{"id":"3013cf98-17de-49d3-8efd-a3e2e331ea75","arxiv_id":"1908.03741","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For partitions λ and μ of the same integer, replacing each square by d^2 squares makes the character value χ_{λ̲}(μ̲) divisible by d!.","lead":"This paper proves that dilating two Young diagrams by the same factor forces the corresponding symmetric group character value to be divisible by the factorial of that factor. The result gives a clean new family of forced divisibility congruences in character tables of symmetric groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the terse distinctness claim (4.9) underlying freeness is valid, and the central congruence theorem holds up under scrutiny.","rationale":"The reader identified the distinctness of zero-counts in (4.9) as the weakest assumption, and that is indeed the most compressed step in the proof of Theorem 2(ii). I reconstructed the missing argument using the residue-class decomposition modulo d and found that the assertion is correct: the first column in each residue class flips from 0 to 1 exactly once, at a transition whose index is unique to that residue, so the zero-counts for the first d columns are distinct. Consequently, the S_d action is free and weight-preserving, and the Murnaghan-Nakayama sum in Proposition 1 is divisible by d!. The remaining steps of the proof (the cascade-to-rim-hook-tableau bijection, the sign formula wt(C) = sgn(pi_C), the block-column action, and the reductions leading to (1.1)-(1.3)) are standard or verified by the text's own derivations. No counterexample or hidden assumption emerged. I therefore agree with the reader's ACCEPT verdict and see no basis to change it.","tokens_in":9238,"tokens_out":14347,"duration_ms":152305,"concrete_test":"Write a brute-force enumeration of cascades for small parameters (e.g. d = 2,3; lambda = (2), (3,1), (2,2); alpha any positive d-divisible sequence summing to d^2|lambda|) and check directly that (4.9) holds for every cascade and that the S_d action defined by (4.1) has orbit size d! with constant weight; this settles whether freeness and weight-preservation actually hold in all allowed cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is Theorem 2(ii), where freeness of the S_d action on C(lambda_d, alpha) rests on the assertion (4.9) that the zero-counts z_i(C) for the first d columns are distinct. This is genuinely the least expanded step in the proof: the paper states it follows from the cascade conditions and d-divisibility, without giving the detailed argument. A careful reconstruction, however, confirms the assertion. In any cascade C of shape lambda_d, the first row is the binary word w(lambda_d), which is constant on blocks of d columns; taking columns congruent to a fixed residue r modulo d yields a copy of w(lambda) for each r in {1,...,d}. Since every transition swaps a 0 and a 1 in columns a_i,b_i with a_i congruent to b_i modulo d, each residue-class subsequence evolves independently. The r-th first column (column r of C) begins with 0 and ends with 1, and changes only at the unique row transition i with a_i = r; it can never change back because b_i = r would require t_i = 0 with s_i < t_i, impossible. Therefore z_r(C) is exactly that transition index, and these indices are distinct for distinct r because each transition has a unique residue class. Equation (4.9) thus holds, the free action is well-founded, and the divisibility argument in Theorem 1 is sound. The other components (Lemma 2 bijection, Lemma 4 sign identity, the block-column action, and weight preservation via conjugation) are internally consistent and are backed by standard facts. I find no load-bearing gap in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9552,"tokens_out":9373,"duration_ms":100031,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Section 4, Eq. (4.9)"},{"comment":"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":"Section 4, definition of L before (4.14)"},{"comment":"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.","section":"Section 4, proof of (1.3)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is well within the scope of math.CO. The compressed step (4.9) is valid, as is the overall proof, but the terseness of this step and the typo in the definition of L should be corrected before publication. I do not see any issue with novelty or attribution; the citations to the relevant literature appear appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nIf 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.\n\nWho 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.","headline":"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.","tokens_in":10060,"tokens_out":1367,"would_cite":true,"duration_ms":18248,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C30","05E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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!.","keywords":["character table","symmetric group","character values","Murnaghan–Nakayama formula","rim hook tableaux","cascades","Young diagrams","divisibility"],"falsifier":"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.","tokens_in":9042,"feed_emoji":"📐","tokens_out":13435,"duration_ms":127760,"temperature":0.7,"pith_summary":"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 $d!$.","feed_headline":"d! divides character values of subdivided Young diagrams","feed_subtitle":"For any prime p, choosing d at least p makes every such character value vanish modulo p.","key_machinery":"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 $d!$.","core_discovery":"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 $d$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"States the Murnaghan–Nakayama formula expressing $\\chi_\\lambda(\\mu)$ as a weighted sum over rim-hook tableaux, the starting point of the proof.","marker":"[15]"},{"why":"Stated alongside [15], supplies the rim-hook removal rule used to compute character values.","marker":"[16]"},{"why":"Supplies the classical bijection between binary sequences and rim-hook removals that underlies the cascade construction in Lemma 1.","marker":"[2]"}],"fun_headline_variants":["Subdividing Young diagrams by d^2 forces character values divisible by d!","Free S_d action on cascades proves d! divides character values","Character table congruences: subdividing boxes yields d! divisibility","d! divides character values of scaled Young diagrams – new proof","From rim-hook tableaux to cascades: a free S_d orbit explains mod d! zero"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Subdividing Young diagrams by d^2 forces character values divisible by d!","Free S_d action on cascades proves d! divides character values","Character table congruences: subdividing boxes yields d! divisibility","d! divides character values of scaled Young diagrams – new proof","From rim-hook tableaux to cascades: a free S_d orbit explains mod d! zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1210,"prompt_tokens":856,"completion_tokens":354,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":253}},"tokens_in":472,"tokens_out":354,"duration_ms":3983,"temperature":1.0,"reasoning_tokens":253,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:03:38.947151+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the Murnaghan–Nakayama formula expressing $\\chi_\\lambda(\\mu)$ as a weighted sum over rim-hook tableaux, the starting point of the proof."},{"cited_title":"Nakayama, On some modular properties of the irreducible rep resentations of a symmetric group I, II","cited_arxiv_id":null,"evidence_quote":"Stated alongside [15], supplies the rim-hook removal rule used to compute character values."},{"cited_title":"Com´ et, Notations for partitions","cited_arxiv_id":null,"evidence_quote":"Supplies the classical bijection between binary sequences and rim-hook removals that underlies the cascade construction in Lemma 1."}],"review_version":1}