Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where $(M_\lambda)_\mu$ denotes the weight space with weight $\mu$? Of course Weyl character formula gives me the generating function, but I am looking for a closed form formula for $\dim (M_\lambda)_\mu$.

Edit : Apparently this is called the Kostant partition function.

  • $\begingroup$ Which formulation of Weyl's character formula do you mean? I recall it usually being given directly, rather than via a generating function. $\endgroup$ – Tobias Kildetoft Mar 18 at 17:01
  • $\begingroup$ @TobiasKildetoft For instance, if you expand $1/\Delta(z)$ into a power series where $\Delta(z)$ is a Weyl denominator, the coefficients of this series are the certain dimensions that I want. $\endgroup$ – Henry Mar 19 at 19:35
  • $\begingroup$ As you write in your edit, it's the Kostant partition function: the number of ways to write $\lambda-\mu$ as a non-negative integer linear combination of positive roots. You won't get a better formula than that in general. $\endgroup$ – Stephen Mar 20 at 22:54
  • $\begingroup$ @Stephen But isn't Kostant partition function piecewise polynomial? If an explicit form of the piecewise polynomial is known for, say, $A_n$, then that would be helpful. $\endgroup$ – Henry Mar 20 at 23:18

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