Simple harmonic oscillators from non-semisimple walled Brauer algebras
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arXiv
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| Format: | Preprint |
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2025
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| author | Ramgoolam, Sanjaye Studziński, Michał |
| author_facet | Ramgoolam, Sanjaye Studziński, Michał |
| contents | Walled Brauer algebras $B_N ( m , n ) $ illuminate the combinatorics of mixed tensor representations of $U(N)$, with $m$ copies of the fundamental and $n$ copies of the anti-fundamental representation. They lie at the intersection of research in representation theory, AdS/CFT and quantum information theory. They have been used to study of correlators in multi-matrix models motivated by brane-anti-brane physics in AdS/CFT. They have been applied in computing and optimising fidelities of port-based quantum teleportation. There is a large $N$ regime, specifically $ N \ge (m+n)$ where the algebras are semi-simple and their representation theory more tractable. There are known combinatorial formulae for dimensions of irreducible representations and associated reduction multiplicities. The large $N$ regime has a stability property whereby these formulae are independent of $N$. In this paper we initiate a systematic study of the combinatorics in the non-semisimple regime of $ N = m +n - l $, with positive $l$. We introduce restricted Bratteli diagrams (RBD) which are useful as an instrument to process known data from the large $N$ regime to calculate representation theory data in the non-semisimple regime. We identify within the non-semisimple regime, a region of $(m,n)$-stability, where $ \min ( m, n ) \ge ( 2l -3) $ and the RBD take a stable form depending on $l$ only and not the choice of $ m,n$ within the region. In this regime, several aspects of the combinatorics of the RBD are controlled by a universal partition function for an infinite tower of simple harmonic oscillators closely related, but not identical, to the partition function of 2D non-chiral free scalar field theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_04234 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple harmonic oscillators from non-semisimple walled Brauer algebras Ramgoolam, Sanjaye Studziński, Michał High Energy Physics - Theory Combinatorics Representation Theory Quantum Physics Walled Brauer algebras $B_N ( m , n ) $ illuminate the combinatorics of mixed tensor representations of $U(N)$, with $m$ copies of the fundamental and $n$ copies of the anti-fundamental representation. They lie at the intersection of research in representation theory, AdS/CFT and quantum information theory. They have been used to study of correlators in multi-matrix models motivated by brane-anti-brane physics in AdS/CFT. They have been applied in computing and optimising fidelities of port-based quantum teleportation. There is a large $N$ regime, specifically $ N \ge (m+n)$ where the algebras are semi-simple and their representation theory more tractable. There are known combinatorial formulae for dimensions of irreducible representations and associated reduction multiplicities. The large $N$ regime has a stability property whereby these formulae are independent of $N$. In this paper we initiate a systematic study of the combinatorics in the non-semisimple regime of $ N = m +n - l $, with positive $l$. We introduce restricted Bratteli diagrams (RBD) which are useful as an instrument to process known data from the large $N$ regime to calculate representation theory data in the non-semisimple regime. We identify within the non-semisimple regime, a region of $(m,n)$-stability, where $ \min ( m, n ) \ge ( 2l -3) $ and the RBD take a stable form depending on $l$ only and not the choice of $ m,n$ within the region. In this regime, several aspects of the combinatorics of the RBD are controlled by a universal partition function for an infinite tower of simple harmonic oscillators closely related, but not identical, to the partition function of 2D non-chiral free scalar field theory. |
| title | Simple harmonic oscillators from non-semisimple walled Brauer algebras |
| topic | High Energy Physics - Theory Combinatorics Representation Theory Quantum Physics |
| url | https://arxiv.org/abs/2509.04234 |