@მამუკაჯიბლაძე Wow, GAP is so powerful!

@მამუკაჯიბლაძე Mine is 4.11, just downloaded from their website. Your result coincides with mine, but I am a little bit curious how you did the calculation. I thought this algebra was isomorphic to the centralizer of $\mathfrak{sl}(2)$ represented on $(\mathbb{C}^2)^{\otimes n}$, and this is where the conjecture comes from. Now it seems that this is only true for $n<6$, while for $n=6$, the centralizer is $\mathfrak{sl}(5)\oplus\mathfrak{sl}(9)\oplus\mathfrak{sl}(5)$.

@მამუკაჯიბლაძე Thanks a lot! I just got GAP installed and verified your result. Everything works fine except that I need to change emb(x) to Image(emb,x), presumably due to different versions. This is really unexpected! Do you think using a group algebra over rationals instead of over complex numbers will make any difference?

@მამუკაჯიბლაძე Would you like to describe how you get the result? I expected that the number I suggested should at least be an upper bound.

@PerAlexandersson I haven’t thought about that yet. It could be a very interesting generalization to this problem.