Open problems on the traceless projection of mixed tensor products: spectral-set equality, the kernel of the walled Brauer centralizer map, and trace-level idempotents

A 2025 paper of Y. Goncharov constructs, in closed factorised form, the projector onto the traceless subspace of a mixed tensor product $V^{\otimes m}\otimes(V^*)^{\otimes n}$ of a finite-dimensional complex vector space and its dual, and identifies its analogue in the walled Brauer algebra in the semisimple regime. The paper explicitly leaves four linked questions open: (i) whether the true spectrum of the trace operator $\mathcal{A}_{m,n}$ coincides, for all $m,n,N\geq 1$, with the combinatorial superset $\widetilde{\mathrm{spec}}(\mathcal{A}_{m,n})$ built from Littlewood–Richardson coefficients; (ii) an explicit description of the kernel of the natural surjection from the walled Brauer algebra onto the centralizer algebra in the non-injective regime $N<m+n$; (iii) whether other choices of a single operator, besides $\mathcal{A}_{m,n}$, are possible whose kernel equals the traceless subspace (giving alternative factorised forms of the traceless projector); and (iv) the construction of a complete family of central idempotents splitting mixed tensor spaces into successive trace levels (traceless, doubly traceless, etc.).

Background

Let $V$ be an $N$-dimensional complex vector space with dual $V^*$, and let $V^{m,n}=V^{\otimes m}\otimes(V^*)^{\otimes n}$ be the space of mixed tensors with $m$ contravariant and $n$ covariant indices. The diagonal action of the general linear group $GL(N)$ on $V^{m,n}$ has a centralizer algebra $C_{m,n}(N)=\mathrm{End}_{GL(N)}(V^{m,n})$, and Schur–Weyl duality relates the two actions: the diagonal $GL(N)$-action and the image of the walled Brauer algebra generate each other's centralizers (a correspondence often called the see-saw between $C_{m,n}(N)$ and $B_{m,n}(\delta)$), so the representation theories of the two sides determine one another. In this mixed (covariant plus contravariant) setting the combinatorial partner of $C_{m,n}(N)$ is the walled Brauer algebra $B_{m,n}(\delta)$: the diagram algebra spanned by diagrams with a vertical wall separating the $m$ leftmost and $n$ rightmost nodes, in which every horizontal edge (arc) must cross the wall and each closed loop evaluates to the parameter $\delta$. Benkart, Chakrabarti, Halverson, Leduc, Lee and Stroomer (1994) showed that the natural action of $B_{m,n}(N)$ on $V^{m,n}$ induces a surjection onto the centralizer, $\mathfrak b:B_{m,n}(N)\to C_{m,n}(N)$, which is an isomorphism when $N\geq m+n$. Cox, De Visscher, Doty and Martin (2008) determined the semisimplicity criterion and block structure of $B_{m,n}(\delta)$: it is semisimple for all but finitely many $\delta$, and at the integer parameters $\delta=N$ it is non-semisimple exactly when $N\leq m+n-1$, the regime where $\mathfrak b$ acquires a non-trivial kernel.

For $1\leq a\leq m$, $1\leq b\leq n$ the contraction (trace) map $\mathrm{tr}_{ab'}:V^{m,n}\to V^{m-1,n-1}$ takes the $a$-th contravariant and $b$-th covariant slots and sums $\delta^i_j$ over them; its adjoint insertion map is $\mathrm{tr}^+_{ab'}$. The traceless subspace $V^{m,n}_0$ is the common kernel of all $mn$ trace maps, and $\mathcal{A}_{m,n}=\sum_{a,b'}\mathrm{tr}^+_{ab'}\mathrm{tr}_{ab'}$ is a diagonalisable endomorphism with $\ker\mathcal{A}_{m,n}=V^{m,n}_0$ and non-negative integer spectrum. Hence the $GL(N)$-equivariant traceless projector $P_{m,n}$ (the unique one annihilating the span $V^{m,n}_1$ of all insertions) is the central idempotent

$P_{m,n}=\prod_{a\in\mathrm{spec}(\mathcal{A}_{m,n})\setminus\{0\}}\Bigl(1-\tfrac{1}{a}\mathcal{A}_{m,n}\Bigr),$

and the same product over any finite superset of the non-zero spectrum gives the same projector. Goncharov (2025) determines $\mathrm{spec}(\mathcal{A}_{m,n})$ completely for all $m,n,N\geq 1$ via a Littlewood–Richardson rule (the Littlewood–Richardson coefficient $c^{\lambda}_{\alpha\beta}$ is the multiplicity of the $GL$-irreducible labelled by the Young diagram $\lambda$ in the tensor product of those labelled by $\alpha$ and $\beta$; the content of a Young diagram $\alpha$ is $c(\alpha)=\sum_{(i,j)\in\alpha}(j-i)$, summed over boxes by their column $j$ and row $i$, and $c(\alpha/\beta)=c(\alpha)-c(\beta)$), the eigenvalues taking the form $Nr+c(\rho/\mu)+c(\sigma/\nu)$, and proves that $\mathfrak b$ is injective if and only if $N\geq m+n$, closing a gap left by Benkart et al. He also introduces the combinatorial superset (eq. (2.71)–(2.72) of the source)

$\widetilde{\mathrm{spec}}(\mathcal{A}_{m,n})=\bigl\{Nr+c(\rho/\mu)+c(\sigma/\nu)\bigr\}\cap\mathbb{N}_0,$

where $r$ ranges over $\{0,1,\ldots,\min(m,n)\}$; $(\mu,\nu)$ over pairs of partitions with $m-|\mu|=n-|\nu|=r$ and $\ell(\mu)+\ell(\nu)\leq N$, $\ell$ denoting the number of parts (these pairs index the cell modules of $B_{m,n}(\delta)$, i.e. its standard-compatible indecomposable modules in the non-semisimple case); and $(\rho,\sigma)$ over pairs of partitions of $m$ and of $n$, each with at most $N$ parts (indexing the irreducible $\mathbb{C}[S_m\times S_n]$-modules); subject to $c^{\rho}_{\mu\beta}c^{\sigma}_{\nu\beta}\neq 0$ for some partition $\beta\subseteq\rho\cap\sigma$ with at most $N$ parts. He shows $\mathrm{spec}(\mathcal{A}_{m,n})\subseteq\widetilde{\mathrm{spec}}(\mathcal{A}_{m,n})$ with equality whenever a certain Littlewood–Richardson multiplicity estimate saturates — in particular for $N\geq m+n-1$, for $m=1$ or $n=1$ with small $N$, and for $N=1$. For ordinary (metric) tensor spaces, the analogous structural questions have been answered: Gavarini (1999) described the kernel of the Brauer-algebra action on tensor space in the orthogonal case, and Helpin's thesis (2023) constructed the trace-level (traceless, doubly traceless, etc.) projection idempotents via the Brauer algebra. The mixed $GL(N)$ case is the subject of the present questions.

Problem Statement

Resolve the four questions left open by Goncharov (arXiv:2511.00303) about the trace calculus of mixed tensor products $V^{\otimes m}\otimes(V^*)^{\otimes n}$ of $GL(N)$ and their walled Brauer centralizers; they share one objective — completing the traceless-projection framework, and the see-saw between $C_{m,n}(N)$ and $B_{m,n}(N)$, uniformly in all integers $m,n,N\geq 1$, including the non-semisimple regime $N<m+n$:

(i) Prove that $\mathrm{spec}(\mathcal{A}_{m,n})=\widetilde{\mathrm{spec}}(\mathcal{A}_{m,n})$ for all $m,n,N\geq 1$, or determine the exact necessary and sufficient condition on $(m,n,N)$ for this equality — equivalently in the open regime $m,n\geq 2$, $2\leq N\leq m+n-2$, either produce a triple $(m,n,N)$ with $\widetilde{\mathrm{spec}}(\mathcal{A}_{m,n})\not\subseteq\mathrm{spec}(\mathcal{A}_{m,n})$ or prove none exists. Here $\widetilde{\mathrm{spec}}(\mathcal{A}_{m,n})$ is the combinatorial superset defined in the Background above (eq. (2.71)–(2.72) of the source paper).

(ii) Describe explicitly the kernel of the surjective algebra homomorphism $\mathfrak b:B_{m,n}(N)\to C_{m,n}(N)$ induced by the action on mixed tensor space, in the non-injective regime $N<m+n$.

(iii) Determine whether, besides $\mathcal{A}_{m,n}$ itself and the trivial constructions obtained as scalar multiples or polynomials in $\mathcal{A}_{m,n}$, other choices of a single operator on $V^{m,n}$ are possible whose kernel is exactly the traceless subspace $V^{m,n}_0$, giving alternative factorised forms of the traceless projector, and analyse such alternative choices.

(iv) Construct the complete family of central idempotents in $C_{m,n}(N)$ that decomposes mixed tensor space into the successive trace levels — the traceless, doubly traceless (tensors all of whose contractions are traceless), etc. subspaces, i.e. the levels of the filtration by iterated trace maps, together with the trace part — and characterise them algebraically, extending to the mixed $GL(N)$ case what Helpin's thesis achieved in the metric case.

The four open questions (i)–(iv) are posed jointly; each can also be attacked and reported on independently.

References

1. Yegor Goncharov, "Traceless projection of mixed tensor products, and walled Brauer algebras", arXiv:2511.00303 (2025). DOI: 10.48550/arXiv.2511.00303, https://arxiv.org/abs/2511.00303 (source of the open questions)
2. D. V. Bulgakova, Y. O. Goncharov, T. Helpin, "Construction of the traceless projection of tensors via the Brauer algebra", arXiv:2212.14496 (2022). DOI: 10.48550/arXiv.2212.14496, https://arxiv.org/abs/2212.14496
3. G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Y. Lee, J. Stroomer, "Tensor Product Representations of General Linear Groups and Their Connections with Brauer Algebras", Journal of Algebra 166 (1994) 529-567. DOI: 10.1006/jabr.1994.1166, https://doi.org/10.1006/jabr.1994.1166
4. A. Cox, M. De Visscher, S. Doty, P. Martin, "On the blocks of the walled Brauer algebra", Journal of Algebra 320 (2008) 169-212. DOI: 10.1016/j.jalgebra.2008.01.026, https://doi.org/10.1016/j.jalgebra.2008.01.026
5. F. Gavarini, "A Brauer Algebra Theoretic Proof of Littlewood's Restriction Rules", Journal of Algebra 212 (1999) 240-271. DOI: 10.1006/jabr.1998.7536, https://doi.org/10.1006/jabr.1998.7536
6. Thomas Helpin, "Irreducible decompositions of tensors via the Brauer algebra and applications to metric-affine gravity", PhD thesis, IDP Tours (2023). arXiv:2407.18019, DOI: 10.48550/arXiv.2407.18019, https://arxiv.org/abs/2407.18019
7. Sanjaye Ramgoolam, Michal Studzinski, "Oscillators from non-semisimple walled Brauer algebras", arXiv:2604.27087 (2026). DOI: 10.48550/arXiv.2604.27087, https://arxiv.org/abs/2604.27087
8. Sanjaye Ramgoolam, Michal Studzinski, "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
9. Marcin Markiewicz, Łukasz Pawela, Zbigniew Puchała, "Choi-level twirling of quantum channels: finite constructions and non-compact transformations", Journal of Physics A: Mathematical and Theoretical 59 (2026) 335305. DOI: 10.1088/1751-8121/ae965b, https://doi.org/10.1088/1751-8121/ae965b (preprint arXiv:2512.23586, https://arxiv.org/abs/2512.23586)