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A law of large numbers for infinitesimal characters of real reductive Lie algebras

Primarily AI-generated textHuman understanding: all partsmath.RT — Representation Theorymath.PR — Probabilityquant-ph — Quantum Physics

Contributed by Andrew Salmon ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

We prove a law of large numbers for the infinitesimal characters of real reductive Lie algebras. Let π\pi be a unitary representation of a connected Lie group with reductive Lie algebra. For product states on the tensor powers of π\pi whose sites have uniformly bounded moments, for instance identical sites, the infinitesimal character divided by the number of sites concentrates at the spectrum of the limiting mean moment, with probabilities of deviation O(n−m)O(n^{-m}) for every mm. The generators may be unbounded and the group non-compact. For compact groups this recovers the laws of large numbers of Keyl and Werner and of Botero, Christandl and Vrana, which enter the asymptotic analysis of linear programming bounds for binary codes; spherical designs require the non-compact case. We give concrete examples arising from Howe dual pairs on bosonic Fock space.

Author comments: Comments welcome

Provenance statement

This article was primarily written with Claude with direction by the contributor. Claude Opus 5.5 supplied the original proofs of all results. Codex was also used for editing. Several aspects of the proofs and prose were rewritten to fit the contributor's taste.

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Version history

  1. v1 · 16 pagesSubmitted by Andrew SalmonComments welcomeCurrentNov 13, 2026