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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:

  1. The A-series: $A_n$ = $SL_{n+1}$
  2. The B-series: $B_n$ = $SO_{2n+1}$
  3. The C-series: $C_n$ = $Sp_{n}$
  4. The D-series: $D_n$ = $SO_{2n}$

Wikipedia says they can for:

  1. Irreducible representations for $SL_n\mathbb{(C)}$

  2. 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 $Sp_n$.

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1 Answer 1

up vote 4 down vote accepted

Your analysis is wrong: $SU_2$ is the universal cover of $SO_3(\mathbb R)$ but (1) no such relation exists for general rank (note that already the indices are not the same between the two groups), and more importantly (2) the types $D_n$ and $B_n$ refer to complex groups $SO_{2n}(\mathbb C)$ and $SO_{2n+1}(\mathbb C)$ which are not in any special relation to any real group $SU_n$, or the corresponding complex group $SL_n(\mathbb C)$, at all. So types $B,D$ are really different from type $A$ (as are types $C$ of course), and one does not get any other classical types for free by doing type $A$. In fact, especially for questions of tableaux, the other classical types are considerbly harder to treat than type $A$.

To answer your question, there do exist generalisations of tableaux for other classical types that capture some aspects of what Young tableaux do for type $A$. However, unfortunately the type of generalisation to use depends on the particular aspect of interest, and so there are about as many different generalisations as there are papers on these subjects. It is a great mess; in some cases there are (rather involved) bijections between certain types of tableaux, or other rather more subtle relations, but not in all cases, and I don't think there is even a good inventory of all the types of tableaux that have been invented.

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Thanks for a clear answer, and correcting my blunder! –  Mozibur Ullah Mar 30 '12 at 20:18
    
Hi Marc - I was wondering if you could share any references that have tableaux generalizations that work for SO(N)? Thanks! –  DJBunk Jun 25 at 13:08

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