Thursday, 24 August 2006

rt.representation theory - Decomposing a tensor product

For SO(n) a calculation using LiE gives:
(using partition notation so W is [2])
and assuming n is not small



For WotimesW, [4],[3,1],[2,2],[2],[1,1],[]
(all with multiplicity one)



and for WotimesWotimesW,
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 Sp(n) by taking conjugate partitions.



There is also a relationship with SL(n).



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 m. in front to denote multiplicity is m. A partition is [a1,a2,a3,...] where aigeaj if i less than j. To convert to a highest weight vector add the appropriate number of 0s to the end.
Then take [a1a2,a2a3,a3a4,...]. This gives a dominant integral weight. The fundamental weights are the partitions [1,,,,1]. If this has length k this corresponds to the k-th exterior power of the vector representation (provided 2k1 less than n).



In particular the trivial representation is [], the vector representation V is [1], the exterior square of V is [1,1], the symmetric square is [2]+[].



For the k-th tensor power of W you will see partitions of 2k2p for 0leplek only and it remains to determine the multiplicities (possibly 0). For SL(n) just take the partitions of 2k (with their multiplicities) and ignore the rest.

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