For a calculation using LiE gives:
(using partition notation so is [2])
and assuming is not small
For , [4],[3,1],[2,2],[2],[1,1],[]
(all with multiplicity one)
and for ,
1.[6] 2.[5,1] 3.[4,2] 1.[3,3] 1.[4,1,1] 2.[3,2,1] 1.[2,2,2] 3.[4] 6.[3,1] 2.[2,2] 3.[2,1,1] 6.[2] 3.[1,1] 1.[]
The same works for by taking conjugate partitions.
There is also a relationship with .
This is taking your question at face value. If it is understanding you're after instead then the best approach is to use crystal graphs.
The notation I have used denotes a representation by a partition. I have put in front to denote multiplicity is . A partition is where if less than . To convert to a highest weight vector add the appropriate number of s to the end.
Then take . This gives a dominant integral weight. The fundamental weights are the partitions . If this has length this corresponds to the -th exterior power of the vector representation (provided less than ).
In particular the trivial representation is , the vector representation is , the exterior square of is , the symmetric square is .
For the -th tensor power of you will see partitions of for only and it remains to determine the multiplicities (possibly ). For just take the partitions of (with their multiplicities) and ignore the rest.
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