A Hardy-Ramanujan formula for Lie algebras. (English) Zbl 1161.17004

Summary: We study certain combinatoric aspects of the set of all unitary representations of a finite-dimensional semisimple Lie algebra \(\mathfrak g\). We interpret the Hardy-Ramanujan-Rademacher formula for the integer partition function as a statement about \(\mathfrak {su}_2\), and explore in some detail the generalization to other Lie algebras. We conjecture that the number Mod\(({\mathfrak g},{d})\) of \(\mathfrak g\)-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 \(\mathfrak g\)-modules. In particular, the fraction \(\mathcal F_{1}({\mathfrak g},{d})\) of \(d\)-dimensional\(\mathfrak g\)-modules that have a one-dimensional submodule is determined by the generating function for Mod\(({\mathfrak g},{d})\). The dependence of \(\mathcal F _{1}({\mathfrak g},{d})\) 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 \(\mathfrak g\)-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 \(\mathfrak {su}_n\).


17B15 Representations of Lie algebras and Lie superalgebras, analytic theory
05A15 Exact enumeration problems, generating functions
81P68 Quantum computation
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