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Morita equivalences between cyclotomic KLR algebras in types $\mathtt{C}_\infty$ and $\mathtt{A}_\infty$

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Pith's one-line read This paper proves that every level one cyclotomic KLR algebra in type C∞ is graded Morita equivalent to a level two cyclotomic KLR algebra in type A∞, transferring the full graded representation theory of the former to the well-understood l

desk verdict A genuine and likely correct Morita equivalence result, but the proof has a load-bearing unproven simplicity assertion that must be fixed before acceptance. read the letter →

arxiv 2511.08779 v2 pith:SCWRVCOV submitted 2025-11-11 math.RT

Chris Bowman , Robert Muth , Liron Speyer , Louise Sutton This is my paper
classification math.RT MSC 16G1017B3720C0805E10
keywords KLRalgebrasquiverHeckeMoritaequivalencetypeC∞A∞cyclotomicquotientsSpechtmodulesdecompositionnumbers
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

Level one cyclotomic KLR algebras in type C∞ are hard: almost nothing is known about their simple modules. This paper proves that each such algebra is graded Morita equivalent to a level two cyclotomic KLR algebra in type A∞, whose structure is well understood. The bridge is an explicit isomorphism between a corner of the C∞ algebra and a tensor product of a simple algebra with the A∞ algebra; the simple factor collapses under Morita equivalence. Consequently, graded decomposition numbers and the full submodule structure of all level one type C∞ KLR algebras can be read off from the type A∞ theory, and the decomposition numbers are characteristic-free anti-spherical Kazhdan–Lusztig polynomials.

What carries the argument

The argument rests on the idempotent 1_{ω,β−ω} that cuts out the rectangular partition ρ, and the algebra homomorphism φ(x1⊗x2) = 1_{ω,β−ω}(x1⊗x2)1_{ω,β−ω} from R^{ΛκC}_ω(sp∞) ⊗ R^{Λκ1+Λκ2}_{β−ω}(sl∞) into the corner of R^{ΛκC}_β(sp∞). Theorem 3.8 proves φ is an isomorphism by showing, via double-coset representatives and residue-word combinatorics, that the only block transpositions that survive are trivial, and by matching cellular basis dimensions. Then a sequence of diagrammatic lemmas (culminating in Proposition 3.19) shows φ sends cell ideals to cell ideals, so the Specht and simple modules match. Since R^{ΛκC}_ω(sp∞) is asserted to be a simple algebra, the corner is Morita equivalent

What would settle it

Take a0 = 2 and κC = 0, so ρ = (2,2), and check whether R^{Λ0}_ω(sp∞) is a simple algebra by computing its two-sided ideals. If it is not simple, the step 'since R^{ΛκC}_ω is simple' fails. Alternatively, compute the graded decomposition matrix of R^{Λ0}_β(sp∞) for β of height ≤ 3 directly and compare with anti-spherical Kazhdan–Lusztig polynomials; any mismatch would destroy the claimed Morita equivalence.

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Extended reading notes

Core claim

The central claim is Theorem 3.21: for any level-one weight ΛκC and any positive root β of type C∞, the cyclotomic KLR algebra R^{ΛκC}_β(sp∞) is graded Morita equivalent to the level-two type A∞ algebra R^{Λκ1+Λκ2}_{β−ω}(sl∞), where ω is the weight of the minimal rectangle ρ containing a0 0-nodes, κ1 = κC + a0, and κ2 = a0. Under the equivalence, simple modules and Specht modules correspond by D^{(λ,μ)} ↦ D^{ρ+(λ,μ')} and S^{(λ,μ)} ↦ S^{ρ+(λ,μ')}. This resolves the level-one type C∞ graded decomposition number problem and gives the complete submodule structure of these algebras.

Load-bearing premise

The proof assumes without proof that the rectangle algebra R^{ΛκC}_ω(sp∞) is a simple algebra (and split over the ground field); if this fails, the corner is not Morita equivalent to the level two A∞ algebra and the main theorem collapses.

Editorial extensions

If this is right

  • Graded decomposition numbers of all level one type C∞ cyclotomic KLR algebras are characteristic-free and equal to anti-spherical (p-)Kazhdan–Lusztig polynomials for maximal finite parabolics of finite symmetric groups.
  • The full Ext-quiver presentations of the basic algebras, and hence the complete submodule structure of Specht modules, become accessible via the known type A∞ theory.
  • Simple and Specht modules of level one type C∞ are naturally labelled by bipartitions (after adding a rectangle and transposing the second component), giving a combinatorial indexing that did not exist before.
  • Off-diagonal entries of the graded decomposition matrix lie in strictly positive degree, matching canonical basis coefficients of the type C∞ highest weight module V(ΛκC).
  • The equivalence is an explicit graded isomorphism on a corner, so it can be used to translate explicit computations (e.g., dimensions, characters) between the two families of algebras.

Reading between the lines

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

  • If the asserted simplicity of R^{ΛκC}_ω(sp∞) is verified in general, the same corner-plus-simple-factor mechanism may extend to higher levels and affine types; the authors predict this 'folding phenomenon' but expect it to be hard.
  • A direct consequence the authors only note in passing: previously observed small-height examples where characteristic 0 decomposition numbers of type C∞ disagreed with canonical basis coefficients cannot occur at level one, so those examples must live at higher levels or in affine type C.
  • The rectangle-cutting construction depends only on the 0-node count a0; one could test whether the Morita equivalence survives replacing the rectangle by other self-conjugate shapes, which would yield finer block decompositions.
  • The 'X is trivial' step suggests a purely combinatorial characterization of when block transpositions die in the corner; making that explicit could give an independent, calculus-free proof of the isomorphism.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proves a graded Morita equivalence between level-one cyclotomic KLR algebras of type C_∞ and level-two cyclotomic KLR algebras of type A_∞. The main construction is an explicit algebra isomorphism φ between an idempotent corner of R^{Λ_{κ_C}}_β(sp_∞) and a tensor product R^{Λ_{κ_C}}_ω(sp_∞) ⊗ R^{Λ_{κ_1}+Λ_{κ_2}}_{β−ω}(sl_∞), where ω is the height of a rectangular partition ρ. The authors then use a claim that the first tensor factor is a simple algebra to pass from this corner isomorphism to a global Morita equivalence. From this they deduce the graded decomposition numbers and full submodule structure of all level-one type C_∞ cyclotomic KLR algebras, identifying simple and Specht modules under the correspondence (λ,μ) ↦ ρ+(λ,μ').

Significance. If correct, the main result is significant: it gives the first complete solution to the level-one type C_∞ graded decomposition number problem over arbitrary fields, and it yields full submodule structure via known type-A results. The explicit nature of the isomorphism φ and the cellular matching are valuable and go well beyond an abstract existence statement. The paper also benefits from being built on established cellular and KLR machinery ([EM24], [KL09], [Rou08]) rather than on the authors' own unpublished preprints. However, the proof has at least one load-bearing unproved assertion, and the central claim should be considered conditional until that assertion is supplied.

major comments (3)
  1. [Proof of Theorem 3.21 (final paragraph)] The step 'since R^{Λ_{κ_C}}_ω(sp_∞) is a simple algebra' is unproved and load-bearing. It is the only bridge from the algebra isomorphism φ of Theorems 3.6 and 3.8 to the claimed Morita equivalence with R^{Λ_{κ_1}+Λ_{κ_2}}_{β−ω}(sl_∞). For an arbitrary field F, 'simple' is insufficient: R must be Morita equivalent to F, i.e. split simple; a non-split central simple algebra would not tensor with R^{A} to a Morita-equivalent algebra. Moreover, simplicity does not follow from the one-cell cellular structure alone, since a one-cell cellular algebra can have a nonzero radical (e.g. the nilHecke algebra). Please prove directly, using the cellular basis of Theorem 2.11, that the unique cell module S^ρ is absolutely irreducible and that the cellular bilinear form is nondegenerate, or cite a precise theorem. Corollary 3.3 and Theorems 3.6/3.8 reduce the whole theorem to exactly this point.
  2. [Proof of Theorem 3.8 (paragraph after (3.10))] The assertion 'X is trivial' is the key step proving that φ is surjective, but it is justified in a single sentence. From the fact that the last entry of j is 0, it follows that a nontrivial block transposition X would move a 0-residue strand into the β−ω component; however, one must also rule out the possibility that the corresponding spanning element vanishes for other reasons, and one must check all double-coset representatives c ∈ [1, r+1]. Please expand this into a complete diagrammatic or algebraic argument showing that every nonzero element in the spanning set (3.9) has X trivial.
  3. [Proof of Theorem 3.21, cellular matching] The proof checks that a cell-ideal generator maps into the corresponding cell ideal and that dimensions agree, but it does not establish that φ is an isomorphism of cellular algebras with respect to the cellular bases of Theorem 2.11. To conclude that D^{(λ,μ)} ↦ D^{ρ+(λ,μ')} and S^{(λ,μ)} ↦ S^{ρ+(λ,μ')}, one must know that the radical of the bilinear form on the cell module is preserved, not merely that the cell ideals are matched. Please state and prove the matching of the full cellular bases, or prove directly that the induced isomorphism on cell modules sends the radical to the radical.
minor comments (4)
  1. [Definition 2.7] The formula for ρ+λ is ambiguous when ℓ(λ) < ℓ(ρ), since it is not stated explicitly that λ is padded with trailing zeroes. Please clarify.
  2. [Theorem 3.6] The inclusion of R^{Λ_{κ_C}}_ω(sp_∞) ⊗ R^{Λ_{κ_1}+Λ_{κ_2}}_{β−ω}(sl_∞) into R^{Λ_{κ_C}}_β(sp_∞) is implicit. Please define it explicitly before writing the map φ, for example by describing how generators are sent to the first |ω| and last |β−ω| strands.
  3. [Lemma 3.4] The symbol k is used both for the index of a node and for a row parameter in 'k = (r_k−1)a_0 + 2r_k'; this makes the proof harder to follow. Please use different letters.
  4. [Proof of Theorem 3.21, displayed equation] The display '1_{ω,β−ω}(y_{tρ}⊗y_{t(λ,μ)})1_{ω,β−ω} = y_{tρ}⊗y_{t(λ,∅)}⊗e(i_μ)' mixes tensor factors with idempotents and is not fully precise. Rewrite with explicit embeddings into R^{Λ_{κ_C}}_β(sp_∞).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Morita equivalence is built from an explicit isomorphism and external cellular-basis and KLR spanning results; the unproven simplicity assertion is a correctness gap, not a circular step.

full rationale

The central claim (Theorem 3.21) is derived from an explicit algebra isomorphism φ (Theorems 3.6 and 3.8) between the corner 1_{ω,β−ω} R^{ΛκC}_β(sp∞) 1_{ω,β−ω} and the tensor product R^{ΛκC}_ω(sp∞) ⊗ R^{Λκ1+Λκ2}_{β−ω}(sl∞), combined with the Morita equivalence of the corner with the full algebra (Corollary 3.3). The proof of φ uses the KLR relations (2.1)–(2.5), the cellular basis theorem from [EM24, Theorem A], and the spanning result from [KL09, Rou08]; these are external sources, not the authors' own unverified claims. Corollary 3.3 uses Proposition 3.1, which is argued from good-node combinatorics and [EM24, Cor 6G.10]. The final step invokes the sentence 'since R^{ΛκC}_ω(sp∞) is a simple algebra' without proof or citation. This is an unproven hypothesis and a potential correctness gap — in particular, over arbitrary fields one would need split simplicity for the tensor-product-to-Morita reduction — but it is not circular: it is not obtained from the target Morita equivalence, nor from a fitted parameter, and the target result does not reduce to it by construction. The author-group preprints cited for Ext-quiver presentations and submodule structures ([BDVHS23], [BDVD+24a,b]) are used only for downstream consequences after Theorem 3.21, not as premises of the proof. No fitted-input-called-prediction, self-definitional, or ansatz-smuggling pattern appears. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters fitted to data. The construction depends only on the input multicharge κC and the coefficient a0 of α0 in β; these are not adjusted to make the proof work. The main axioms are prior structural theorems about KLR algebras plus one unproved assertion about the rectangle algebra.

assumptions (6)
  • domain assumption R^Λ_β(g) is a graded cellular algebra with cellular basis {c_st} (Theorem 2.11, [EM24, Theorem A]).
    Used throughout to define Specht modules, cell ideals, and to compare dimensions.
  • domain assumption The simple modules of R^Λ_β(g) are exactly {D^λ | λ∈K^ℓ_β(g)} (Theorem 2.12, [EM24, Theorem C]).
    Provides the classification of simples used in Proposition 3.1 and Theorem 3.21.
  • domain assumption A Kleshchev-partition criterion: e(j)D^ν ≠ 0 if j is a good-node sequence from ∅ to ν ([EM24, Cor 6G.10]).
    Used in the proof of Proposition 3.1 to show the truncation idempotent hits every simple module.
  • domain assumption The idempotent-truncated KLR algebra has the spanning set (3.9) from [KL09, Theorem 2.5] and [Rou08, Theorem 3.7].
    Basis of the surjectivity argument in Theorem 3.8.
  • ad hoc to paper The algebra R^{ΛκC}_ω(sp∞) is simple.
    Asserted without proof or citation in the final paragraph of the proof of Theorem 3.21; needed to reduce the tensor product to a Morita equivalence. Could be a standard fact, but the paper does not justify it.
  • standard math For a finite-dimensional simple F-algebra A, A ⊗ B is Morita equivalent to B.
    Used implicitly to pass from R^Λ_ω ⊗ R^A to R^A in the proof of Theorem 3.21; requires splitness of R^Λ_ω, which is not established.

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Pith. "Pith review of Morita equivalences between cyclotomic KLR algebras in types $\mathtt{C}_\infty$ and $\mathtt{A}_\infty$." pith.science (2026). https://pith.science/paper/SCWRVCOV

@misc{pith2026251108779,
  author       = {Pith},
  title        = {Pith review of: Morita equivalences between cyclotomic KLR algebras in types $\mathttC_\infty$ and $\mathttA_\infty$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCWRVCOV}},
  note         = {Machine review of arXiv:2511.08779}
}
abstract

We prove that level one cyclotomic KLR algebras in type $\mathtt{C}_\infty$ are graded Morita equivalent to level two cyclotomic KLR algebras in type $\mathtt{A}_\infty$. We hence deduce the graded decomposition numbers and full submodule structures of all level one cyclotomic KLR algebras in type $\mathtt{C}_\infty$.

Figures

Figures reproduced from arXiv: 2511.08779 by the authors.

Figure 1
Figure 1. An arbitrary partition ν labelling a Specht module of the level 1 type C∞ KLR algebra. In grey we highlight the rectangular subpartition ρ and in pink and blue we highlight the partitions λ and µ which label a Specht module S(λ, µ ′ ) of the level 2 type A∞ KLR algebra. The charges for the Specht modules are κC ∈ Z⩾0 and (κ1, κ2) ∈ Z 2 >0 , respectively. Acknowledgements. Firstly, we would like to thank Andrew Matha… view at source ↗
Figure 2
Figure 2. The Dynkin diagrams of types A∞ (above) and C∞ (below) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. The element ψ Tρ+λ T ρ+λ yT ρ+λ corresponding to the tableaux in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: For κC = 0, ρ = (44 ), and λ = (3, 2, 1), we depict examples of the C-residues of ρ + λ, the tableau T ρ+λ , and the tableau Tρ+λ respectively. 3. The isomorphism theorem For the remainder of the paper we fix β = P i∈I aiαi ∈ Q+ n , and Λ = ΛκC ∈ P +. Let ρ = ((a0) κC+…
Figure 5
Figure 5. Figure 5: The leftmost tableau is of maximal degree, 3 (note that every orange tile has degree +1 and every green has degree 0). The next three tableaux are all possible tableaux of degree 1; in each case there is a unique pair of orange/green nodes of degree 0/−1; these are 18/…
Figure 6
Figure 6. Figure 6: The diagram D formed of a pair of (possibly decorated) thick strands, coloured grey and pink, which double-cross each other. eT−1(k) ⊗ eT−1(k+1) (as required, since deg(T −1 (k + 1)) = 0 = deg(T −1 (k))). This is illustrated via an example in Figures 7 and 8. 0 1 2 3 4…
Figure 7
Figure 7. Figure 7: On the left we depict the residues of ρ+λ. The two semistandard tableaux are examples of S = T ρ+λ and T = s4(S) as in Case 1. Here yS = y2y9 and yT = y2. 0 1 2 1 0 1 2 1 0 3 4 2 1 0 1 2 1 0 1 2 1 0 3 4 2 1 = 0 1 2 1 0 1 2 1 0 3 4 2 1 0 1 2 1 0 1 2 1 0 3 4 2 1 [PITH_F…
Figure 8
Figure 8. Figure 8: We depict ψ Tν T (ψ T S ySψ S T )ψ T Tν = ψ Tν T yTψ T Tν for S, T as in [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: On the left we depict the C-residues of ρ + λ. The two semistandard tableaux are examples of S and T = Tρ+λ = s3(S) as in Case 2. Here yS = y2 and yT = y2y7. 0 1 2 1 0 1 2 1 0 3 4 2 1 0 1 2 1 0 1 2 1 0 3 4 2 1 = − 0 1 2 1 0 1 2 1 0 3 4 2 1 0 1 2 1 0 1 2 1 0 3 4 2 1 [P…
Figure 10
Figure 10. Figure 10: We depict ψ Tν S ySψ S Tν = −ψ Tν T yTψ T Tν for S, T as in [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: We depict an example of Case (3b) corresponding to the tableaux S, T pictured in [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: On the left we depict the C-residues of ρ + λ. The two semistandard tableaux are examples of S = s6(T) as in Case 3. Finally, we are able to deduce the main result of the paper (as discussed in the introduction), a graded Morita equivalence between cyclotomic quiver H…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION output.nonempty.mrnumber duplicate missing pop "" 'skip if duplicate empty 'pop " " swap * " " * write if FUNCTION fin.entry add.period write newline INTEGERS nameptr namesleft numnames FUNCTION format.language language e...

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.