Explicit combinatorics of the non-semisimple walled Brauer algebra regime $N=m+n-l$: modified multiplicities, kernel, matrix units, and the $S_m\times S_n$-commuting subalgebra

The walled Brauer algebra $B_N(m,n)$ governs Schur-Weyl duality for the unitary group $U(N)$ acting on mixed tensor space $V_N^{\otimes m}\otimes\overline{V}_N^{\otimes n}$ ($m$ fundamental and $n$ anti-fundamental factors). For $N\ge m+n$ the algebra is semisimple, and its representation combinatorics — irreducible representation dimensions and multiplicities given by path counting on Bratteli diagrams — is stable and completely known. For $N=m+n-l<m+n$ with $l>0$ the algebra becomes non-semisimple: the tensor-space representation acquires a non-trivial kernel, and the multiplicities drop to modified values $\widehat{d}_{m,n,N}(\gamma)$ for which no general formula is known. Ramgoolam and Studziński (arXiv:2509.04234, 2025) introduced restricted Bratteli diagrams, computed $\widehat{d}_{m,n,N}(\gamma)$ for $1\le l\le 4$, proved an $(m,n)$-stability region $m,n\ge 2l-3$ in which the diagram combinatorics depends only on $l$, and showed that node counting is governed by a universal oscillator partition function. The open problems, posed explicitly in that paper's Discussion and Outlook, concern completing the explicit combinatorial description of the non-semisimple regime: (1) a general closed-form formula or efficient combinatorial/algorithmic rule for $\widehat{d}_{m,n,N}(\gamma)$ for arbitrary $(m,n,N)$; (2) an explicit construction of the kernel $I_N(m,n)=\ker\rho_{N,m,n}$ and of matrix units $Q^{\gamma}_{IJ}$ for the semisimple quotient $\widehat{B}_N(m,n)\cong\mathcal{A}^N_{m,n}$; and (3) the extension of the restricted-Bratteli-diagram analysis to the subalgebra of $B_N(m,n)$ commuting with $S_m\times S_n$, which controls orthogonal bases of multi-matrix invariants at finite $N$. These three directions are open as posed in the cited Ramgoolam–Studziński paper.

Background

Classical Schur-Weyl duality links the action of the unitary group $U(N)$ on tensor powers of its fundamental representation $V_N$ to the symmetric group algebra. For mixed tensor space $V_N^{\otimes m}\otimes\overline{V}_N^{\otimes n}$, built from $m$ copies of the fundamental and $n$ copies of the dual (anti-fundamental) representation, the corresponding duality partner is the walled Brauer algebra $B_N(m,n)$: the complex associative algebra spanned by walled Brauer diagrams — perfect matchings of $2(m+n)$ vertices, with $m$ vertices to the left and $n$ to the right of a vertical wall on each of the top and bottom rows, where horizontal edges connect vertices on the same side of the wall and contraction edges cross it — multiplied by stacking, with the parameter $N$ counting closed loops. This algebra and its Schur-Weyl duality with mixed tensor representations of the general linear group were introduced in the study of mixed tensor products (Benkart, Chakrabarti, Halverson, Leduc, Lee and Stroomer, 1994). The representation $\rho_{N,m,n}:B_N(m,n)\to\mathrm{End}(V_N^{\otimes m}\otimes\overline{V}_N^{\otimes n})$ has as its image the commutant of the $U(N)$ action (the algebra of all operators on this space that commute with every element of $U(N)$), namely the algebra $\mathcal{A}^N_{m,n}$ of partially transposed permutation operators (permutation operators whose matrix is transposed on the last $n$ tensor factors), which is why these algebras also appear in quantum information theory.

In the stable (large-$N$) regime $N\ge m+n$, the algebra $B_N(m,n)$ is semisimple (the semisimplicity theory of Hecke and walled Brauer algebras is due to Andersen, Stroppel and Tubenhauer, 2017). Its irreducible representations are labelled by Brauer triples $\gamma=(k,\gamma_{+},\gamma_{-})$: a number $k\le\min(m,n)$ of contraction pairs, a Young diagram (a left-justified array of boxes encoding an integer partition) $\gamma_{+}$ with $m-k$ boxes and a Young diagram $\gamma_{-}$ with $n-k$ boxes. Their dimensions $d_{m,n}(\gamma)$ are computed by counting paths on the Bratteli diagram, the layered graph whose vertices at level $L$ are the Brauer triples for the first $L$ tensor factors and whose edges are the allowed box-addition/removal moves. The mixed tensor space decomposes as a direct sum $\bigoplus_{\gamma}V^{U(N)}_{\gamma}\otimes V^{B}_{\gamma}$ of irreducible $U(N)$ modules and irreducible $B_N(m,n)$ modules, with the multiplicity of each $U(N)$ irreducible equal to $d_{m,n}(\gamma)$; all of these data are independent of $N$ in the stable regime (rank-stability).

For $N<m+n$, write $N=m+n-l$ with $l\ge 1$. Here $B_N(m,n)$ is non-semisimple: the map $\rho_{N,m,n}$ has a non-trivial kernel $I_N(m,n)$, and the quotient $\widehat{B}_N(m,n)=B_N(m,n)/I_N(m,n)$ is a semisimple algebra isomorphic to the commutant $\mathcal{A}^N_{m,n}$. The irreducibles of the quotient are labelled by the admissible Brauer triples, those with $c_1(\gamma_{+})+c_1(\gamma_{-})\le N$, where $c_1$ of a Young diagram is the length of its first column (its height). The tensor-space multiplicities become $\widehat{d}_{m,n,N}(\gamma)=d_{m,n}(\gamma)-\delta_{m,n,N}(\gamma)$: unchanged for some (unmodified) triples and strictly smaller for the others. The abstract structure of non-semisimple walled Brauer algebras — their blocks (indecomposable representation sectors) and cell modules (the standard modules of a cellular algebra, a combinatorially presented algebra structure) — has been determined (Cox, De Visscher, Doty and Martin, 2008), and constructive bases for the image algebra $\mathcal{A}^N_{m,n}$ in the style of Gelfand-Tsetlin bases (bases built by stepwise restriction of representations along a nested chain of subgroups, making matrix coefficients explicit) or group-adapted matrix units (families of elements $Q^{\gamma}_{IJ}$ satisfying $Q^{\gamma}_{IJ}Q^{\gamma'}_{KL}=\delta_{\gamma\gamma'}\delta_{JK}Q^{\gamma}_{IL}$, from which the algebra's structure is read off) exist in related regimes (Grinko, Burchardt and Ozols, 2023; Studziński, Młynik, Mozrzymas, Horodecki and Grinko, 2025). However, no general formula for $\widehat{d}_{m,n,N}$ and no explicit, uniform construction of the kernel or of matrix units tied to the Brauer-diagram presentation in the non-semisimple regime were available.

Ramgoolam and Studziński (2025) initiated the systematic study of this regime. They introduced restricted Bratteli diagrams (RBD): the stable-regime Bratteli diagram restricted to the nodes admissible at a given $(m,n,N)$, with nodes colored green (depth $0$, corresponding to irreducibles with modified multiplicities) and red (depths $1\le d\le l-1$). They computed $\widehat{d}_{m,n,N}(\gamma)$ for general $(m,n)$ with $1\le l\le 4$; they proved $(m,n)$-stability, namely that for $m,n\ge 2l-3$ the RBD — and hence the pattern of which triples have modified dimensions — depends only on $l$; and they showed that the counting of red and green nodes by depth is governed by

$\mathcal{Z}_{\mathrm{univ}}(x)=\frac{x}{(1-x)(1-x^{2})}\prod_{i=1}^{\infty}\frac{1}{(1-x^{i})^{2}},$

the partition function of an infinite tower of simple harmonic oscillators (each factor $(1-x^{i})^{-1}$ is one bosonic mode of energy $i$, and the squaring gives two copies), which is close to but distinct from the partition function of a two-dimensional non-chiral free scalar field; its coefficient sequence is OEIS (Online Encyclopedia of Integer Sequences) A000714. The constructions are motivated from two sides: in gauge-string duality (the AdS/CFT correspondence, which relates certain gauge theories to string theories on curved backgrounds), orthogonal bases of invariants of two complex matrices $Z,\overline{Z}$ (describing brane-anti-brane systems, i.e. pairs of oppositely oriented D-branes, the extended objects of string theory) require matrix units for the subalgebra of $B_N(m,n)$ commuting with the product group $S_m\times S_n$ of permutations acting separately on the $m$ and $n$ tensor factors; in quantum information, many systems ($m+n$ large) with small local dimension $N$ (e.g. qubits, $N=2$) land deep in the non-semisimple regime, and knowledge of the kernel and of the irreducible decomposition is the route to symmetry reductions of semidefinite programs (convex optimisation problems over positive-semidefinite matrix variables) and to port-based teleportation analyses (a teleportation variant that avoids corrective unitaries by using many entangled ports).

What remains missing is the transition from these finite-$l$ computations and structural theorems to a complete description: the $l\le 4$ results do not assemble into a general rule for $\widehat{d}_{m,n,N}$; the kernel is characterized only abstractly; and the $S_m\times S_n$-commuting subalgebra has not been analysed in the non-semisimple regime at all.

Problem Statement

Let $B_N(m,n)$ be the walled Brauer algebra acting on mixed tensor space $V_N^{\otimes m}\otimes\overline{V}_N^{\otimes n}$ through the representation $\rho_{N,m,n}$, and consider the genuinely non-semisimple regime $N=m+n-l$ with $l\ge 1$. As posed in the Discussion and Outlook of Ramgoolam and Studziński (arXiv:2509.04234), the problem is to complete the explicit combinatorial description of this regime. The source poses the following three directions as separate open problems, each of interest in its own right:

1. (General multiplicity rule) A general closed-form formula, or an efficient combinatorial or algorithmic rule, for the modified multiplicities $\widehat{d}_{m,n,N}(\gamma)$ — the multiplicities of the irreducible representations $V_{\gamma}$ of the semisimple quotient $\widehat{B}_N(m,n)\cong\mathcal{A}^N_{m,n}$, equivalently of the corresponding $U(N)$ irreducibles in mixed tensor space — for arbitrary $(m,n,N)$ with $N<m+n$. This must in particular go beyond the range $l\le 4$ treated in the source, and is sought for both the $(m,n)$-stable region $m,n\ge 2l-3$, where the restricted Bratteli diagram depends only on $l$, and the small-$(m,n)$ cases outside the stable region.

2. (Explicit kernel and matrix units) An explicit algebraic construction, in the walled Brauer diagram basis or an equivalently concrete presentation, of the kernel ideal $I_N(m,n)=\ker\rho_{N,m,n}$ — as posed by the source, for $N=m+n-l$ with $l$ small and Brauer triples $\gamma$ with $c_1(\gamma_{+})+c_1(\gamma_{-})$ close to $m+n$ — and of matrix units $Q^{\gamma}_{IJ}$ for the semisimple quotient $\widehat{B}_N(m,n)$, which the source poses without that restriction, so that the relation between $B_N(m,n)$ and its semisimple quotient becomes constructive.

3. ($S_m\times S_n$-commuting subalgebra) The extension of the restricted-Bratteli-diagram combinatorics to the subalgebra of $B_N(m,n)$ commuting with the subgroup $S_m\times S_n$ of permutation symmetries acting separately on the $m$ fundamental and $n$ anti-fundamental tensor factors, in the non-semisimple regime $N<m+n$, including the resulting modified-dimension formulae. As in the source, orthogonal bases of multi-matrix ($Z,\overline{Z}$-type) invariants at finite $N$, built from matrix units for this subalgebra, are the motivating application of this direction rather than a required deliverable.

Each of these three directions is separately open in the source, and each contributes to the broader objective of turning the restricted-Bratteli-diagram framework from a finite-$l$ computational device into a complete, explicit, uniform description of the non-semisimple regime of the walled Brauer algebra and of its tensor-space representation; the source does not present them as a single conjunctive completion requirement. The source's more speculative outlook directions — a physical interpretation of the oscillator partition function $\mathcal{Z}_{\mathrm{univ}}$ in 2d Yang-Mills/string-theory or brane-anti-brane terms, and quantum-information applications such as symmetry-adapted circuit design or semidefinite-program reductions — are motivations for this objective, not parts of its completion boundary.

References

1. S. Ramgoolam, M. Studziński, Simple harmonic oscillators from non-semisimple walled Brauer algebras, arXiv:2509.04234 (2025), DOI 10.48550/arXiv.2509.04234, https://arxiv.org/abs/2509.04234
2. S. Ramgoolam, M. Studziński, Oscillators from non-semisimple walled Brauer algebras, arXiv:2604.27087 (2026), DOI 10.48550/arXiv.2604.27087, https://arxiv.org/abs/2604.27087
3. M. Horodecki, M. Studziński, M. Mozrzymas, Iterative construction of $\mathfrak{S}_p\times\mathfrak{S}_p$ group-adapted irreducible matrix units for the walled Brauer algebra, arXiv:2509.17698 (2025), DOI 10.48550/arXiv.2509.17698, https://arxiv.org/abs/2509.17698
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