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 (α/d) exp(βd ^γ ) for d » 1, which (if true) has profound consequences for other combinatorial invariants of g-modules. In particular, the fraction F ₁(g, d) of d-dimensional g-modules that have a one-dimensional submodule is determined by the generating function for Mod(g, d). The dependence of F 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 consider the total number of repeated irreducible 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.

Keywords:

• Primary 17B15
• Secondary 05A99
• Enumerative combinatorics
• representation theory