Question:


Suppose that $V\_1$ and $V\_2$ are two finite dimensional representations of lie algebra $\mathfrak{g}$ generated by highest weight vectors $v\_1^\*$ and $v\_2^\*$ of weights $\mu\_1$ and $\mu\_2$ respectively.


Now, suppose that $V\_3$ is an irreducible representation of $V\_1 \otimes V\_2$ such that the highest weight of $V\_3$ has weight $\mu$. I want to show that $\mu = \mu\_1 + \lambda $ for some weight $\lambda$ of $V\_2$.


Approach: I know that the weight spaces of the tensor $V\_1 \otimes V\_2$ are spanned by tensor products of weight vectors of $V\_1$ and $V\_2$. Then $v\_1 \otimes v\_2$ is a highest weight vector of $V\_1 \otimes V\_2$ of highest weight $\mu\_1+ \mu\_2$. If $w$ is a highest weight vector of $V\_3$, then it is a sum of the form $v\_{1,k} \otimes v\_{2,k}$ with $v\_{1,k}$ of weight $\lambda\_{1,k}$ and $v\_{2,k}$ of weight $\lambda\_{2,k}$ such that $\lambda\_{1,k} + \lambda\_{1,k} =\mu.$


The goal is to show that for one of these tensors $\lambda\_{1,k} = \mu\_1.$ Considering a simpler case when $w = v\_1 \otimes v\_2$ with $v\_1$ of weight $\lambda\_1$, $v\_2$ of weight $\lambda\_2$ and $\lambda\_1 +\lambda\_2 = \mu$. Suppose that $\lambda\_1 \neq \mu\_1$, how can I get a contradiction here? I believe if this part is cleared, we can then next suppose that $w = \sum\_{k}v\_{1,k} \otimes v\_{2,k}$ and also that $\lambda\_{1,k} \neq \mu\_1$ for all $k$ ( the sum being over some indexing set) and obtain a contradiction ( I believe this approach answers this question).


P.S: I apologize this is not a research question but I think its an interesting question. I have been struggling to come up with a proof and that is why I am seeking help. I will be glad if anyone interested in this question to please provide me a proof/explanation to this question. Thanks



Question on irreducible representation of tensor products
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*](http://www.mathunion.org/ICM/ICM2010.3/Main/icm2010.3.1226.1261.pdf) 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*](http://www.ams.org/mathscinet-getitem?mr=0109192), 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.]


