl\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}= S^{(4)} \oplus S^{(3,1)}$

The right term is $\bigl( X \boxtimes Y \bigr)\bigl\uparrow_{S_2\wr S_{2} \times S_2\wr S_{2}}^{S_2\wr S_4} \oplus \bigl( (X\otimes S^{(1,1)}) \boxtimes Y \bigr)\bigl\uparrow_{S_2\wr S_{2} \times S_2\wr S_{2}}^{S_2\wr S_4} $, where $X={S^{(1,1)}}^{\widetilde{\otimes 2}}$ and $Y={S^{(2)}}^{\widetilde{\otimes 2}}$.

Is this correct? Should I write this differently? I suspect my "$\uparrow_{S_2\wr S_{1} \times S_2\wr S_{3}}^{S_2\wr S_4}$" to be unappropriate.