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What is an example of two irreducible (closed) subgroups $ H_1,H_2 $ of $ SU(n) $ which are isomorphic but not conjugate?

For $ SU(2) $ there are no examples like this since isomorphic (closed) subgroups of $ SU(2) $ are always conjugate.

If we drop the assumption of irreducibility then $ diag(\zeta_3,\zeta_3,\zeta_3) $ and $ diag(1,\zeta_3,\zeta_3^2) $ both generate cyclic $ 3 $ subgroups of $ SU(3) $ which are isomorphic but not conjugate.

Ian Gershon Teixeira
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  • https://mathoverflow.net/questions/53955/which-compact-groups-have-nonisomorphic-irreducible-representations-of-the-same – Moishe Kohan Apr 22 '23 at 19:28
  • @MoisheKohan Interesting link! Could you help me better understand the precise relationship between a compact group $ H $ having non-isomorphic irreps of the same dimension and the answer to my question? For example, the finite group $ H=SL(2,5) $ has two non-isomorphic irreps of dimension $ 2 $, but nonetheless it is the case that all subgroups of $ SU(2) $ isomorphic to $ SL(2,5) $ are conjugate. – Ian Gershon Teixeira Apr 22 '23 at 20:20
  • Just think about it a bit more and keep in mind that simple compact connected Lie groups have only inner automorphisms. – Moishe Kohan Apr 22 '23 at 20:31
  • @MoisheKohan I thought the simple https://en.wikipedia.org/wiki/Simple_Lie_group simply connected compact connected Lie groups of types $ A_n, n \geq 2 $ and $ D_n, n \geq 4 $ and $ E_6 $ all had nontrivial outer automorphisms? But I see that the two $ C_n $ fundamental reps from your link should work as examples. – Ian Gershon Teixeira Apr 22 '23 at 21:01
  • Sorry, I forgot about the automorphisms of the Dynkin diagram. But the link will give you examples not coming from diagram automorphisms. – Moishe Kohan Apr 22 '23 at 23:54

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The minimal counterexample is the almost quasisimple group $ 2.S_5 \cong SL(2,5).2 $. This group has two faithful irreps of degree $ 4 $ that have character values in $ \mathbb{Z}[\sqrt{-3}] $, these irreps are just related by complex conjugation and they have the same image in $ SU(4) $. However $ 2.S_5 \cong SL(2,5).2 $ has a third and final faithful irrep of degree $ 4 $, which maps into the $ Sp(2) $ subgroup of $ SU(4) $ and has character values in $ \mathbb{Z} $. The $ 2.S_5 $ subgroup of $ SU(4) $ defined over $ \mathbb{Z}[\sqrt{-3}] $ is not conjugate to the $ 2.S_5 $ subgroups of $ SU(4) $ defined over $ \mathbb{Z} $. I think these two isomorphic irreducible but not conjugate $ 2.S_5 $ subgroups of $ SU(4) $ are probably the smallest example of the phenomenon I asked about.

I think the two non conjugate $ Sp(3) $ subgroups of $ SU(6) $ are the smallest connected counterexample.

$ SO(7) $ has two non conjugate irreps in $ SU(7) $. This is not minimal but it is interesting.

Ian Gershon Teixeira
  • 10,176