Cellularity of the p-Canonical Basis for Symmetric Groups

Lars Thorge Jensen

REVIEW 2 major objections 4 minor 15 references

Cellularity of the p-Canonical Basis for Symmetric Groups

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Pith's one-line read The p-canonical basis of the Hecke algebra of a symmetric group is a cellular basis for every prime p, and its two-sided p-cell preorder equals the classical Kazhdan–Lusztig two-sided cell preorder.

desk verdict Solid and novel results, but the cellularity proof has a repairable generator mix-up that needs fixing before this should be accepted. read the letter →

arxiv 2009.11715 v1 pith:UF53QXCY submitted 2020-09-24 math.RT math.COmath.GR

Lars Thorge Jensen This is my paper · ORCID

classification math.RTmath.COmath.GR MSC 20C0820C3005E1020F55

keywords p-canonicalbasisp-cellsIwahori-HeckealgebrasymmetricgroupcellularKazhdan-LusztigcellsRobinson-SchenstedcorrespondencePerron-Frobeniustheorem

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

The paper proves that the p-canonical basis of the Hecke algebra of a symmetric group is not just a basis but a cellular basis, for every prime p, and for p=0, where it recovers the classical Kazhdan–Lusztig basis. The cell datum is concrete: two-sided p-cells are indexed by partitions of n with the dominance order, left and right p-cells by standard tableaux, and the basis element of a permutation is indexed by its Robinson–Schensted pair. The paper also proves that the two-sided p-cell preorder on S_n coincides with the Kazhdan–Lusztig two-sided cell preorder for all p, so the two-sided cell poset does not change with the characteristic. For every finite Weyl group it proves Property A: left (or right) p-cells inside the same two-sided p-cell are incomparable under the left (or right) p-cell preorder. The payoff is that modular characteristic still yields the full cellular-algebra structure that organizes the representation theory of the Hecke algebra, together with new p-special modules and p-families.

What carries the argument

The load-bearing mechanism is the Perron–Frobenius theorem applied to cell modules. Because the structure coefficients $p\mu^z_{x,y}$ of the p-canonical basis are Laurent polynomials with non-negative coefficients, the matrix of the positive element $a_c=\sum_{w\in W} c_w \otimes p H_w$ acting on a two-sided p-cell module has strictly positive entries on each left-cell block (Lemma 4.5). Perron–Frobenius then produces, for each two-sided p-cell $J$, an idempotent $e_J$ with positive coefficients that projects onto each left-cell component (Theorem 4.6); the block-diagonal form of this projector forces left p-cells in $J$ to be incomparable and yields the p-special module of $J$. For symmetric groups, the Robinson–Schensted correspondence identifies left, right, and two-sided p-cells with Q-symbols, P-symbols, and shapes, so the two-sided p-cell preorder can be compared with dominance order on partitions; a separate lemma shows that on a left cell module the multiplication coefficients depend only on the P-symbol, which is exactly what the cell datum axioms require.

What would settle it

Compute the p-canonical multiplication table for a small finite Weyl group (for example type $B_2$ at $p=2$, as in the paper's own example) and check whether there exist $x,y$ in the same two-sided $p$-cell with $x\le_L^p y$ but $x\not\sim_L^p y$; such a pair would refute Theorem 4.4. For the symmetric-group preorder, test adjacent partitions $\lambda<\mu$ in dominance order: if for some $n$ and $p$ the two-sided $p$-cell of $\lambda$ is not strictly below the $p$-cell of $\mu$ in the $p$-preorder, Theorem 5.7 fails. Both checks are finite computations from the structure coefficients $p\mu^z_{x,y}$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the p-canonical basis of the Iwahori–Hecke algebra of S_n satisfies the cell datum axioms for cellular algebras, not only in characteristic zero. Concretely, the quadruple consisting of the partitions of n under dominance order, the standard anti-involution, standard tableaux of each shape, and the map (P,Q) ↦ p H_w gives a cell datum (Theorem 5.14). Consequently the two-sided p-cell preorder on S_n equals the Kazhdan–Lusztig two-sided cell preorder for every prime p (Theorem 5.7), extending the order-preserving bijection between two-sided cells and partitions in dominance order to p-cells. Using the previously established fact that p-cells and Kazhdan–Lusztig cells coincide as sets, the paper shows that for all finite Weyl groups Property A holds (Theorem 4.4): if two elements lie in the same two-sided p-cell and one is ≤_L^p the other, then they lie in the same left p-cell, and analogously on the right. Along the way it introduces p-special modules and p-families, the Perron–Frobenius analogues of the special modules and families attached to Kazhdan–Lusztig cells.

Load-bearing premise

The central claims assume two previously established results—that p-canonical structure coefficients are non-negative Laurent polynomials and that p-cells of symmetric groups are classified by the Robinson–Schensted Q-symbol—and if either prior theorem had a gap, the cell-datum and preorder conclusions would lose their foundation.

Editorial extensions

If this is right

For every prime $p$, the Hecke algebra of $S_n$ with the $p$-canonical basis is a cellular algebra in the sense of the cell datum axioms, so it comes with cell modules, cell representations, and the usual cellular-algebra toolkit.
The two-sided $p$-cell preorder on $S_n$ is the dominance order on partitions, the same poset as for Kazhdan–Lusztig two-sided cells; hence the two-sided cell poset does not depend on the characteristic.
For every finite Weyl group, left $p$-cells inside a common two-sided $p$-cell are incomparable: if $x\le_L^p y$ and $x,y$ lie in the same two-sided $p$-cell, then $x$ and $y$ lie in the same left $p$-cell, and analogously on the right.
The $p$-canonical cell modules give a supply of modules whose graded structure is controlled by the $p$-canonical structure coefficients rather than the ordinary Kazhdan–Lusztig polynomials, so they can differ from Specht modules in positive characteristic.
The $p$-families and $p$-special modules are new invariants of two-sided $p$-cells; in examples they refine classical families, so they carry information about the characteristic $p$ that the classical families lose.

Reading between the lines

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

The paper states as an open question whether the left and right $p$-cell preorders on $S_n$ coincide with the Kazhdan–Lusztig left and right cell preorders. If they do, the cell datum constructed here would be isomorphic to the classical one, not merely at the two-sided level; a finite check of the left $p$-cell preorder on standard tableaux for small $n$ and small primes would be a first test.
The proof uses only positivity of the $p$-canonical structure coefficients and the Robinson–Schensted description of cells, so the same Perron–Frobenius route could plausibly produce cellular data from any positive basis whose cells are known, such as Hecke algebras of other finite types once their $p$-cell classifications are in hand.
The paper leaves the connection between $p$-special modules, $p$-families, and the modular representation theory of finite reductive groups explicitly unexplored; a concrete next step would be to compute $p$-families in small rank and compare them with decomposition numbers of $p$-modular characters.

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Formalized claims in Lean

Claim #1: On the paper's own terms, the central discovery is that the p-canonical basis of the Iwahori–Hecke algebra of S_n satisfies the cell datum axioms for cellular algebras, not only in characteristic zero. Concretely, the quadruple consisting of the partitions of n under dominance order, the standard anti-involution, standard tableaux of each shape, and the map (P,Q) ↦ p H_w gives a cell datum (Theore

/-- @claim 1 On the paper's own terms, the central discovery is that the p-canonical basis of the Iwahori–Hecke algebra of S_n satisfies the cell datum axioms for cellular algebras, not only in characteristic zero. Concretely, the quadruple consisting of the partitions of n under dominance order, the standard anti-involution, standard tableaux of each shape, and the map (P,Q) ↦ p H_w gives a cell datum (Theore -/
def central_claim : Prop :=

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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 is worth taking seriously. The main results are genuinely new: the two-sided p-cell preorder on symmetric groups coincides with the Kazhdan-Lusztig preorder, the p-canonical basis extends to a cell datum, and Property A holds for p-cells in finite Weyl groups. The Perron-Frobenius machinery from Kildetoft-Mazorchuk is adapted cleanly, and the Robinson-Schensted combinatorics in Section 5 is mostly well handled. Credit where due: the paper is honest about what rests on imported results and what remains open, and the exposition is careful without being bloated.

There is, however, a real gap in the proof of Theorem 5.14, and the stress-test note is on target. Equation (3) is stated for H_{s_i} * pH_x, but the right-hand side is the expansion of pH_{s_i} * pH_x. The correct relation is H_{s_i} = pH_{s_i} - v * pH_e, so a diagonal term -v * pH_x is missing. The concrete S_2 example in the note is convincing: formula (4) predicts H_s * C^λ ≡ 0 mod H(<λ) for the identity cell, while the actual value is -v * C^λ. So condition (iii) of Definition 5.8 is not verified as written.

The good news is that this is repairable without changing the theorem statement. Since the pH_{s_i} generate H (invertibly over Z[v,v^-1]), one can prove condition (iii) for the pH generators and then extend to all elements. But the proof in the paper needs a nontrivial correction, and this should be fixed before publication.

Other soft spots are minor. The imported positivity from [JW17] and the p-cell classification from [Jen20] are load-bearing, but they are published and the paper's new claims do not appear to be circular. Proposition 5.6 is compressed, and a referee will want that expanded. The paper also openly flags the unresolved left-preorder question, which is the right thing to do.

Who is this for? Modular representation theorists and people working on Hecke algebras and Soergel bimodules. They will cite it once the proof is patched. This deserves a serious referee; it should not be desk-rejected. My recommendation is to send it out, with a request to fix the cellularity argument.

Referee Report

2 major / 4 minor

Summary. The paper studies the p-canonical basis of Iwahori-Hecke algebras for finite Weyl groups, with emphasis on symmetric groups. It has three main theorems: (1) for finite Weyl groups, left (or right) p-cells inside the same two-sided p-cell are incomparable with respect to the left (or right) p-cell preorder (Theorem 4.4, Property A); (2) for symmetric groups, the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder, equivalently with the dominance order on partitions (Theorem 5.7); and (3) the p-canonical basis can be extended to a cell datum for the Hecke algebra of a symmetric group (Theorem 5.14). The proofs use the Perron-Frobenius theorem following Kildetoft-Mazorchuk, the positivity of p-canonical structure coefficients imported from Jensen-Williamson, and the Robinson-Schensted classification of p-cells for symmetric groups imported from the author's earlier paper. Along the way the paper introduces p-special modules and p-families and proves several structural properties of p-cells. Conjecture 3.6, relating the two-sided p-cell preorder to the Perron-Frobenius function, is explicitly stated as open, with computer verification for rank at most 4.

Significance. If the results are correct, they settle a natural open question: although p-cells and Kazhdan-Lusztig cells coincide as sets for symmetric groups, the equality of the two-sided p-cell preorder with the Kazhdan-Lusztig preorder was not previously known. The extension of the p-canonical basis to a cell datum is also a new and useful structural result, and the proof of Property A for all finite Weyl groups is a genuine generalization of a classical theorem. The paper is clearly written and largely self-contained; explicit credit is due for the careful Perron-Frobenius arguments, the worked example in type B_2, and the transparent identification of imported results. The main results are not machine-checked and depend on two substantial imported inputs: positivity of p-canonical structure coefficients [JW17] and the classification of p-cells for S_n [Jen20]. These dependencies are stated honestly. One proof error in the cellularity argument, described below, affects the written verification of Theorem 5.14 but appears repairable within the paper's framework.

major comments (2)

[§5.2, Eq. (3)] Equation (3) is asserted for H_{s_i} p_H_x, but the right-hand side is the p-canonical expansion of p_H_{s_i} p_H_x, not of H_{s_i} p_H_x. Since H_{s_i} = p_H_{s_i} - v p_H_e, the displayed formula omits the term -v p_H_x. Consequently the verification of condition (iii) of Definition 5.8 as written is invalid. For example, in S_2, with p_H_s = H_s + v H_e, one computes H_s p_H_e = p_H_s - v p_H_e, whereas the displayed formula would give only p_H_s; the missing term changes the asserted cell-module action. The theorem is likely repairable: one should verify condition (iii) for the generators p_H_{s_i}, whose products are governed by Proposition 2.4(iii), and then use H_{s_i} = p_H_{s_i} - v p_H_e to extend the condition to all elements. This correction is local in the proof, but as it stands the proof of Theorem 5.14 is not complete.
[§5.1, Proposition 5.6] The proof of Proposition 5.6 is the main combinatorial input for Theorem 5.7, but it is extremely compressed. In particular, Case 2 constructs y and s, then sets t := w_0 s w_0 and x := y w_0, asserting that x lies in J_μ and xt lies in J_ν by invoking Theorem 5.5; the verification of these Robinson-Schensted statements is not written out. Since the proposition is load-bearing for the equivalence in Theorem 5.7, this step needs a fuller argument or a precise reference that establishes the effect of right and left multiplication by w_0 on P- and Q-symbols in the specific configuration used.

minor comments (4)

[§3, Theorem 3.2] In the proof of Theorem 3.2, the displayed inequality after passing to the limit has M_{d_i}|_{L_j} on the right-hand side; this should be M_d|_{L_j} in the limiting expression. As written, the inequality does not state the intended continuity argument.
[§5.1, Proposition 5.6] In Case 1 of the proof of Proposition 5.6, the notation v_j is used for the parts of the partition ν, which is easy to confuse with the formal variable v; the symbol ν_j would be clearer. In Case 2, the phrase “two adjacent columns l and l+1” appears to be a slip for rows, given the preceding discussion of row lengths.
[§3, Conjecture 3.6] Conjecture 3.6 is explicitly left unproved, with the paper stating that current techniques do not suffice. It is not used in the later sections, but the paper should clearly flag in the introduction that this conjecture remains open, so that readers do not mistake it for a proved result.
[§5.2, Definition 5.8] In Definition 5.8, the notation C(S,T) with S,T in M(λ) is clear, but the string “write Cλ S,T = C(S,T)” loses the superscript λ in the displayed text; this is a minor formatting issue that should be corrected for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the p-cell classification and positivity inputs are prior parameter-free theorems, and the new preorder, Property A, and cellularity results are derived rather than assumed.

full rationale

The derivation chain is not circular in the sense of the seven listed patterns. The main new claims are Theorem 4.4 (Property A for finite Weyl groups), Theorem 5.7 (coincidence of the two-sided p-cell preorder with the Kazhdan-Lusztig preorder for symmetric groups), and Theorem 5.14 (cellularity of the p-canonical basis). Each is proved from inputs that are not identical to the conclusion: Property A is proved by Perron-Frobenius arguments applied to the p-canonical structure coefficients, following the independent framework of Kildetoft-Mazorchuk; Theorem 5.7 combines the Robinson-Schensted classification of p-cells (Theorem 5.1, cited from the author's earlier [Jen20]) with Geck's dominance-order description of the Kazhdan-Lusztig two-sided preorder and the paper's own Proposition 5.6; Theorem 5.14 uses Corollary 4.8, Theorem 5.7, and the new independence lemma Lemma 5.13. The cited inputs [Jen20, Theorem 4.33] and [JW17, Proposition 4.2] are self-citations, but they are parameter-free theorems with stated assumptions that do not include the target conclusions of this paper, and they are externally checkable; under the hard rules this counts as independent support rather than circularity. No fitted parameter is renamed as a prediction, no known result is merely renamed, and no uniqueness theorem is imported to forbid alternatives. The proof of Theorem 5.14 does contain a possible correctness issue: equation (3) is written for the standard generator H_{s_i} but appears to describe multiplication by pH_{s_i}, omitting the diagonal term -v pH_x. That is a proof-repair concern, not a circularity: it does not make the claimed theorem equivalent to its input by construction. The paper also explicitly leaves Conjecture 3.6 and the left p-cell preorder comparison open, confirming that those stronger statements are not silently assumed as inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The paper defines p-special modules and p-families, but these are derived objects with existence proofs based on cell modules and Perron-Frobenius theory, not ad hoc inventions requiring independent falsification. The axioms listed are the main theorems imported from prior literature that the new proofs depend on.

assumptions (7)

standard math Perron-Frobenius theorem for positive matrices (Theorem 2.6)
Used throughout Sections 3 and 4 to produce Perron-Frobenius eigenvalues, eigenvectors, and idempotents for cell modules.
domain assumption Structure coefficients pµ^z_{x,y} of the p-canonical basis are non-negative Laurent polynomials with non-negative integer coefficients (Proposition 2.4(iii), cited from [JW17])
This positivity is the foundation for applying Perron-Frobenius theory to p-cells; without it, matrices M_{c,C} in Lemma 4.5 need not be positive.
domain assumption Kildetoft-Mazorchuk results on positively based algebras (Corollary 3, Theorem 5, Theorem 6, Propositions 13, 14, 17 of [KM16])
The framework of p-special modules, apices, and cell module properties in Sections 3-4 is imported from [KM16] and applied to the p-canonical basis.
domain assumption Classification of p-cells for symmetric groups via Robinson-Schensted: left p-cells by Q-symbol, right p-cells by P-symbol, two-sided cells by shape (Theorem 5.1, cited from [Jen20, Theorem 4.33])
This is the main cell-level input for all of Section 5, directly used to identify the poset M(λ) and the modules in Lemma 5.13 and Theorem 5.14.
standard math Soergel categorification: existence of indecomposable objects kB_w, the character map ch, and the p-canonical basis properties (Theorem 2.2 and Proposition 2.4, imported from [EW16] and [JW17])
Background for the definition of the p-canonical basis and the non-negative structure coefficients.
standard math Classical Robinson-Schensted and Knuth theorems: Symmetry Theorem, Knuth equivalence, tableau descent sets, w0-multiplication rules (Theorems 5.5, 5.9, 5.10, Lemma 5.12)
Used in Proposition 5.6, Lemma 5.13, and the proof of cellularity to relate RS data with descent sets, star operations, and multiplication by w0.
domain assumption Demazure surjectivity of the Cartan realization (Assumption 2.1)
Technical condition for the diagrammatic Soergel category to behave correctly; automatic for simply-laced rank at least 2 and when 2 is invertible in k, and satisfied in the type A cases considered in Section 5.

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Pith. "Pith review of Cellularity of the p-Canonical Basis for Symmetric Groups." pith.science (2026). https://pith.science/paper/UF53QXCY

@misc{pith2026200911715,
  author       = {Pith},
  title        = {Pith review of: Cellularity of the p-Canonical Basis for Symmetric Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UF53QXCY}},
  note         = {Machine review of arXiv:2009.11715}
}

read the original abstract

For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder. Moreover, we show that left (or right) p-cells inside the same two-sided p-cell for Hecke algebras of finite crystallographic Coxeter systems are incomparable (Property A).

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Works this paper leans on

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Über Matrizen aus positiven Element en

Sitzungsber. Königl. Preuss. Akad. Wiss. (1908), pp. 471–476. [Fro09] G. Frobenius. “Über Matrizen aus positiven Element en”. In:

work page 1908

[2] [2]

Sitzungsber. Königl. Preuss. Akad. Wiss. (1909), pp. 514–518. [Ful97] W. Fulton. Young tableaux. Vol

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Why the Kazhdan-Lusztig basis of th e Hecke al- gebra of type A is cellular

IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2000, pp. 179–238. [Wil03] G. Williamson. “Why the Kazhdan-Lusztig basis of th e Hecke al- gebra of type A is cellular”. A vailable at http://people.mpim-bonn.mpg.de/geordie/Hecke.pdf. 10/2003. [Wil17] G. Williamson. “Schubert calculus and torsion expl osion”. In: J. Amer. Math. Soc. 30.4 (2017). ...

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CRM Monograph Series. American Mathematical Society, Prov - idence, RI, 2003, pp. vi+136. [Lus79b] G. Lusztig. “Unipotent representations of a ﬁnite Chevalley group of type E8”. In: Quart. J. Math. Oxford Ser. (2) 30.119 (1979), pp. 315–338. [Lus81] G. Lusztig. “On a theorem of Benson and Curtis”. In: J. Algebra 71.2 (1981), pp. 490–498. REFERENCES 25 [Lu...

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