Homology of Rook-Brauer Algebras and Motzkin Algebras

Khoa Ta

REVIEW 2 major objections 4 minor 9 references

Homology of Rook-Brauer Algebras and Motzkin Algebras

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that when $\epsilon$ is invertible, the homology of the Rook-Brauer algebra is isomorphic to symmetric group homology in all degrees, and that Motzkin algebra homology vanishes in positive degrees.

desk verdict A real attempt at a sharper homology theorem for Rook-Brauer and Motzkin algebras, but Lemma 4.6 contains an explicit false idempotent claim that brings the main proof down. read the letter →

arxiv 2505.21977 v1 pith:3QCZNIKF submitted 2025-05-28 math.RA math.RT

Khoa Ta This is my paper

classification math.RAmath.RT MSC 16E3018G1520J06

keywords Rook-BraueralgebrasMotzkinhomologicalstabilityTorgroupsinductiveresolutiondiagramsymmetricgrouphomology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Rook-Brauer algebras and Motzkin algebras are diagram algebras built from partitions into blocks of size at most two, and their homology is interpreted here as Tor with the trivial module. This paper proves that whenever $\epsilon$ is invertible in the ground ring, with $\delta$ arbitrary, the inclusion of the symmetric group algebra into the Rook-Brauer algebra induces an isomorphism in homology in every degree. It also proves that the Motzkin algebras have vanishing positive-degree homology. Together these results establish homological stability for both families, with a sharp stable range for the Rook-Brauer algebras. The earlier proofs of these statements required both parameters to be invertible; this paper removes the condition on $\delta$.

What carries the argument

The central device is inductive resolution, a way of proving Tor vanishing by resolving a module by modules that already have vanishing positive Tor. The paper constructs such resolutions from left ideals and link-state modules: $A_x$ (diagrams in which a chosen node is isolated), $B_{X,x}$ (diagrams in which a chosen node connects to another node in $X$), $M_{\{a,b\}}$ (diagrams in which two nodes are connected), and, for the Motzkin algebras, $Y_P$ (modules built from a planar right-link state via splices and deletions). The load-bearing checks are the lemmas that identify the quotients $A_{X,x}$, $M_{X,\{a,b\}}$, and $Y_{P,\{x,y\}}$ as direct summands of simpler modules, using right multiplication by idempotents such as $\epsilon^{-1}T_x$, $\epsilon^{-1}T_aV_{ab}$, and $\epsilon^{-k_0}\gamma$.

What would settle it

Compute $\mathrm{Tor}^{RBr_2}_1(1,1)$ over $R=\mathbb{Z}$ with $\epsilon=1$ and, say, $\delta=0$; Theorem 2.1 predicts $H_1(S_2)=\mathbb{Z}/2$. A more targeted check is to test Lemma 4.6 directly: for $X=\{1,2\}$ in $RBr_2$, verify that right multiplication by $\epsilon^{-1}T_1$ sends $J_{\{2\}}$ into itself and is idempotent on the quotient; a single diagram violating this would break the inductive step.

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Extended reading notes

Core claim

The central discovery is that the homological behaviour of the Rook-Brauer and Motzkin algebras is controlled entirely by their permutation diagrams. With $\epsilon$ invertible in a unital commutative ring $R$ and $\delta$ arbitrary, the inclusion $\iota: RS_n \to RBr_n(R,\delta,\epsilon)$ induces an isomorphism $$\iota_*: H_*(S_n;1) \to \mathrm{Tor}^{RBr_n(R,\delta,\epsilon)}_*(1,1)$$ for all degrees, and $\mathrm{Tor}^{M_n(R,\delta,\epsilon)}_*(1,1)$ is $1$ in degree zero and $0$ in positive degrees. The proof obtains this by building inductive resolutions of the modules $RBr_n/J_X$ and $M_n/J_X$ and then applying a Shapiro-type comparison to pass from the symmetric group algebra to the full Rook-Brauer algebra. Homological stability follows directly from the known sharp stable range for symmetric groups.

Load-bearing premise

The argument rests on the claim that certain modules built from diagrams split off as clean direct pieces of simpler modules, with the splitting given by multiplying on the right by specific idempotent elements such as $\epsilon^{-1}T_x$; those splitting checks are stated briefly, and if any fails, the vanishing result and both main theorems collapse.

Editorial extensions

If this is right

Rook-Brauer homology equals symmetric group homology in every degree whenever $\epsilon$ is invertible, so the higher Tor groups are independent of $\delta$.
Motzkin algebras have no positive-degree homology with trivial coefficients; the only nonzero Tor group is $\mathrm{Tor}^{M_n}_0(1,1)\cong 1$.
Rook-Brauer algebras satisfy homological stability for $n\ge 2i+1$, and this stable range is sharp.
Motzkin algebras also satisfy homological stability.
The inclusion and projection maps between $RS_n$ and $RBr_n$ give mutually inverse isomorphisms on homology, so the symmetric group part is a homology direct summand.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension is to check whether the vanishing theorem survives with nontrivial coefficients, such as the sign representation; if the direct-summand splitting is as robust as it appears, the Motzkin and Rook-Brauer homology should remain concentrated in the symmetric-group component.
Because $\delta$ drops out of the homology, one might expect the higher Tor groups to be invariant under deforming $\delta$, and the same inductive-resolution construction may provide explicit resolutions for other one-parameter diagram algebras.
The results suggest that the homological theory of these diagram algebras is inherited from the symmetric group, so representation-stability phenomena for $S_n$ should transfer to the Rook-Brauer algebras without extra parameter conditions.
A testable weakening would be to replace invertibility of $\epsilon$ by the assumption that $\epsilon$ is a non-zero-divisor; the splittings by $\epsilon^{-1}T_x$ would then require more care and may reveal the exact boundary of the theorem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper aims for a genuine improvement over Fisher–Graves: proving the homology of Rook-Brauer and Motzkin algebras under only ε-invertibility, with δ arbitrary. The inductive-resolution strategy is appropriate, and the Y_P modules introduced for Motzkin algebras are a plausible new device. If the proof worked, this would be a real extension.

It does not work as written. Lemma 4.6 asserts that right multiplication by ε^{-1}T_aV_ab is idempotent and that αT_aV_ab = εα for α ∈ M_{a,b}. The second assertion is false. Take n=3, ε=1, δ=0, a=1, b=2, and α=V_{12}. Composing V_{12}T_1 gives the diagram D with blocks {−1,−2}, {1}, {2}, {−3,3}. Multiplying D on the right by V_{12} produces a middle-only loop and leaves the outer diagram V_{12}, so the product is δV_{12}=0. Thus αT_aV_ab = 0 ≠ εα. The same computation shows (T_aV_ab)^2 = δT_aV_ab, so e is idempotent only when δ=ε, not in general.

This is not a missing detail; it is a concrete error. The splitting of M_{X,{a,b}} as a direct summand of RBr_n/J_{X−{a,b}} fails, and with it Lemma 4.7 and the inductive resolution in Theorem 4.3 collapse. The main theorems 2.1 and 2.2 are therefore unproven. The paper correctly diagnoses Boyde's gap and organizes the material well, and the Y_P idea may be salvageable, but the central proof has a load-bearing false lemma.

The reader's conditional verdict is too generous. I would still send this to a serious referee because the question matters and the approach is sound in outline, but the referee should be asked to scrutinize Lemma 4.6 and the analogous Motzkin splitting claims. I would not cite it as a proof of the theorems in its current form.

Referee Report

2 major / 4 minor

Summary. The paper claims that, for a unital commutative ring R with ε invertible and δ arbitrary, the homology of the Rook-Brauer algebra RBr_n(R,δ,ε), defined as Tor^{RBr_n}_*(1,1), is isomorphic to the homology of the symmetric group in all degrees, and that the homology of the Motzkin algebra M_n(R,δ,ε) vanishes in positive degrees. The proof follows the inductive-resolution framework of Boyd–Hepworth–Patzt: it constructs resolutions of the modules RBr_n/J_X and M_n/J_X using auxiliary modules A_{X,x}, B_{X,x}, M_{X,{a,b}}, and Y_{P,{x,y}}, and proves vanishing of positive Tor groups by induction. The main theorems are then derived from an analogue of Shapiro's Lemma for Rook-Brauer algebras.

Significance. If true, the result would be a substantial strengthening of the Fisher–Graves theorem, removing the invertibility assumption on δ. The inductive-resolution strategy is appropriate and the paper correctly identifies the relevant modules. Some parts of the argument, such as the splitting of A_{X,x} via right multiplication by ε^{-1}T_x, are sound. However, the central splitting lemma for the modules M_{X,{a,b}} is false under the paper's own multiplication rule, and the proof of exactness of the key resolutions is not adequately justified. Since these points are load-bearing for the inductive vanishing theorem, the main results are not established by the manuscript.

major comments (2)

[Section 4.1, Lemma 4.6 (third bullet)] The element e = ε^{-1}T_aV_ab is not idempotent and does not split the inclusion M_{a,b} → RBr_n. Using the multiplication rule in §3.1, one computes V_ab T_a V_ab = δ V_ab. Hence e^2 = (δ/ε)e, and for α = V_ab one has αe = (δ/ε)α, not α. In the concrete case n=3, a=1, b=2, ε=1, δ=0, e^2=0 and the proposed retraction sends V_ab to 0. Therefore M_{X,{a,b}} is not shown to be a direct summand of RBr_n/J_{X−{a,b}}, and the inductive step of Theorem 4.3—which needs this summand to conclude Tor_{>0}(1,M_{X,{a,b}})=0—collapses. Since Theorem 5.1 and Theorem 2.1 depend on Theorem 4.3, the main result is not established.
[Section 4.1, Proposition 4.8 and Section 4.2, Proposition 4.15] The proof of exactness at degree 1 is not valid as written. The assertion that the map A_{X,x} ⊕ B_{X,x} → RBr_n/J_{X−{x}} is injective is justified only by the sentence that A_x and B_X,x have no basis elements in common. For submodules spanned by disjoint basis subsets, an element of J_{X−{x}} can be a sum of a vector from A_x and a vector from B_X,x without either summand lying in J_{X−{x}}. One needs a proof that J_{X−{x}} ∩ (A_x + B_X,x) = (J_{X−{x}}∩A_x) ⊕ (J_{X−{x}}∩B_X,x), or an explicit basis argument for the quotient. The same gap appears in the Motzkin analogue, Proposition 4.15, which is used in Theorem 4.17 and hence in Theorem 2.2.

minor comments (4)

[Corollary 2.3] There is a typo: 'satisify' should be 'satisfy'.
[Lemma 4.16] In the proof, 'the map in injective' should be 'the map is injective'.
[Theorem 5.1] The proof is delegated entirely to the argument in [3] with the comment that it follows 'exactly as that of Theorem 4.1 of [3]'. Since the algebra and the hypotheses differ, the paper should at least state the spectral sequence or the explicit chain comparison maps that yield the mutually inverse isomorphisms.
[Definition 4.2] The notation J_m for J_{\{n-m+1,\ldots,n\}} is potentially confusing because the subscript is a number rather than a subset; a brief clarification would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the main theorems follow from the external inductive-resolution criterion of [1], explicit resolution lemmas, and Nakaoka's stability theorem; the only self-citation ([9]) is not load-bearing.

full rationale

Walking the derivation chain, Theorem 2.1 is proved via the Shapiro-analogue Theorem 5.1, whose proof invokes Theorem 4.3 and Lemma 5.3; Theorem 4.3 itself is proved by induction on |X| using Proposition 4.8 and the external inductive-resolution principle Theorem 4.1 from [1]. Theorem 2.2 follows from Theorem 4.17 by the same external principle. No step fits a parameter to the claimed output, and the target isomorphisms are never assumed as inputs: delta and epsilon are arbitrary ring parameters, and the vanishing statements are derived rather than posited. The one self-citation, Proposition 3.4 from [9], gives a basis description for induced modules in Section 3.1; it is not referenced in the proofs of Theorem 4.3, Theorem 5.1, or the main theorems, so it is not load-bearing even if one worried about self-citation. The skeptical concern about Lemma 4.6's asserted idempotent e = epsilon^{-1} T_a V_ab would be, if valid, a correctness defect in a splitting lemma, not a circular dependency: a failed or unproved direct-summand claim is not equivalent to assuming the theorem being proved. Thus no quoted passage exhibits a step that reduces the paper's conclusions to their own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters: the proof is a pure derivation. The main external inputs are the inductive resolution theorem of [1] and Nakaoka's stability theorem. The only self-citation, [9], supplies a basis description that is not load-bearing.

assumptions (4)

standard math Inductive resolution principle (Theorem 3.1 of [1]): if Q_* -> N is a resolution, Tor^A_*(M,Q_j)=0 for all j, and M tensor_A Q_* -> M tensor_A N is a resolution, then Tor^A_*(M,N)=0 in positive degrees.
Assumed without proof; this is the engine of all vanishing results in Section 4.
standard math Nakaoka's homological stability for symmetric groups: H_i(S_{n-1}) -> H_i(S_n) is an isomorphism for n at least 2i+1.
Used in Corollary 2.3 to transfer Rook-Brauer homology to stability with sharp range.
domain assumption Trivial module conventions: permutation diagrams act as the identity and all other diagrams as zero on the trivial module 1.
This defines the Tor groups that are called homology; standard for diagram algebras and used in Lemmas 4.4, 4.6 and 4.12.
domain assumption Basis description for RBr_n tensor_{RBr_m} 1 (Proposition 3.4, cited from [9]) in terms of box-connected diagrams.
Stated in Section 3; the proof of Theorem 5.1 uses induced modules, though the explicit basis is not used in the final argument.

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Cite this review

Pith. "Pith review of Homology of Rook-Brauer Algebras and Motzkin Algebras." pith.science (2026). https://pith.science/paper/3QCZNIKF

@misc{pith2026250521977,
  author       = {Pith},
  title        = {Pith review of: Homology of Rook-Brauer Algebras and Motzkin Algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3QCZNIKF}},
  note         = {Machine review of arXiv:2505.21977}
}

abstract

Using the technique of inductive resolution introduced in arXiv:2303.07979, we prove that the homology of Rook-Brauer Algebra, interpreted as appropriate Tor-group, is isomorphic to that of symmetric group for all degrees under the assumption that $\epsilon$ in $R$ is invertible; furthermore, we also prove the homology of the Motzkin algebras vanishes in positive degrees under the same assumption. These results thereby establish homological stability of both algebras.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

1, 1-27, available athttps://msp.org/pjm/2023/327-1/p01.xhtml

Rachel Jane Boyd, Richard Hepworth, and Peter Patzt,The homology of the partition algebras, Pacific Journal of Mathematics327(2023), no. 1, 1-27, available athttps://msp.org/pjm/2023/327-1/p01.xhtml

work page 2023
[2]

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Rachel Jane Boyd and Richard Hepworth,The homology of the Temperley-Lieb algebras, Geometry and Topology 28(2024), no. 3, 1437-1499, available athttps://msp.org/gt/2024/28-3/gt-v28-n3-p11-s.pdf

work page 2024
[3]

Rachel Jane Boyd, Richard Hepworth, and Peter Patzt,The homology of the Brauer algebras, Sel. Math. New Ser.27(2021), no. 85, available athttps://link.springer.com/article/10.1007/s00029-021-00697-4

work page doi:10.1007/s00029-021-00697-4 2021
[4]

Ann.391(2025), 2173–2207, available at https://link.springer.com/article/10.1007/s00208-024-02960-3

Guy Boyde,Idempotents and homology of diagram algebras, Math. Ann.391(2025), 2173–2207, available at https://link.springer.com/article/10.1007/s00208-024-02960-3

work page doi:10.1007/s00208-024-02960-3 2025
[5]

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work page doi:10.1016/j.jalgebra.2023.09.017 2024
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work page arXiv 2024
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,Cohomology of diagram algebras(2024), available athttps://arxiv.org/abs/2412.14887

work page Pith review arXiv 2024
[8]

Minoru Nakaoka,Decomposition Theorem for Homology Groups of Symmetric Groups, Annals of Mathematics, vol 71 (1960), 16-42, available athttps://www.jstor.org/stable/1969878

work page arXiv 1960

Show all 9 references

[9]

48550/arXiv.2409.10885

Anthony Muljat and Khoa Ta,Noetherianity of Diagram Algebras(2024), available athttps://doi.org/10. 48550/arXiv.2409.10885. CSU FULLERTON, 800 N. STATECOLLEGEBLVD. FULLERTONCA 92831 Email address:khoa.ta@fullerton.edu

2024 doi arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

1, 1-27, available athttps://msp.org/pjm/2023/327-1/p01.xhtml

Rachel Jane Boyd, Richard Hepworth, and Peter Patzt,The homology of the partition algebras, Pacific Journal of Mathematics327(2023), no. 1, 1-27, available athttps://msp.org/pjm/2023/327-1/p01.xhtml

work page 2023

[2] [2]

3, 1437-1499, available athttps://msp.org/gt/2024/28-3/gt-v28-n3-p11-s.pdf

Rachel Jane Boyd and Richard Hepworth,The homology of the Temperley-Lieb algebras, Geometry and Topology 28(2024), no. 3, 1437-1499, available athttps://msp.org/gt/2024/28-3/gt-v28-n3-p11-s.pdf

work page 2024

[3] [3]

Rachel Jane Boyd, Richard Hepworth, and Peter Patzt,The homology of the Brauer algebras, Sel. Math. New Ser.27(2021), no. 85, available athttps://link.springer.com/article/10.1007/s00029-021-00697-4

work page doi:10.1007/s00029-021-00697-4 2021

[4] [4]

Ann.391(2025), 2173–2207, available at https://link.springer.com/article/10.1007/s00208-024-02960-3

Guy Boyde,Idempotents and homology of diagram algebras, Math. Ann.391(2025), 2173–2207, available at https://link.springer.com/article/10.1007/s00208-024-02960-3

work page doi:10.1007/s00208-024-02960-3 2025

[5] [5]

Peter Patzt,Representation stability for diagram algebras, Journal of Algebra, vol 638 (2024), 625-669, available athttps://doi.org/10.1016/j.jalgebra.2023.09.017

work page doi:10.1016/j.jalgebra.2023.09.017 2024

[6] [6]

Andrew Fisher and Daniel Graves,Cohomology of Tanabe algebras(2024), available athttps://arxiv.org/ abs/2410.00599

work page arXiv 2024

[7] [7]

,Cohomology of diagram algebras(2024), available athttps://arxiv.org/abs/2412.14887

work page Pith review arXiv 2024

[8] [8]

Minoru Nakaoka,Decomposition Theorem for Homology Groups of Symmetric Groups, Annals of Mathematics, vol 71 (1960), 16-42, available athttps://www.jstor.org/stable/1969878

work page arXiv 1960