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Jim Humphreys
Jim Humphreys
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The question is definitely not trivial, but is not currently at research level because the answer has been known for a long time. A good modern source available online here is the 2010 ICM write-up by Shrawan Kumar. He attributes the result to the late Bert Kostant in aKostant's seminal (though in retrospect overcomplicated) 1959 paper on, with free access online here, deriving a closed formula for weight multiplicities in irreducible finite dimensional representations. See Proposition (3.2) in Kumar's recent survey, where there is also a natural further refinement giving an upper bound on the multiplicity of each irreducible summand of such a tensor product.

[By the way, I included a version of this result as Exercise 24.12 in my 1972 graduate text on Lie algebras, though for some reason I can't recall now the exercise stated that it should be deduced from Steinberg's tensor product formula. That seems misleading.]

The question is definitely not trivial, but is not currently at research level because the answer has been known for a long time. A good source available online here is the 2010 ICM write-up by Shrawan Kumar. He attributes the result to the late Bert Kostant in a seminal 1959 paper on weight multiplicities. See Proposition (3.2) in Kumar's recent survey, where there is also a natural further refinement giving an upper bound on the multiplicity of each irreducible summand of such a tensor product.

[By the way, I included a version of this result as Exercise 24.12 in my 1972 graduate text on Lie algebras, though for some reason I can't recall now the exercise stated that it should be deduced from Steinberg's tensor product formula. That seems misleading.]

The question is definitely not trivial, but is not currently at research level because the answer has been known for a long time. A good modern source available online here is the 2010 ICM write-up by Shrawan Kumar. He attributes the result to the late Bert Kostant's seminal (though in retrospect overcomplicated) 1959 paper, with free access online here, deriving a closed formula for weight multiplicities in irreducible finite dimensional representations. See Proposition (3.2) in Kumar's recent survey, where there is also a natural further refinement giving an upper bound on the multiplicity of each irreducible summand of such a tensor product.

[By the way, I included a version of this result as Exercise 24.12 in my 1972 graduate text on Lie algebras, though for some reason I can't recall now the exercise stated that it should be deduced from Steinberg's tensor product formula. That seems misleading.]

Source Link
Jim Humphreys
Jim Humphreys
  • 52.9k
  • 4
  • 120
  • 241

The question is definitely not trivial, but is not currently at research level because the answer has been known for a long time. A good source available online here is the 2010 ICM write-up by Shrawan Kumar. He attributes the result to the late Bert Kostant in a seminal 1959 paper on weight multiplicities. See Proposition (3.2) in Kumar's recent survey, where there is also a natural further refinement giving an upper bound on the multiplicity of each irreducible summand of such a tensor product.

[By the way, I included a version of this result as Exercise 24.12 in my 1972 graduate text on Lie algebras, though for some reason I can't recall now the exercise stated that it should be deduced from Steinberg's tensor product formula. That seems misleading.]