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For partitions $\lambda, \mu \vdash n$, the Kostka-Foulkes polynomial $K_{\lambda,\mu}(q)$, a $q$-analog of the Kostka coefficient $K_{\lambda,\mu}$, has a combinatorial description, due to Lascoux and Schützenberger, as $$ K_{\lambda,\mu}(q) = \sum_{T} q^{\mathrm{charge}(T)},$$ where the sum is over semistandard Young tableaux of shape $\lambda$ and content $\mu$. In the case of standard tableaux, charge is basically the same as major index (maybe we need to consider cocharge instead); at any rate, what is true is that if $\mu=(1,1,\ldots,1)$, then we have the $q$-hook length formula (due to Stanley) which gives a product formula for $K_{\lambda,\mu}(q)$: $$ K_{\lambda,\mu}(q) = \sum_{\textrm{$T$ a SYT of shape $\lambda$}} q^{\mathrm{maj}(T)}= \frac{[n]!_q}{\prod_{u\in\lambda} [h(u)]_q}.$$

The Kostka-Foulkes polynomials are the Type A version of Lusztig's $q$-analog of weight multiplicity (see papers of Lusztig and Joseph-Letzter-Zelikson cited below for the definition of these). As I understand it, giving a combinatorial description of these in other types as a statistical generating function over tableaux analogous to the charge formula is a pretty difficult problem. This problem is not fully resolved but the theory of crystals has led to significant progress (see the recent papers of Lecouvey-Lenart https://arxiv.org/abs/1707.03314 and Jang-Kwon https://arxiv.org/abs/1908.11041).

My question however is not about statistical generating functions, however. It is rather about product formulas.

Question: Are there any examples of Lusztig's $q$-analog of weight multiplicity, beyond the SYT $q$-hook length formula explained above, for which we have a product formula?

(Note that when we pass to the more general root system formulation, content $\mu=(1,1,\ldots,1)$ actually corresponds to weight $\mu=0$ since we mod out by $(1,1,\ldots,1)$ to form the weight lattice in Type A. But also note that not every Type A $q$-analog of weight multiplicity of the form $K_{\lambda,0}(q)$ has a hook-length formula because we also need that $\lambda$ is a partition with the right number of boxes.)

Lusztig, George, Singularities, character formulas, and a (q)-analog of weight multiplicities, Astérisque 101-102, 208-229 (1983). ZBL0561.22013.

Joseph, Anthony; Letzter, Gail; Zelikson, Shmuel, On the Brylinski-Kostant filtration, J. Am. Math. Soc. 13, No. 4, 945-970 (2000). ZBL0991.17006.

EDIT: I just realized that the question still makes sense even removing the "q" from it. That is...

q=1 version of question: Are there other product formulas for weight multiplicities of simple Lie algebras, beyond the Type A cases corresponding to SYTs?

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For some skew shapes, there might be product formulas, I suggest to look at the Pak-Panova-Morales series of papers.

When it comes to major-index generating functions of posets, there is a q-analog proved for trees and forests, due to A. Bjorner and M. Wachs, which has a nice product formula. There is also the class of D-complete posets, which allow for product formulas.

For shifted shapes, see e.g. this document.

Although, the connection with charge is perhaps not clear.

Coincidentally, I am currently working on a project which will answer some of your questions regarding charge for certain Kostka-Foulkes polynomials...

EDIT: For things closer to weights, see e.g. this twisted version of Kostka-Foulkes polynomials, which gives weight multiplicities of tensor products of twisted representations into twisted representations (it seems).

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  • $\begingroup$ Are any of these actually $q$-analogs of weight multiplicity? $\endgroup$ – Sam Hopkins Aug 30 at 9:34
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    $\begingroup$ I think maybe my question was unclear. It was intended as: is there a root system $\Phi$ and choice of weights $\lambda$, $\mu$, such that we have a product formula for Lusztig's $q$-analog of weight multiplicity $K_{\lambda,\mu}(q)$, beyond $\Phi=A_n$ and the $\lambda$,$\mu$ corresponding to Standard Young Tableaux (for which the $q$-hook length formula applies)? $\endgroup$ – Sam Hopkins Aug 30 at 19:50

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