Irrep composition for Lie groups

[Joliet Jake]

Hi everyone,

 

I have tried to figure this out myself but I couldn't find any useful references. Does anyone have a clue?

 

Does any Lie group G based on the simple Lie algebras A_n, B_n, C_n, D_n, E₆, E₇, E₈, F₄, G₂ on the complex numbers have the following properties:

 

i) it's simply connected, or it coincides with its universal covering group;

 

ii) there exists an irrep r with dimension < dim(G) (i.e. number of generators) such that r (and r^*) are contained in the composition r x r^*

 

Ideally r should be the fundamental, hence the ^*, but if this holds with a real irrep smaller than the adjoint it would also do.

 

For SU(n) this does not work, the first irrep to be contained in the composition with itself is the adjoint, i.e. either you give up the first hypothesis and take G=SU(n)/Z_n or the second and take dim =dim(G).

 

And if this is never possible, is there a theorem to prove it?

 

Or does one need to abandon the realm of A_n, B_n, C_n, D_n, E₆, E₇, E₈, F₄, G₂ and go to more complex structures?

 

Cheers,

 

Joliet Jake

[Joliet Jake]

E₈ is the closest to my requirements I could make out up to now, since it's simply connected and its smallest dimensional representation is the adjoint, but still doesn't meet them fully...

 

Actually, to boil down my question to a calculation, if anyone can deal with Young tableaus for exceptional groups (I am familiar with su(n) only), could she/he confirm the following compositions:

 

for G₂ : 7 x 7 = 1 + 7 + 14 +27;

 

for F₄ : 26 x 26 = 1 + 26 + 52 + 273 + 324;

 

for E₆ : 27 x 27^* = 1 + 78 + 650;

 

for E₇ : 56 x 56 = 1 + 133 + 1463 + 1539;

 

for E₈ : 248 x 248 = 1 + 248 + 3875 + 27000 + 30380;

 

which will make G₂ and F₄ fulfill the requirements, with E₈ "borderline". I'm not sure if A_n , B_n , C_n , D_n would...

[Joliet Jake]

I found the above relations on Cvitanovic, chapters 16 onwards... I knew I eventually had to learn it!