Congruences in character tables of symmetric groups

Alexander R. Miller

Congruences in character tables of symmetric groups

T0 review · 0 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that subdividing every square of two equal-size Young diagrams by a d×d grid forces the corresponding symmetric-group character value to be divisible by d!.

desk verdict Genuinely new congruence for dilated character values, proved cleanly by a free S_d action on cascades; the one terse step checks out, so send it to review. read the letter →

arxiv 1908.03741 v2 pith:IMGK6C4A submitted 2019-08-10 math.CO math.RT

Alexander R. Miller This is my paper · ORCID

classification math.COmath.RT MSC 20C3005E10

keywords charactertablesymmetricgroupvaluesMurnaghan–NakayamaformularimhooktableauxcascadesYoungdiagramsdivisibility

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 establishes a congruence that holds for every pair of partitions: if λ and μ are Young diagrams with the same number of squares, and each square is blown up into a d×d block to form $\underline{\lambda}$ and $\underline{\mu}$, then the irreducible character value $\chi_{\underline{\lambda}}(\underline{\mu})$ of the symmetric group is a multiple of $d!$. This is uniform in both partitions, so it gives a stable way to produce divisible entries in character tables: once $d \geq p$, every such value vanishes modulo the prime $p$. The proof reorganizes the Murnaghan–Nakayama rim-hook expansion into a sum over new combinatorial objects called cascades, and exhibits an action of the symmetric group $S_d$ on them that has no fixed points and leaves each term's sign unchanged. Because the action is free, terms split into orbits of size $d!$, forcing the whole sum to be divisible by $d!$.

What carries the argument

The central object is a cascade: a binary matrix whose first row encodes the shape $\lambda$, each successive row is obtained from the previous one by swapping a single 0 with a right-lying 1, and whose last row is a block of 1s followed by 0s. A cascade encodes a rim-hook removal sequence; its weight is $(-1)^{\text{number of crossings}}$, and Lemma 4 identifies this weight with the sign of the permutation $\pi_C$ that the cascade induces on its lattice paths. The action in Theorem 2 is given by right multiplication by a block-diagonal matrix $\Phi(\sigma)$ with $d \times d$ permutation matrices, so it permutes the columns within each block of $d$. That action is free, weight-preserving, and compatible with the cascade-to-tableau bijection, which is exactly what forces the Murnaghan–Nakayama sum to be divisible by $d!$.

What would settle it

For $d=2$, take $\lambda=(2)$ and $\mu=(1,1)$; the theorem predicts that $\chi_{\underline{\lambda}}(\underline{\mu}) = \chi^{(4,4)}((2,2,2,2))$ is even. A direct Murnaghan–Nakayama computation, or a short character-table program, checking whether this integer is odd would settle Theorem 1: an odd value refutes it, while an even value is consistent.

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Extended reading notes

Core claim

The central discovery is that the congruence $\chi_{\underline{\lambda}}(\underline{\mu}) \equiv 0 \pmod{d!}$ follows from a symmetry of the rim-hook sums themselves. After translating rim-hook tableaux into cascades—matrices whose rows record successive swaps of a 0 with a right-lying 1—the paper shows that permuting columns in blocks of $d$, by multiplying on the right by a block-diagonal permutation matrix, sends cascades of shape $\underline{\lambda}$ and content $d.\mu$ (each part of $\mu$ multiplied by $d$) to cascades of the same shape and content. This $S_d$ action is free: no nontrivial permutation fixes a cascade, because the first $d$ columns of every allowed cascade contain pairwise distinct numbers of zeros. The action also preserves the weight of a cascade, which equals the sign of the permutation induced on its lattice paths, and the bijection between cascades and rim-hook tableaux carries the action over to tableaux. A free orbit of size $d!$ with equal weights makes the weighted sum, and therefore the character value, divisible by $d!$. A separate cascade argument gives the exact vanishing $\chi_{\underline{\lambda}}(d^2.\mu) = 0$ when the common size of $\lambda$ and $\mu$ is not divisible by $d$.

Load-bearing premise

The divisibility proof rests on the claim that, in every matrix used in the argument, the first $d$ columns each contain a different number of zeros; if two columns could ever match, the $S_d$ action might have fixed points and the orbit-counting argument would collapse.

Editorial extensions

If this is right

For any prime $p$, choosing $d \geq p$ in the main congruence makes $\chi_{\underline{\lambda}}(\underline{\mu})$ divisible by $p$ for every pair of equal-size partitions, so every row and column position indexed by subdivided shapes is $p$-divisible.
The more general congruence $\chi_{\underline{\lambda}}(d.\mu) \equiv 0 \pmod{d!}$ covers cases where only one side is subdivided and the other side has parts scaled by $d$; the equal-size statement is the special case where $\mu$ is itself subdivided.
When the common size $n$ of $\lambda$ and $\mu$ is not divisible by $d$, the character value $\chi_{\underline{\lambda}}(d^2.\mu)$ is exactly zero, a vanishing result stronger than divisibility.
The free $S_d$ action transfers to rim-hook tableaux, giving a weight-preserving symmetry of the objects appearing directly in the Murnaghan–Nakayama formula.

Reading between the lines

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

The same orbit-counting mechanism may yield congruences modulo products of factorials if the $d \times d$ subdivision is replaced by a rectangular $a \times b$ subdivision and the block permutation group is $S_a \times S_b$; the paper does not address this.
Because the divisibility is uniform in the pair of partitions, it suggests that the proportion of $d!$-divisible entries in character tables of $S_n$ tends to 1 as $n$ grows, extending known prime-specific density results; this is not claimed here.
The distinct-zero-counts condition behind freeness is the delicate step; identifying exactly which shapes and contents produce cascades with equal column zero-counts could generalize the method to other character values or other Coxeter groups.

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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 headline is simple: this paper proves a real, new congruence theorem. For any two partitions lambda and mu of the same positive integer, if you subdivide every square of both Young diagrams into d^2 squares, the resulting character value is divisible by d!. The paper also proves the adjoint statement for scaled cycle types and a clean vanishing result. None of these are in the earlier literature, and the proof does not lean on the author's prior results or on any fitted choices. It is a direct, self-contained argument from Murnaghan–Nakayama, so the math should age well.

What makes it work is a new combinatorial object, cascades, which translate rim hook tableaux into binary matrices and lattice paths. That gives a lattice-path version of Murnaghan–Nakayama and, more importantly, a weight-preserving action of S_d on the relevant cascades. When the action is free, the weighted sum in Murnaghan–Nakayama is a sum over orbits each contributing a multiple of d!, which is exactly the divisibility claim. The construction is explicit, the examples are helpful, and the paper is careful about the bookkeeping.

The soft spot is the freeness step in Theorem 2(ii). The paper asserts, in equation (4.9), that the zero-counts in the first d columns of any cascade are distinct, and says this follows from the cascade conditions and d-divisibility. That is terse, and it is the load-bearing point: without distinctness the action could have fixed points and the orbit argument would collapse. I checked the reconstruction, and the assertion is valid. Because the first row of the dilated shape is constant on blocks of d columns, each residue class evolves independently, and each transition can change a given residue class at most once. That makes the zero-counts exactly the transition indices, which are distinct. So the freeness is sound. The rest of the proof, including the conjugation identity for the induced permutation, checks out.

If I have any minor complaint, it is that the distinctness step deserves a sentence or two of elaboration for the reader, and the reproduced cascades in the example are dense enough to slow reading. Neither affects correctness.

Who is this for? People working on character values of symmetric groups, divisibility phenomena, and anyone who likes clean combinatorial proofs of known-but-explicit sums. It complements the recent probabilistic results of Peluse and Soundararajan without overlapping them: those concern almost all entries for a fixed prime, while this gives an explicit infinite family of p-divisible entries for every prime. I would cite it and would bring it to a reading group on character theory. A serious referee should engage with it; it deserves peer review, not desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper proves a striking divisibility theorem for character values of symmetric groups. If λ and μ are partitions of the same positive integer, and λ_d and μ_d are obtained by subdividing each square of their Young diagrams into d^2 congruent squares, then the character value χ_{λ_d}(μ_d) is divisible by d!. The proof is self-contained and combinatorial: the author translates rim hook tableaux into new objects called cascades, proves a cascade version of the Murnaghan–Nakayama formula (Proposition 1), and then constructs a free, weight-preserving action of S_d on the relevant cascades (Theorem 2). This action forces the character value to be a multiple of d!. The paper also proves a more general congruence χ_{λ_d}(d.μ) ≡ 0 mod d! for any partition μ of d n, and a vanishing result χ_{λ_d}(d^2.μ)=0 when n is not divisible by d.

Significance. If correct, this is an elegant and unexpected stability result in the character theory of symmetric groups: simultaneous d-refinement of both the shape and the cycle type makes every character value divisible by d!. In particular, for any prime p and any d≥p, χ_{λ_d}(μ_d) ≡ 0 mod p for all partitions λ, μ, complementing the recent probabilistic results of Peluse and of Peluse–Soundararajan with a deterministic statement that holds for every single entry in the refined table. The paper introduces cascades, a lattice-path analogue of Comét's binary notation, and a weight-preserving free group action that is likely to be of independent interest. The proof is genuinely parameter-free and derives the congruences directly from the Murnaghan–Nakayama formula, with no fitted coefficients and no reliance on the author's earlier results. The explicit example of an S_d-orbit in Section 4 makes the new action very concrete and verifiable.

minor comments (3)

[Section 4, Eq. (4.9)] The distinctness of the zero-counts z_i(C) for 1≤i≤d is a load-bearing step for the freeness of the action, and the current one-sentence justification is too compressed. I recommend adding the short argument: for each residue r∈{1,...,d}, the r-th first column starts at 0 and ends at 1, and it changes value only at a transition with a_i≡r mod d; since a_i<b_i and b_i is in the same residue class, the column can be the 0-side only when it equals a_i, so it flips exactly once, at the unique transition with a_i=r, and it can never flip back. Hence z_r(C) is exactly the row index of that transition, and these indices are distinct for distinct r because each transition has a unique residue class.
[Section 4, definition of L before (4.14)] The text sets L = dn + dℓ(λ), but the word w(λ_d) has length dλ1 + dℓ(λ) = d·l, not d(n+ℓ(λ)). As written, the permutation γ in (4.14) is defined on a larger index set than the actual columns of the cascade. The subsequent argument is unaffected because γ is only applied to genuine column indices and its restriction to those indices is a permutation, but the displayed definition of L should be corrected to L = dλ1 + dℓ(λ).
[Section 4, proof of (1.3)] The construction of C′ from the columns 1, d+1, 2d+1, ... and the deletion of redundant rows is stated in a single sentence. Since this is the crux of the vanishing statement, I suggest expanding it to explain why the resulting matrix is a cascade of shape λ and why its content is of the form d.μ′ for some partition μ′, which then yields the contradiction n = d|μ′| when n is not divisible by d.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the congruence is derived from the standard Murnaghan–Nakayama formula through a new free S_d action on cascades.

full rationale

The paper's derivation chain is self-contained rather than circular. Lemma 1 is an external standard fact credited to Comét, Lemma 2 proves a bijection between cascades and rim hook tableaux using that fact, Lemma 4 proves that cascade weight equals the sign of the associated permutation, and Proposition 1 is a direct reformulation of the standard Murnaghan–Nakayama formula (2.2). The central new ingredient, Theorem 2, defines a permutation action on cascades and proves it is an action, is free, and preserves weight. The freeness proof relies on the distinctness assertion (4.9), which is terse, but it is an internal proof obligation about zero-counts in cascade columns, not a consequence of the theorem being proved; even if it were incorrect, the failure would be a gap rather than circular reasoning. The final proof of Theorem 1 uses the orbit-size divisibility consequence of the free weight-preserving action, with no fitted parameters and no dependence on any prior result of the author for the main congruence. The self-citations in the introduction and historical discussion are contextual and are not load-bearing for the proof. Therefore there is no identifiable circular step.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper rests on standard theorems (Murnaghan-Nakayama, Comét's lemma) and internal definitions. There are no free parameters, fitted constants, or unsupported postulates.

assumptions (3)

standard math Murnaghan-Nakayama formula
Used in Eq (2.2) and Proposition 1 to express χ_λ(μ) as a weighted sum over rim hook tableaux.
standard math Comét's word bijection (Lemma 1)
Relates binary sequence swaps to rim hook removal; used to define cascades and prove Lemma 2.
standard math Orbit-counting for free actions
A free action of S_d partitions a set into orbits of size d!, so the signed sum is divisible by d!; used in the proof of Theorem 1.

invented entities (1)

Cascades
purpose: Binary matrix encoding of rim hook tableaux that admits a free weight-preserving S_d action.
Defined in Section 3, Definition 1. They are new combinatorial objects with no external falsifiable handle; their validity is established by the proofs that use them.

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Cite this review

Pith. "Pith review of Congruences in character tables of symmetric groups." pith.science (2026). https://pith.science/paper/IMGK6C4A

@misc{pith2026190803741,
  author       = {Pith},
  title        = {Pith review of: Congruences in character tables of symmetric groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMGK6C4A}},
  note         = {Machine review of arXiv:1908.03741}
}

abstract

If $\lambda$ and $\mu$ are two non-empty Young diagrams with the same number of squares, and $\boldsymbol\lambda$ and $\boldsymbol\mu$ are obtained by dividing each square into $d^2$ congruent squares, then the corresponding character value $\chi_{\boldsymbol\lambda}(\boldsymbol\mu)$ is divisible by $d!$.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Burnside, On an arithmetical theorem connected with roots o f unity, and its application to group-characteristics

W. Burnside, On an arithmetical theorem connected with roots o f unity, and its application to group-characteristics. Proc. Lond. Math. Soc. 1 (1904) 112–116

work page 1904

[2] [2]

Com´ et, Notations for partitions

S. Com´ et, Notations for partitions. Math. Tables Aids Comput. 9 (1955) 143–146

work page 1955

[3] [3]

Frobenius, ¨Uber die Charaktere der symmetrischen Gruppe

G. Frobenius, ¨Uber die Charaktere der symmetrischen Gruppe. Sitzungsbericht e der K¨ oniglich Preußischen Akademie der Wissenschaften zu Berlin, 1900 , 516–534

work page 1900

[4] [4]

P. X. Gallagher, Degrees, class sizes and divisors of character v alues. J. Group Theory 15 (2012) 455–467

work page 2012

[5] [5]

P. X. Gallagher, M. J. Larsen, and A. R. Miller, Many zeros of many characters of GL( n, q ). Int. Math. Res. Not. IMRN, in press

work page

[6] [6]

Gluck, Parity in columns of the character table of Sn

D. Gluck, Parity in columns of the character table of Sn. Proc. Amer. Math. Soc. 147 (2019) 1005–1011

work page 2019

[7] [7]

I. M. Isaacs, Character theory of ﬁnite groups . Dover, New York, 1994

work page 1994

[8] [8]

M. J. Larsen and A. R. Miller, The sparsity of character tables of high rank groups of Lie type. Preprint arXiv:2006.00847

work page Pith review arXiv 2006