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In the context of the Simple Lie Algebras Representations, let $\rho$ be half-the-sum of the positive roots and let $V_\rho$ be the irreducible representation of highest weight $\rho$.

Let$\mu$ be a dominant weight smaller than $\rho$. I'm looking for "non classical" formulas for the dimension of the eigenspace relative to $\mu$ in the representation $V_\rho$.

By "non classical" I mean some thing "different from the Kostant Formula or its corollaries", in particular I'm looking, if it exists, for strictly combinatorial one, obtained by coloring opportune Young Diagrams or involving other strictely combinatorial objects.

In particular I'm interested in the cases of simple algebras of type $A_n$ and $C_n$ where there is a well developed Young-diagrammatic theory.

Thank you in advance.

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    $\begingroup$ This may not be easy, but in any case the more standard computational formulas are those of Freudenthal and perhaps Kostant (but not Steinberg). Keep in mind the easy case of type $A_2$, where the 0 weight occurs with multiplicity 2. Can this be readily predicted from a combinatorial viewpoint? $\endgroup$ Aug 15, 2017 at 20:23

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