Statistics for random representations of Lie algebras
Walter Bridges, Kathrin Bringmann, Caner Nazaroglu
TL;DR
This work develops a comprehensive statistical framework for random finite-dimensional representations of $\frak{sl}_{r+1}(\bf C)$ with large dimension, extending partition statistics (the $r=1$ case) to higher rank via a Boltzmann ensemble. By encoding irreps with dominant weights $\bm{k}$ and dimensions $a_r(\bm{k})$, the authors introduce a canonical model $Q_q$ with independent Geometric multiplicities, identify a saddle point $s_n$ so that $E_{q_n}(\dim)=n$, and prove an equivalence of ensembles to transfer asymptotic distributions from the Boltzmann model to the uniform measure $P_n$. They derive precise asymptotics for the partial sums of irreps, lower bounds on $a_r(\bm{k})$ on circles, and limiting distributions for small multiplicities, the maximal dimension and height (both yielding Gumbel limits after suitable centering), the limit shape of weight profiles, and the limiting distribution for the total number of irreps via a moment-generating function characterization. The results generalize classical partition phenomena to higher-rank Lie algebras and point to deep connections with circle-method techniques and Witten zeta functions, while offering a roadmap for extensions to other algebras and measures.
Abstract
In this paper we investigate how a typical, large-dimensional representation looks for a complex Lie algebra. In particular, we study the family $\mathfrak{sl}_{r+1}(\mathbb{C})$ of Lie algebras for $r \geq 2$ and derive asymptotic probability distributions for the multiplicity of small irreducible representations, as well as the largest dimension, the largest height, and the total number of irreducible representations appearing in the decomposition of a representation sampled uniformly from all representations with the same dimension. This provides a natural generalization to the similar statistical studies of integer partitions, which forms the case $r=1$ of our considerations and where one has a rich toolkit ranging from combinatorial methods to approaches utilizing the theory of modular forms. We perform our analysis by extending the statistical mechanics inspired approaches in the case of partitions to the infinite family here.