Hidden symmetries in subgroups of Lie groups?

[arivero]

TL;DR Summary

Wonder all the ways to hide SU(3) inside of SU(5)

I was wondering, consider the group SU(5) and I break it down to SU(3) x SU(2).
To break it, I have taken some choice of roots. But is the choice unique, or by selecting some roots have I left some other SU(3) that could also exist?
If so, is there a way to see the remnants? Not sure how. Perhaps the 15 or the 24 of SU(5) happend to have a lot of equilateral triangles and by taking a choice some of them survive into the representations of SU(3) and some of them dissapear but they are really there and they could be visible somehow.

 

[jambaugh]

Disclaimer: I've been away from this for a while so do cross check and take my assertions here as tentative.

I think what you are talking about is the quotient space (topological space) for these manifolds. The quotient
SU(5)/SU(3) expressed by the short exact sequence 0-->SU(3)-->SU(5)-->SU(5)/SU(3)-->0. (This is a sequence in the category of manifolds I believe.)

Another way to get a handle on this is to consider the representation spaces. Let Hⁿ be the n-dimensional complex Hilbert space. Then U(n) is its automorphism group. You can then understand U(5)/U(3) or rather U(5)/U(3)xU(2) in terms of the Grasmannian of 3 dimensional subspaces (which also identify their respective 2 dimensional orthogonal subspaces) of H⁵. Note that once you remove the central element of U(5) you will still have U(3)xU(2)/Z₂ subgroups but with a central correlation (there's an element central to U(3) and U(2) in SU(5). Indeed this centralizer in SU(5) will specify the specific SU(3)xSU(2) subgroup as its centralizer.

In the matrix representation you'll need two mutually commuting projective operators P₃ and P₂ such that trace(P_n)=n (they project onto n-dimensional subspaces n=2,3. The defining central element is then:
X= 3/5 P₂-2/5 P₃ up to sign. Since by construction X is hermitian and traceless it is a generator for the U(1) subgroup {g=exp(iXt); all real t}. This then is the object of interest and its orbit under the SU(5) group gives you that space of all SU(2)xSU(3) subgroups or rather SU(2)xSU(3)xU(1) subgroups (that is provided the quotient space is connected but I think that is the case here.)

Well, that's my 10min stab at your question. I imagine others here may have better answers.