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This is exercise 22.1 in Humphreys Introduction to Lie Algebras and Representation Theory book. Let $\mathfrak{g}$ be a semisimple complex Lie Algebra and let $\lambda$ be a dominant weight also let $\alpha$ be a root of $\mathfrak{g}$. Prove without using Freudenthal's formula, $V(\lambda)$ , the irreducible representation of highest weight $\lambda$, the dimension of each component in of the $\alpha$ string though $\lambda$ is $1$ i.e. the dimension of the $\lambda- k \alpha$ weight space of $V(\lambda)$ is 1 for $0\leq k \leq <\lambda,\check{\alpha}>$.

My thoughts on this are that the fact that the Weyl group preserves the dimension of weight spaces we immediately have that the start (as its highest weight) and end of the string are dim 1. I am not sure how to proceed, I want to try use $\mathfrak{sl}_2$ theory but haven't thought of how.

Up to this point we have covered the structure theory of lie algebras, abstract root systems and most recently the construction of the $V\lambda$ using Verma modules.

Edit: Simple root $\alpha$

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  • $\begingroup$ Sorry have edited - hope thats clearer $\endgroup$
    – Rick
    Commented Jan 11 at 20:03
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    $\begingroup$ Good question rick I was wondering this too $\endgroup$ Commented Jan 12 at 18:10
  • $\begingroup$ Yes I misunderstood thank you $\endgroup$
    – Rick
    Commented Jan 13 at 20:25

1 Answer 1

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In my edition (3rd printing, 1980) Exercise 22.1 prescribes $\alpha\in\Delta$, so $\alpha$ is supposed to be a simple root. This is necessary for the claim to hold. Otherwise the adjoint representation of $\mathfrak{sl}_3$ is possibly the simplest counterexample: $\lambda=\alpha_1+\alpha_2$, choose $\alpha=\alpha_1+\alpha_2$, when $\dim V(\lambda)_{\lambda-\alpha}=\dim V(\lambda)_0=2$.

But when $\alpha$ is a simple root the claim can be seen as follows. If $N^-=\prod_{\alpha\prec 0}L_\alpha$ is the subalgebra spanned by the negative root spaces, then every standard cyclic module $Z(\lambda)$ is isomorphic to $\mathfrak{U}(N^-)$ as a $\mathfrak{U}(N^-)$-module. PBW then implies that the weight space $Z(\lambda)_{\lambda-j\alpha}$, $j\ge0$, is $1$-dimensional, spanned by $y_\alpha^j\otimes v^+$ (see section 20.3 for a description of this module). This is because $j\alpha$ is the only weight $j\alpha$ linear combination $\sum_{\alpha\in\Phi^+}n(\alpha)\alpha\in\Lambda$ of positive roots with non-negative integer coefficients, $n(\alpha)\in\Bbb{Z}_{\ge0}$.

The module $V(\lambda)$ is a quotient of $Z(\lambda)$, so it follows that the dimensions of all the weight spaces $V(\lambda)_{\lambda-k\alpha}$ are at most one. On the other hand, the representation theory of $\mathfrak{sl}_{2,\alpha}$ dictates that those weight spaces are non-trivial up to $k\le\langle\lambda,\check{\alpha}\rangle$.

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