# Irreducible dimensions generating function for Lie algebra $\mathfrak{sl}_n$

Let $$\lambda = \sum_{i = 1}^{n - 1} m_i \omega_i$$ be the highest weight of irreducible representation $$V(\lambda)$$ of Lie algebra $$\mathfrak{sl}_n$$. As we know from the Weyl formula, $$\dim V(\lambda) = \prod\limits_{i \leq j} \frac{m_i + ... + m_j + j - i + 1}{j - i + 1}.$$ Consider the multivariate generating function in variables $$z_1, ..., z_{n - 1}$$ $$g_n(z) = \sum\limits_{\lambda} \dim V(\lambda)z^{\lambda}.$$ For example, $$g_2(z_1) = \frac{1}{1 - z_1},\hspace{1cm} g_3(z_1, z_2) = \frac{1 - z_1z_2}{(1 - z_1)^2(1 - z_2)^2},$$ $$g_4(z_1, z_2, z_3)$$ is less compact, but i can provide it.

The denomenator is always given by $$\prod_{i=1}^{n - 1} (1 - z_i)^{i(n - i - 1) + 1}$$. I computed the number of terms in the numerator (weighted by coefficients), it is given by the sequence $$1, 2, 36, 9836, ...$$ which does not appear on OEIS.

Question: Is there a combinatorial interpretation for numerator of this generating function? Even if someone presents combinatorial objects counted by the sequence $$1, 2, 36, 9836, ...$$, then it would already be interesting.

Motivation: If we restrict to representations of the form $$V(m_i\omega_i)$$ (corresponding to rectangular diagrams), or equivalenty, if we set all variables except for $$z_i$$ to zero, then the numerator is given by d-dimensional Narayana polynomial $$\mathcal{N}_{i, n - i}(z_i)$$ which has many good interpretations.

• See math.mit.edu/~rstan/pubs/pubfiles/44.pdf for a combinatorial formula for the "character generator" as a sum of rational functions. Aug 7, 2021 at 14:39
• @RichardStanley It is indeed very close to what i want. Getting a sum with a common denominator would be the best. Also, in this work you mention that no analogue is known for $\mathfrak{sp}$ and $\mathfrak{so}$. Has that changed? Aug 8, 2021 at 13:24
• To my knowledge that has not changed. Aug 9, 2021 at 22:31
• @RichardStanley Okay, thank you! Aug 14, 2021 at 11:29