Refined asymptotic formula for representation counts

Derive an improved asymptotic formula for the number p_r(n) of n-dimensional representations of \(\mathfrak{sl}_{r+1}(\mathbb{C})\), generalizing Romik’s asymptotic formula for \(r=2\), by computing the saddle-point parameter \(q_n\) to sufficient accuracy and conducting a more precise analysis of the associated Witten zeta functions or of the expansion in Proposition 1.1.

Background

The paper relates the enumeration function $p_r(n)$ to the Boltzmann model and, using equivalence of ensembles, obtains the asymptotic $$p_r(n) \sim n^{-\frac{r+2}{r+3}}(2\pi D_{r,\mathrm{var}})^{-1/2}q_n^{-n}\prod_{k\in\mathbb{N}^r}(1-q_n^{a(k)})^{-1}$$. The authors explain that a more accurate evaluation of the saddle-point parameter $q_n$ would permit this estimate to be sharpened into an asymptotic formula with further terms.

The proposed refinement is intended to extend the known result of Romik for $r=2$ to the general family of representation-counting functions. The authors identify the likely technical requirements as a more careful study of Witten zeta functions or additional terms in the asymptotic expansion established in Proposition 1.1. The problem is explicitly left unresolved.