Can Young tableaux, or generalisations thereof, determine and parametrise (uniquely) all the irreducible representations of each simple Lie group over the complex numbers, ignoring the 5 exceptions?
There are four families of lie groups:
- The A-series: $A_n$ = $SL_{n+1}$
- The B-series: $B_n$ = $SO_{2n+1}$
- The C-series: $C_n$ = $Sp_{n}$
- The D-series: $D_n$ = $SO_{2n}$
Wikipedia says they can for:
Irreducible representations for $SL_n\mathbb{(C)}$
Irreducible representations for $SU_n$
Now $SU_n$ is the universal cover of $SO_n$, so its representation theory subsumes that of $SO_n$. So we have the $A,B,D$-Series, this leaves only the $C$-Series, that is for the symplectic groups $Sp_n$.