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I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$. It is possible that they have been already computed; in this case I will appreciate a reference. Otherwise, How could I compute them?

These particular representations are as follows: let $\pi_{k,p}$ be the irreducible representation of $SO(2m)$ with highest weight $$ \Lambda_{k,p}:=k \varepsilon_1+(\varepsilon_1+\varepsilon_2+...+\varepsilon_p) = (k+1) \varepsilon_1+\varepsilon_2+...+\varepsilon_p, $$ for $k\geq0$ and $1\leq p<m$. Thus, for $\mu=\sum_{j=1}^m a_j\varepsilon_j$ with $a_j\in\mathbb Z$, I want a formula for $$\dim V_\pi(\mu).$$

Extremal cases: If $k=0$, $\pi_{0,p}$ is the $p$-exterior representation $\bigwedge^p(\mathbb C^{2m})$ of the standard representation. One can check that $$ \dim V_\pi(\mu) = \binom{m-p+2r}{r} $$ if $\mu=\sum a_j \varepsilon_j$ with $|a_j|\leq 1$ for all $j$ and $\|\mu\|_1:=\sum |a_j|= p-2r$ for some $r\in\mathbb Z_{\geq0}$, and $\dim V_\pi(\mu) = 0$ otherwise.

If $p=1$, then $\pi_{k,1}$ can be realized as the vector space of harmonic homogeneous polynomials of degree $k+1$ in $m$ variables. One can check that $$ \dim V_\pi(\mu) = \binom{r+m-2}{m-2} $$ if $\mu=\sum a_j \varepsilon_j$ with $\|\mu\|_1=k+1-2r$ for some $r\in\mathbb Z_{\geq0}$, and $\dim V_\pi(\mu) = 0$ otherwise.

One idea: For $k\geq1$ and $2\leq p\leq m$, I have tried by using Steinberg's formula for the decomposition of $\pi_{k,0}\otimes\pi_{0,p}$ and then induction on $k$ and $p$. I failed even for $p=2$.

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  • $\begingroup$ A quick google search discovered an article by Koike, that expresses all you want in terms of Littlewood-Richardson adn Kostka coefficients. sciencedirect.com/science/article/pii/0021869387901001 $\endgroup$ – Vít Tuček Sep 3 '14 at 12:27
  • $\begingroup$ @Vít: you are right, the formula is expressed in terms of other coefficients. I hope there exists a more explicit closed formula, since these representations are very particular. $\endgroup$ – emiliocba Sep 3 '14 at 14:07
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    $\begingroup$ I'm not optimistic about finding closed formulas for all weight multiplicities in these cases. In general, Freudenthal's formula is probably best for direct computations when other methods are inadequate. (By the way, it's unclear to me why you specify $p < m$ rather than $p \leq m$. Maybe I'm misunderstanding your notation.) $\endgroup$ – Jim Humphreys Sep 3 '14 at 20:25
  • $\begingroup$ I considered $p<m$ only because the case $p=m$ splits ($\Lambda_{k,m}$ and $(k+1)\varepsilon_1+ \varepsilon_2 +\dots+ \varepsilon_{m-1} - \varepsilon_m$), so I didn't want to explain these technicalities. $\endgroup$ – emiliocba Sep 4 '14 at 7:37
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For $\mu$ a weight, let $||\mu||_1$ denote the one-norm of $\mu$ (the sum of the absolute values of its entries) and let $Z(\mu)$ be the number of zero coordinates of $\mu$. Let $k\geq0$ and $1\leq p\leq n$. Write $r(\mu)=(k+p-||\mu||_1)/2$. If $r(\mu)$ is a non-negative integer, then \begin{align*} m_{\pi_{\Lambda_{k,p}}}(\mu) &= \sum_{j=1}^{p} (-1)^{j-1} \sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor} \binom{n-p+j+2t}{t} \sum_{\beta=0}^{p-j-2t} 2^{p-j-2t-\beta} \binom{n-Z(\mu)}{\beta} \binom{Z(\mu)}{p-j-2t-\beta} \\ &\quad \sum_{\alpha=0}^\beta \binom{\beta}{\alpha} \sum_{i=0}^{j-1} \binom{r(\mu)-i-p+\alpha+t+j+n-2}{n-2}, \end{align*} and $m_{\pi_{k,p}}(\mu)=0$ otherwise.

This is Theorem 4.1 in this article, where there are also analogous weight multiplicity formulas for the rest of the classical complex Lie algebras.

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I suspect that the answer you seek can be deduced from the Kostant Multiplicity Formula. It gives the multiplicity of a weight in an irreducible complex representation of a compact Lie group $G$. One reference would be: A Formula for the Multiplicity of a Weight, by Kostant.

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    $\begingroup$ I think in general, it's quite difficult to use Kostant's formula to obtain a closed formula for the multiplicity. $\endgroup$ – emiliocba Sep 4 '14 at 10:18

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