Oscillators from non-semisimple walled Brauer algebras

TL;DR

This paper studies the non-semisimple regime of walled Brauer algebras, introducing restricted Bratteli diagrams to compute dimension corrections and revealing a connection to harmonic oscillator partition functions.

Contribution

It introduces restricted Bratteli diagrams to systematically analyze non-semisimple walled Brauer algebras and uncovers a harmonic oscillator structure in their dimension counting.

Findings

Restricted diagrams exhibit stability properties in certain regimes.

Dimension corrections are computed via paths in the diagrams.

Partition functions of harmonic oscillators govern the generating functions.

Abstract

The walled Brauer algebras $B_N(m,n)$ govern Schur--Weyl duality for unitary groups $U(N)$ acting on mixed tensor spaces $V_N^{\otimes m}\otimes \overline{V}_N^{\otimes n}$ and play an important role in applications ranging from AdS/CFT to quantum information theory. In the stable regime $N\ge m+n$ the algebra is semisimple and its representation theory is well understood. For $N<m+n$, however, $B_N(m,n)$ becomes non-semisimple. The representation of the algebra on tensor space has a non-trivial kernel and the corresponding quotient algebra is semisimple, with representation dimensions differing from those in the stable regime. We introduce \emph{restricted Bratteli diagrams}, obtained by modifying the standard Bratteli diagrams for $B_N(m,n)$. This construction provides a systematic way to use representation-theoretic data from the stable regime to compute the dimension modifications arising in the non-semisimple regime. In the regime $N=m+n-l$, with $l$ small compared to $m,n$, we show that the restricted diagrams exhibit a stability property and enable an efficient counting of the paths responsible for these dimension corrections. Remarkably, the resulting generating functions are governed by the partition function of an infinite tower of simple harmonic oscillators. We briefly discuss implications for the construction of orthogonal bases of matrix invariants in gauge theory and related applications in quantum information theory.