Simple harmonic oscillators from non-semisimple walled Brauer algebras

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.