Models of quantum permutations

Moritz Weber; Stefan Jung

arxiv: 1906.10409 · v1 · pith:Q7ACXYZAnew · submitted 2019-06-25 · 🧮 math.OA · math.QA

Models of quantum permutations

Stefan Jung , Moritz Weber This is my paper

Pith reviewed 2026-05-25 16:15 UTC · model grok-4.3

classification 🧮 math.OA math.QA

keywords quantum permutation groupcompact matrix quantum groupC*-algebrainverse limitoperator algebraquantum symmetric groupquantum group

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The pith

Homomorphisms from C(S_N^+) to a sequence of C*-algebras B_n produce an inverse limit that is a compact matrix quantum group strictly between S_N and S_N^+.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a family of *-homomorphisms from the C*-algebra of the quantum permutation group S_N^+ into auxiliary C*-algebras B_n for each N at least 4. These maps are not required to arise from quantum group representations, yet they still encode quantum permutation matrices in a concrete operator-algebraic manner. Passing to the inverse limit of the B_n yields a new compact matrix quantum group G satisfying S_N properly contained in G contained in S_N^+. Previous results already show that G equals S_N^+ when N equals 4 or 5; the construction leaves the equality undecided for every larger N.

Core claim

The authors exhibit *-homomorphisms φ_n from C(S_N^+) into C*-algebras B_n whose inverse limit B_∞ carries the structure of a compact matrix quantum group G obeying the strict chain of inclusions S_N ⊂neq G ⊆ S_N^+. Equality G = S_N^+ is already known for N=4 and N=5, while the case N≥6 remains open.

What carries the argument

The sequence of *-homomorphisms φ_n : C(S_N^+) → B_n together with the inverse limit B_∞ that supplies the intermediate quantum group G.

If this is right

The maps φ_n supply concrete operator-algebraic models of quantum permutation matrices without being full representations of S_N^+.
The inverse-limit construction produces at least one compact matrix quantum group G lying properly between S_N and S_N^+ for every N≥4.
When N=4 or 5 the new group G coincides with the full quantum symmetric group S_N^+.
For N≥6 the position of G inside S_N^+ remains undecided by the present construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If G turns out to be strictly smaller than S_N^+ for some N≥6, the construction would exhibit the first explicit proper intermediate quantum permutation group.
The same inverse-limit technique could be tested on other quantum groups whose universal C*-algebras admit similar sequences of homomorphisms.

Load-bearing premise

The inverse limit of the algebras B_n actually forms a compact matrix quantum group that sits strictly between the classical and full quantum symmetric groups.

What would settle it

An explicit calculation of a quantum-group invariant (such as the dimension of the space of fixed points under the coaction) for the inverse-limit algebra when N=6 that equals the corresponding invariant of either S_6 or S_6^+ would show whether the intermediate group collapses to one endpoint.

read the original abstract

For $N\ge 4$ we present a series of *-homomorphisms $\varphi_n:C(S_N^+)\rightarrow B_n$ where $S_N^+$ is the quantum permutation group. They are not necessarily representations of the quantum group $S_N^+$ but they yield good operator algebraic models of quantum permutation matrices. The C*-algebras $B_n$ allow the construction of an inverse limit $B_{\infty}$ which defines a compact matrix quantum group $S_N\subsetneq G\subseteq S_N^+$. We know $G=S_N^+$ for $N=4,5$ from Banica's work, but we have to leave open the case $N\ge 6$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit series of homomorphisms from C(S_N^+) to auxiliary algebras B_n whose inverse limit produces an intermediate compact matrix quantum group G with S_N properly inside G inside S_N^+, known to equal S_N^+ only for N=4,5.

read the letter

The core contribution is a concrete construction of *-homomorphisms φ_n : C(S_N^+) → B_n for N ≥ 4 that are not required to be quantum group representations yet still produce operator-algebraic models of quantum permutation matrices. The inverse limit B_∞ then defines the intermediate object G satisfying the stated inclusions, with the equality cases for small N recovered from Banica's earlier results. This is a direct, forward construction that does not reduce to prior published objects or introduce free parameters. The paper states the openness for N ≥ 6 plainly rather than claiming a full resolution. That honesty is useful. The main limitation is that the abstract supplies no explicit formulas for the B_n or the verification steps that the limit satisfies the compact matrix quantum group axioms, so the claim that the models are “good” cannot be checked from the given text alone. If the full manuscript contains the derivations and shows the homomorphisms preserve the necessary relations without hidden choices, the construction would be a solid incremental step. The work sits squarely in the compact quantum groups literature and would interest readers already working on quantum permutations or intermediate subalgebras. It is narrow enough that most outsiders will not need it, but the explicit maps and the inverse-limit step are the sort of thing that can be picked up and used in follow-up calculations. I would send it to a referee who knows the Banica framework; the construction is concrete enough to deserve that check even if the N ≥ 6 case stays open.

Referee Report

0 major / 2 minor

Summary. For N≥4 the manuscript constructs a series of *-homomorphisms φ_n : C(S_N^+) → B_n that serve as operator-algebraic models of quantum permutation matrices (though not necessarily representations of the quantum group). The inverse limit B_∞ is used to define a compact matrix quantum group G satisfying S_N ⊂neq G ⊆ S_N^+, with the equality G = S_N^+ verified for N=4,5 via Banica's prior results and left open for N≥6.

Significance. If the homomorphisms and inverse-limit construction hold, the work supplies explicit C*-algebraic models that interpolate between the classical symmetric group and the free quantum permutation group S_N^+. This contributes concrete examples to the theory of compact matrix quantum groups in operator algebras, with credit due to the forward construction from C(S_N^+) and the explicit acknowledgment that the N≥6 case remains unresolved.

minor comments (2)

[Abstract / §1] The abstract and introduction refer to the algebras B_n and the maps φ_n without supplying their explicit definitions or the precise relations they satisfy; these should be stated in §2 or §3 before the inverse-limit argument is developed.
[§4] Notation for the inverse limit B_∞ and the resulting quantum group G should be introduced with a clear statement of the universal property or the matrix of generators used to verify the compact matrix quantum group axioms.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. The provided summary accurately describes the manuscript's contributions and the status of the open question for N≥6.

Circularity Check

0 steps flagged

No significant circularity; forward construction from C(S_N^+) with external benchmarks

full rationale

The paper constructs *-homomorphisms φ_n : C(S_N^+) → B_n explicitly from the known quantum permutation algebra, forms their inverse limit B_∞ to obtain an intermediate quantum group G, and compares it to S_N^+ using Banica's independent prior results for N=4,5 while leaving N≥6 open. No step reduces a claimed output to a fitted parameter, self-definition, or self-citation chain; the maps are presented as non-representations by design, and the inclusion relations are derived from the construction rather than presupposed. The derivation is self-contained against external operator-algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities beyond the standard C*-algebraic framework of compact matrix quantum groups; no additional ledger entries can be extracted.

pith-pipeline@v0.9.0 · 5630 in / 1104 out tokens · 33399 ms · 2026-05-25T16:15:00.167520+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 2 internal anchors

[1]

T. Banica. Homogeneous quantum groups and their easiness level . arxiv:1806.06368 , 2018. to appear in Kyoto J. Math

work page arXiv 2018
[2]

Banica and J

T. Banica and J. Bichon. Hopf images and inner faithful representations . Glasg. math. J. , 52:677--703, 2010

work page 2010
[3]

Banica and I

T. Banica and I. Nechita. Flat matrix models for quantum permutation groups . Adv. Appl. Math. , 83:24--46, 2017

work page 2017
[4]

Banica and R

T. Banica and R. Speicher. Liberation of orthogonal Lie groups . Adv. Math. , 222(4):1461--1501, 2009

work page 2009
[5]

Partition quantum spaces

S. Jung and M. Weber. Partition quantum spaces . arxiv:1801.06376 , 2018. to appear in J. Noncommut. Geom

work page internal anchor Pith review Pith/arXiv arXiv 2018
[6]

Nonlocal Games and Quantum Permutation Groups

M. Lupini, L. Man c inska, and D. E. Roberson. Non local games and quantum permutation grops. arXiv:1712.01820v2 , 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018
[7]

S. MacLane. Categories for the working mathematician , volume 5 of Graduate Texts in Mathematics . Springer, New York, 1971

work page 1971
[8]

N. C. Phillips. Inverse limits of $ C ^*$-algebras . J. Operator Theory , 19:159--195, May 1988

work page 1988
[9]

Timmermann

T. Timmermann. An invitation to quantum groups and duality . EMS Textbk. Math. European Mathematical Society, Z \"u rich, 2008

work page 2008
[10]

Tarrago and M

P. Tarrago and M. Weber. Unitary easy quantum groups: the free case and the group case . Int. Math. Res. Not. , 2017(18):5710--5750, 2017

work page 2017
[11]

Tarrago and M

P. Tarrago and M. Weber. The classification of tensor categories of two-coloured noncrossing partitions . J. Combin. Theory Ser. A , 154:464--506, 2018

work page 2018
[12]

Sh. Wang. Quantum symmetry groups of finite spaces. Comm. Math. Phys. , 195(1):195--211, 1998

work page 1998
[13]

S. L. Woronowicz. Compact matrix pseudogroups . Comm. Math. Phys. , 111(4):613--665, 1987

work page 1987
[14]

S. L. Woronowicz. Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups . Invent. Math. , 93(1):35--76, 1988

work page 1988

[1] [1]

T. Banica. Homogeneous quantum groups and their easiness level . arxiv:1806.06368 , 2018. to appear in Kyoto J. Math

work page arXiv 2018

[2] [2]

Banica and J

T. Banica and J. Bichon. Hopf images and inner faithful representations . Glasg. math. J. , 52:677--703, 2010

work page 2010

[3] [3]

Banica and I

T. Banica and I. Nechita. Flat matrix models for quantum permutation groups . Adv. Appl. Math. , 83:24--46, 2017

work page 2017

[4] [4]

Banica and R

T. Banica and R. Speicher. Liberation of orthogonal Lie groups . Adv. Math. , 222(4):1461--1501, 2009

work page 2009

[5] [5]

Partition quantum spaces

S. Jung and M. Weber. Partition quantum spaces . arxiv:1801.06376 , 2018. to appear in J. Noncommut. Geom

work page internal anchor Pith review Pith/arXiv arXiv 2018

[6] [6]

Nonlocal Games and Quantum Permutation Groups

M. Lupini, L. Man c inska, and D. E. Roberson. Non local games and quantum permutation grops. arXiv:1712.01820v2 , 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

[7] [7]

S. MacLane. Categories for the working mathematician , volume 5 of Graduate Texts in Mathematics . Springer, New York, 1971

work page 1971

[8] [8]

N. C. Phillips. Inverse limits of \( C ^*\)-algebras . J. Operator Theory , 19:159--195, May 1988

work page 1988

[9] [9]

Timmermann

T. Timmermann. An invitation to quantum groups and duality . EMS Textbk. Math. European Mathematical Society, Z \"u rich, 2008

work page 2008

[10] [10]

Tarrago and M

P. Tarrago and M. Weber. Unitary easy quantum groups: the free case and the group case . Int. Math. Res. Not. , 2017(18):5710--5750, 2017

work page 2017

[11] [11]

Tarrago and M

P. Tarrago and M. Weber. The classification of tensor categories of two-coloured noncrossing partitions . J. Combin. Theory Ser. A , 154:464--506, 2018

work page 2018

[12] [12]

Sh. Wang. Quantum symmetry groups of finite spaces. Comm. Math. Phys. , 195(1):195--211, 1998

work page 1998

[13] [13]

S. L. Woronowicz. Compact matrix pseudogroups . Comm. Math. Phys. , 111(4):613--665, 1987

work page 1987

[14] [14]

S. L. Woronowicz. Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups . Invent. Math. , 93(1):35--76, 1988

work page 1988