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Simple harmonic oscillators from non-semisimple walled Brauer algebras

Sanjaye Ramgoolam, Michał Studziński

TL;DR

This work develops a systematic framework for the non-semisimple regime of walled Brauer algebras $B_N(m,n)$ by introducing restricted Bratteli diagrams (RBD) that partition irreps into modified and unmodified dimensions when $N=m+n-l$ with small $l$. A key finding is the $(m,n)$-stability: for $m,n\ge 2l-3$, the RBD form depends only on $l$, enabling universal counting rules. The modified dimensions are computed explicitly for $l=2,3,4$ through detailed RBD analysis and connections to partial transposition kernels, revealing a deep link to a universal partition function ${\cal Z}_{\rm univ}(x)$ of an infinite tower of simple harmonic oscillators. This oscillator perspective unifies red/green node counting across depths and relates the combinatorics of non-semisimple representations to two-dimensional field-theoretic structures, with potential implications for AdS/CFT, matrix invariants, and quantum information tasks that exploit Brauer symmetry.

Abstract

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.

Simple harmonic oscillators from non-semisimple walled Brauer algebras

TL;DR

This work develops a systematic framework for the non-semisimple regime of walled Brauer algebras by introducing restricted Bratteli diagrams (RBD) that partition irreps into modified and unmodified dimensions when with small . A key finding is the -stability: for , the RBD form depends only on , enabling universal counting rules. The modified dimensions are computed explicitly for through detailed RBD analysis and connections to partial transposition kernels, revealing a deep link to a universal partition function of an infinite tower of simple harmonic oscillators. This oscillator perspective unifies red/green node counting across depths and relates the combinatorics of non-semisimple representations to two-dimensional field-theoretic structures, with potential implications for AdS/CFT, matrix invariants, and quantum information tasks that exploit Brauer symmetry.

Abstract

Walled Brauer algebras illuminate the combinatorics of mixed tensor representations of , with copies of the fundamental and 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 regime, specifically 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 regime has a stability property whereby these formulae are independent of . In this paper we initiate a systematic study of the combinatorics in the non-semisimple regime of , with positive . We introduce restricted Bratteli diagrams (RBD) which are useful as an instrument to process known data from the large regime to calculate representation theory data in the non-semisimple regime. We identify within the non-semisimple regime, a region of -stability, where and the RBD take a stable form depending on only and not the choice of 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.

Paper Structure

This paper contains 35 sections, 4 theorems, 211 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Key concepts and terminology
  3. Technical background and motivation
  4. Bratteli diagrams
  5. Relating Kernels : symmetric group algebras and Brauer algebras to tensor space
  6. The restricted Bratteli diagrams : definition and properties
  7. Modified dimensions for $B_N(m,n)$ with $N=m+n-l$ and small values of $l$
  8. Modified dimensions for $l=2$
  9. Modified dimensions for $l=3$
  10. Case of $B_N(2,n)$ and $N=n-1$
  11. Case of $B_N(3,n)$ and $N=n$
  12. Case of $B_N(m,n)$ and $N=m+n-3$
  13. Modified dimensions for $l=4$
  14. Case of $B_N(2,n)$ and $N=n-2$
  15. Case of $B_N(3,n)$ and $N=n-1$
  16. ...and 20 more sections

Key Result

Lemma 1

For the maps $\rho_{N,m+n},\rho_{N,m,n}$ from eq:mapRhom+n0, eq:mapRhoBmn respectively, the following equality holds:

Figures (15)

  • Figure 1: Example of graphical composition of two diagrams $b_1,b_2 \in B_{N}(4,4)$. Identifying a closed loop (in red) results in multiplying the diagram by a scalar $N \in \mathbb{C}$. We see that the composition $b_1 \circ b_2$ remains within $B_{N}(4,4)$.
  • Figure 2: The diagram represent relation between diagrammatic representation of a cycle $\sigma=(26543)$ partially transposed with respect to systems $4,5,6$ and an element of the Walled Brauer algebra $B_N(3,3)$.
  • Figure 3: The graphic presents the coloured Bratteli diagram (CBD) for $m=3,n=2$ and $N=2$. The nodes in red are associated with Brauer representation triples which do not obey the finite $N$ constraint \ref{['finiteNconst']}, while green nodes do obey the constraint.
  • Figure 4: The graphic presents the restricted Bratteli diagram (RBD) for $m=n=6$ and $N=9$. The red nodes are associated with Brauer triples which do not obey the finite $N$ constraint and admit paths to green nodes in the final layer. The green nodes are associated with triples labelling irreps with modified dimensions.
  • Figure 5: Commutativity diagram for algebra elements $\sigma \in \mathbb{C}[S_{m+n}]$ under action of maps $\rho_{N,m+n},P^t_{m,n}$, and $\rho_{N,m,n}$ given through \ref{['eq:mapRhom+n0']}, \ref{['eq:mapPt']}, and \ref{['eq:mapRhoBmn']} respectively.
  • ...and 10 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Example 2
  • Example 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Proposition 8
  • Proposition 9