A Hardy-Ramanujan Formula for Lie Algebras

Abstract

We study certain combinatoric aspects of the set of all unitary representations of a finite-dimensional semisimple Lie algebra $\g$. We interpret the Hardy--Ramanujan--Rademacher formula for the integer partition function as a statement about $\su_2$, and explore in some detail the generalization to other Lie algebras. We conjecture that the number $\Mod{\g}{d}$ of $\g$-modules in dimension $d$ is given by $(\alpha/d) \exp(\beta d^\gamma)$ for $d \gg 1$, which (if true) has profound consequences for other combinatorial invariants of $\g$-modules. In particular, the fraction $\FracMod{1}{\g}{d}$ of $d$-dimensional\linebreak $\g$-modules that have a one-dimensional submodule is determined by the generating function for $\Mod{\g}{d}$. The dependence of $\FracMod{1}{\g}{d}$ on $d$ is complicated and beautiful, depending on the congruence class of $d$ mod $n$ and on generating curves that resemble a double helix within a given congruence class. We also summands in the direct sum decomposition as a function on the space of all $\g$-modules in a fixed dimension, and plot its histogram. This is related to the concept (used in quantum information theory) of noiseless subsystem. We identify a simple function that is conjectured to be the asymptotic form of the aforementioned histogram, and verify numerically that this is correct for $\su_n$.