Statistics for random representations of Lie algebras
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909524947894272 |
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| author | Bridges, Walter Bringmann, Kathrin Nazaroglu, Caner |
| author_facet | Bridges, Walter Bringmann, Kathrin Nazaroglu, Caner |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistics for random representations of Lie algebras Bridges, Walter Bringmann, Kathrin Nazaroglu, Caner Representation Theory Combinatorics Number Theory Probability 60C05, 17B10, 11P81 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. |
| title | Statistics for random representations of Lie algebras |
| topic | Representation Theory Combinatorics Number Theory Probability 60C05, 17B10, 11P81 |
| url | https://arxiv.org/abs/2503.02822 |