{"id":"39d53e61-eb62-4770-a1e0-9e389c1133c1","arxiv_id":"2506.04007","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New conjugacy-class sum formulas enumerate double cosets and self-inverse double cosets in S_n, GL_n(F_q), and type B, yielding new OEIS terms and large polytope counts.","lead":"A mathematician found cleaner formulas for counting double cosets and their self-inverse versions in symmetric, general linear, and type B Coxeter groups. The formulas turn the count into sums over conjugacy classes, making large enumerations (like polytope symmetries) feasible.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's ACCEPT verdict rests primarily on Theorem 2.1, whose proof I re-derived line by line. The crucial identity in (2.9) holds because every g with g^2 ∈ C_i has |C_G(g^2)| = |G|/|C_i|, and the number of square roots is uniform on C_i; thus |{g : g^2 ∈ C_i}| = |C_i| · Sq(C_i) and the sum collapses exactly as claimed. The reduction to (2.10) is also correct, and the final sum over classes is immediate. The paper's supporting computation of Sq(C_i) for GL_n depends on Proposition 3.3, and the reader rightly notes that its proof is terse. However, the even-characteristic rule itself is correct and easily checked, so it does not undermine the central formula or any stated application beyond a minor expositional gap. Since I found no flaw in the main theorem and no unresolved risk to the paper's central enumerative claims, the appropriate outcome is to keep the reader's ACCEPT verdict unchanged.","tokens_in":16792,"tokens_out":21427,"duration_ms":184869,"concrete_test":"Verify Proposition 3.3 computationally for q = 2 and block sizes k = 1 through 8: construct J_k(ρ)^2 with ρ = 1, compute its Jordan normal form over F_2, and confirm the block sizes are ⌈k/2⌉ and ⌊k/2⌋; then recompute |Θ^{GL_4(F_2)}_{P_4}| using the paper's algorithm and compare with Table 5's entry 19.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.1 is the central claim, and its proof is sound. The double-counting in (2.8)–(2.10) correctly reduces the fixed-point character sum to (1/|H|) Σ_{h∈H} |{g : g^2 = h}|, and the passage to conjugacy classes is valid because the number of square roots is constant on each class C_i. Spot checks against S_3, C_4, and the trivial-subgroup edge case confirm the formula produces integer self-inverse double coset counts. The only previously flagged weakness, Proposition 3.3's even-characteristic Jordan-block rule, is a supporting lemma for GL_n square-root enumeration rather than a premise of Theorem 2.1. Its proof gap is expositional: in characteristic 2, J_k(ρ)^2 has ρ^2 on the diagonal and 1 on the second superdiagonal, so the nilpotent part is similar to J_{⌈k/2⌉}(0) ⊕ J_{⌊k/2⌋}(0), which is standard. No load-bearing concern about the central claim was identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a character-theoretic formula for the number of self-inverse double cosets in H\\G/H, namely |Theta^G_H| = (1/|H|) sum_i |H cap C_i| Sq(C_i), where C_i are the conjugacy classes of G and Sq(C_i) is the number of square roots of any element of C_i. The proof is based on Frame's double-coset formula and the Frobenius-Schur indicator, and the paper also shows equivalence with the character-multiplicity form. The remaining sections develop the auxiliary data needed to apply the formula: square-root counts and class-intersection counts for symmetric groups and general linear groups, including a Jordan-block rule for GL_n over even characteristic. These tools are then applied to cycles, polygons, Boolean functions, polytope vertex permutations, parabolic double cosets in symmetric groups and type-B Coxeter groups, and matrices up to row/column permutations or scalar multiplications. The paper also states a conjecture that |GL_lambda \\ GL_n / GL_mu| is a monic polynomial in q with positive integer coefficients.","tokens_in":16964,"tokens_out":49272,"duration_ms":443308,"significance":"If the results are correct, Theorem 2.1 is a clean and useful reformulation of the self-inverse double-coset count, and the paper gives a substantial menu of applications with explicit numerical tables. The derivation of the main theorem is self-contained and the equivalence with Frame's character formula is shown. The paper is also honest about the empirical status of Conjecture 4.10, which is checked only up to n=8. The main weakness is not in the central theorem but in one of the applied identities: Equation (4.10) in Section 4.3 is false as written, and the derivation of the subsequent sequence formula depends on it. Since the numerical sequence A068313 appears to be computed from the right-hand side of that false identity, the published values may be correct, but the displayed mathematics needs correction. The paper would also benefit from more detail in Proposition 3.3, on which the GL_n square-root algorithm relies.","major_comments":[{"comment":"The displayed identity is false. For n=3 and lambda=mu=(2,1), the left side sum over nu of K_{nu,(2,1)} K_{nu,(2,1)} equals 1^2 + 1^2 = 2, while the right side equals 1, which is also the actual number of (0,1)-matrices with row and column sums (2,1). The preceding sentence correctly says that (0,1)-matrices correspond to pairs of semistandard Young tableaux with conjugate shapes, so the left side should be sum_nu K_{nu,lambda} K_{nu',mu}. The same correction is needed in the left side of Eq. (4.11). The right-hand side of (4.11) appears to give the corrected total, so the table values may be unaffected, but the derivation as printed is invalid.","section":"4.3, Eq. (4.10)"},{"comment":"The rule for q odd needs to specify that irreducible factors are counted with multiplicity and that the partitions attached to all roots of phi mapping to the same factor must be merged. For example, over F_3, phi=X^2+1 is irreducible with roots i and -i, and both roots square to -1, so phi^2=(X+1)^2; therefore a class with f(phi)=(2) squares to a class with f^2(X+1)=(2,2), not (2). As written, 'f^2 maps each irreducible factor of phi^2 to lambda' is ambiguous and, if read as referring to distinct factors only, gives incorrect values for Sq(C_f) in the GL_n algorithm of Section 3.2.","section":"3.2, Proposition 3.3"}],"minor_comments":[{"comment":"The passage from Eq. (2.8) to Eq. (2.10) skips the intermediate identity |{g : g^2 in C_i}| = |C_i| * Sq(C_i); the equality is correct, but stating it explicitly would make the proof easier to follow.","section":"2, Proof of Theorem 2.1"},{"comment":"For the even-characteristic case, the displayed matrix is helpful but the similarity transformation to J_{ceil(k/2)}(rho^2) direct-sum J_{floor(k/2)}(rho^2) is not shown; a sentence or a reference to the standard nilpotent-block argument would strengthen the proof.","section":"3.2, Proposition 3.3"},{"comment":"The proof states that diagonal matrices with eigenvalue multiplicities lambda belong to (prod(q-i)) * (prod m_k!) conjugacy classes; the count should be (prod(q-i)) / (prod m_k!), as the displayed formula in the theorem requires.","section":"4.6, Proof of Theorem 4.8"},{"comment":"The layout of the figure caption and the two mapping displays is garbled in the manuscript: for the GL_7(F_3) example, the displayed 'square' mapping appears to have total degree 15 rather than 7. Please redraw the figure so that each mapping and its square are clearly separated.","section":"Figure 1"},{"comment":"The values for the icosahedron, dodecahedron, 24-cell, 600-cell and 120-cell are stated without any indication of which group and subgroup were used or how the computation was performed; a sentence describing the setup and the source of the numbers would improve reproducibility.","section":"4.2, Polytope examples"},{"comment":"The set I in the numerical example contains the duplicate entry s_4; it should be s_3, s_4, s_6, ... .","section":"4.4, Example after Proposition 4.6"}],"recommendation":"major_revision","confidential_remarks":"The central theorem and its proof are sound, and the paper is a useful contribution to the enumerative toolbox for double cosets. The main issue is the false identity in Eq. (4.10); it is likely a typo (a missing conjugate partition), but it is load-bearing for the derivation of the A068313 formula. I did not re-run the numerical computations, so I cannot independently confirm the large tables. Providing the Sage code as supplementary material would substantially increase confidence in the extensive numerical claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a solid, useful paper. The central formula (2.7) is not a brand-new theorem—it is equivalent to Frame's indicator sum—but the reformulation as a sum over H is genuinely convenient, and the paper explicitly proves the equivalence rather than just asserting it. The genuinely new computational content is in Section 3: the square-root count for GL_n(F_q) via Jordan block squaring, including the even-characteristic rule, and the cycle index computations for parabolic type-B subgroups. Those are the parts I would actually cite in my own work.\n\nWhat the paper does well: it gives explicit, usable formulas for |H∩C_i| and Sq(C_i) in symmetric groups, general linear groups, and type-B Coxeter groups, then applies them to produce new OEIS terms, counts of polytope symmetry classes, and a concrete conjecture (4.10) with evidence up to n=8. The applications are described in enough detail to reproduce, and the algorithms are implemented in Sage. No constants are fitted and no predictions are tuned to data; the empirical content is clearly labeled as conjecture.\n\nThe soft spots are real but minor. Proposition 3.3's even-characteristic Jordan-block rule is the biggest one: it is stated with a matrix display and a few sentences, not a full derivation. It is a standard fact, and it supports the GL_n square-root enumeration rather than Theorem 2.1, so the main argument does not depend on it. Proposition 4.4's involution count is also compressed—the orbit argument is plausible but not fully written out—and it is peripheral to the paper's main thread. The paper itself notes that there is no closed formula for Sq(C_f) in GL_n; the algorithm described is fine, but readers should know that the GL_n counts come from enumeration up to the relevant n, not from a single closed form.\n\nBottom line: the math is sound as far as I can see. The main theorem is proved cleanly, the equivalence with Frame's formula checks out, and the soft spots are local and fixable. This is not a paper that needs to be torn down; it needs a careful referee who knows Jordan normal form and cycle indices. I recommend sending it to a good combinatorics journal, with a request that the referee check the GL_n square-root algorithm and the type-B cycle index formulas. I would accept after minor revision.","headline":"A genuinely useful reformulation of Frame's double-coset count with new computational machinery for GL_n and type B; minor proof gaps, but the central theorem is sound and the paper deserves a serious referee.","tokens_in":17529,"tokens_out":1667,"would_cite":true,"duration_ms":17969,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A15","05E10","20C15","20E45","20G40","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any finite group G and subgroup H, the number of self-inverse double cosets in H\\G/H equals (1/|H|) times the sum over the conjugacy classes C_i of G of |H∩C_i|·Sq(C_i).","keywords":["double cosets","self-inverse double cosets","conjugacy classes","square roots in finite groups","cycle index","symmetric group","general linear group over finite field","Coxeter group type B"],"falsifier":"The central formula can be tested directly on a small group by enumerating all double cosets and marking the self-inverse ones, then comparing with the right-hand side of Theorem 2.1; any mismatch refutes it. The even-characteristic GL_n rule can be tested by computing $J_3(1)^2$ over $\\mathbb{F}_4$ and checking whether it really has Jordan blocks of sizes $2$ and $1$, or by brute-force square-root counts in $\\mathrm{GL}_4(\\mathbb{F}_2)$ against the algorithm's output.","tokens_in":16557,"feed_emoji":"🔢","tokens_out":10426,"duration_ms":103270,"temperature":0.7,"pith_summary":"Double cosets H\\G/H arise whenever one counts objects up to two-sided symmetry, and this paper focuses on the ones that are their own inverse. The main result expresses the number of such self-inverse double cosets as $\\frac{1}{|H|}\\sum_i |H\\cap C_i|\\,\\mathrm{Sq}(C_i)$, a sum over conjugacy classes of $G$ that needs only the class sizes inside $H$ and the number of square roots each class contains. This replaces a character-theoretic computation with data that can be assembled from cycle indexes and class data, and the paper shows how to assemble them for symmetric groups, general linear groups over finite fields, and type-B Coxeter groups. The paper demonstrates the method on cycles, polygons, Boolean functions, contingency matrices, and matrices over finite fields.","feed_headline":"Class-sum formula counts self-inverse double cosets","feed_subtitle":"The count becomes one sum over conjugacy classes, computable for symmetric and general linear groups.","key_machinery":"The machinery is the square-root counting function $\\mathrm{Sq}(C_i)$ together with class-intersection data $|H\\cap C_i|$. For the symmetric group, conjugacy classes are cycle types $\\lambda$ and the paper gives the explicit formula $\\mathrm{Sq}(C_\\lambda)$ from Equation (3.2), while $|H\\cap C_\\lambda|$ is encoded in the cycle index $Z_H(x_1,x_2,\\dots)$. For $\\mathrm{GL}_n(\\mathbb{F}_q)$, conjugacy classes are parametrized by functions $f$ from irreducible monic polynomials to partitions; the centralizer size $z_f$ gives $|C_f|$, and the square of a class is computed by Proposition 3.3, which says a Jordan block $J_k(\\rho)$ under squaring becomes $J_k(\\rho^2)$ if $q$ is odd and splits into two blocks of sizes $\\lceil k/2\\rceil$ and $\\lfloor k/2\\rfloor$ if $q$ is even. These pieces plug directly into Theorem 2.1 and Formula (2.13), turning abstract character data into computable enumerations.","core_discovery":"The central claim is Theorem 2.1: if $G$ is a finite group with conjugacy classes $C_1,\\dots,C_r$ and $H$ is a subgroup, the number $|\\Theta^G_H|$ of self-inverse double cosets in $H\\backslash G/H$ is $$\\frac{1}{|H|}\\sum_{i=1}^r |H\\cap C_i|\\cdot \\mathrm{Sq}(C_i),$$ where $\\mathrm{Sq}(C_i)$ is the number of square roots in $G$ of any element of $C_i$. The proof passes through the identity $|\\Theta^G_H| = \\frac{1}{|H|}\\sum_{h\\in H} |\\{g\\in G : g^2=h\\}|$, so the count is literally a sum over elements of $H$ of square-root counts. The paper also gives the companion formula $|H_1\\backslash G/H_2| = \\frac{1}{|H_1||H_2|}\\sum_i \\frac{|G|}{|C_i|}|H_1\\cap C_i||H_2\\cap C_i|$, and then computes the three ingredients $|C_i|$, $\\mathrm{Sq}(C_i)$, and $|H\\cap C_i|$ for $S_n$ and $\\mathrm{GL}_n(\\mathbb{F}_q)$. The result is a computational route for enumerating self-inverse double cosets from class data alone.","pith_inferences":["Because the proof identifies self-inverse double cosets as exactly those containing an element whose square lies in $H$, one could search for them in large groups by backtracking over elements $g$ with $g^2\\in H$, without enumerating conjugacy classes.","The same sum-over-classes template may apply to other families with known centralizer structures, such as other finite Coxeter groups or classical groups, where a splitting rule analogous to Proposition 3.3 would supply $\\mathrm{Sq}(C_i)$.","The polynomiality conjecture, if true, suggests a hidden algebraic or geometric meaning for these double-coset counts, for instance as Poincaré-type polynomials or Hall-type polynomials; the paper does not pursue that interpretation.","A testable extension is to seek a closed form for $\\mathrm{Sq}(C_f)$ for $\\mathrm{GL}_n(\\mathbb{F}_q)$ analogous to the symmetric-group formula (3.2), with Proposition 3.3 serving as the core of such a formula."],"forward_implications":["For any finite group where conjugacy classes and square-root counts are known, Theorem 2.1 gives $|\\Theta^G_H|$ from $|H\\cap C_i|$ alone, without computing irreducible characters or Frobenius-Schur indicators.","In the symmetric group, the formula yields the generating function (3.3) for the sum of character table entries, and new values of sequences for parabolic double cosets without computing Kostka numbers.","For cycles and polygons, Propositions 4.1 and 4.2 count self-inverse objects up to $Z_n$ and $D_n$ symmetry, and Proposition 4.3 gives the doubling relation $|\\Theta^{S_n}_{Z_n}| = 2|\\Theta^{S_n}_{D_n}|$ when $n\\equiv 3 \\pmod 4$.","For finite fields, the algorithm computes $|P_n\\backslash \\mathrm{GL}_n(\\mathbb{F}_2)/P_n|$ and $|\\Theta^{\\mathrm{GL}_n(\\mathbb{F}_2)}_{P_n}|$ up to $n=9$, and the conjectured polynomiality of $|\\mathrm{GL}_\\lambda\\backslash\\mathrm{GL}_n(\\mathbb{F}_q)/\\mathrm{GL}_\\mu|$ is verified by examples such as $q^4+7q^3+32q^2+89q+117$."],"supporting_citations":[{"why":"Supplies the character formula $|\\Theta^G_H|=\\sum_i c_i\\mu_i^H$ and the trace formula (2.6) that Theorem 2.1 rewrites.","marker":"[5]"},{"why":"Supplies the identity $\\mathrm{Sq}=\\sum_i c_i\\chi_i$ connecting square-root counts to characters, used to show equivalence with the earlier formula.","marker":"[9]"},{"why":"Supplies the centralizer-size formula $z_f=|G|/|C_f|$ for conjugacy classes of $\\mathrm{GL}_n(\\mathbb{F}_q)$, a key computation step.","marker":"[10]"},{"why":"Supplies the classification of conjugacy classes of $\\mathrm{M}_n(\\mathbb{F}_q)$ and $\\mathrm{GL}_n(\\mathbb{F}_q)$ by $\\mathbb{F}_q[X]$-module decompositions, on which the GL_n parametrization rests.","marker":"[12]"},{"why":"Supplies the bijection between parabolic double cosets in $S_n$ and contingency matrices with prescribed row and column sums, one of the paper's main applications.","marker":"[3]"}],"fun_headline_variants":["Self-inverse double coset count from one class sum","Double cosets: self-inverse tally via square roots","Class-sum formula for self-inverse double cosets","Counting self-inverse double cosets with conjugacy sums","Square-root sums enumerate self-inverse double cosets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a linear-algebra fact about matrices over fields with $q$ even: squaring a $k\\times k$ Jordan block produces two Jordan blocks of sizes $\\lceil k/2\\rceil$ and $\\lfloor k/2\\rfloor$; the general-linear-group square-root algorithm and all its applications depend on this fact.","fun_headline_variants_meta":{"raw":{"variants":["Self-inverse double coset count from one class sum","Double cosets: self-inverse tally via square roots","Class-sum formula for self-inverse double cosets","Counting self-inverse double cosets with conjugacy sums","Square-root sums enumerate self-inverse double cosets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000531,"raw_usage":{"total_tokens":2558,"prompt_tokens":948,"completion_tokens":1610,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":1531}},"tokens_in":564,"tokens_out":1610,"duration_ms":10375,"temperature":1.0,"reasoning_tokens":1531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:48:37.175068+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The central formula can be tested directly on a small group by enumerating all double cosets and marking the self-inverse ones, then comparing with the right-hand side of Theorem 2.1; any mismatch refutes it. The even-characteristic GL_n rule can be tested by computing $J_3(1)^2$ over $\\mathbb{F}_4$ and checking whether it really has Jordan blocks of sizes $2$ and $1$, or by brute-force square-root counts in $\\mathrm{GL}_4(\\mathbb{F}_2)$ against the algorithm's output.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the character formula $|\\Theta^G_H|=\\sum_i c_i\\mu_i^H$ and the trace formula (2.6) that Theorem 2.1 rewrites."},{"cited_title":"Academic Press, 1976","cited_arxiv_id":null,"evidence_quote":"Supplies the identity $\\mathrm{Sq}=\\sum_i c_i\\chi_i$ connecting square-root counts to characters, used to show equivalence with the earlier formula."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the centralizer-size formula $z_f=|G|/|C_f|$ for conjugacy classes of $\\mathrm{GL}_n(\\mathbb{F}_q)$, a key computation step."},{"cited_title":"Macdonald.Symmetric functions and Hall polynomials","cited_arxiv_id":null,"evidence_quote":"Supplies the classification of conjugacy classes of $\\mathrm{M}_n(\\mathbb{F}_q)$ and $\\mathrm{GL}_n(\\mathbb{F}_q)$ by $\\mathbb{F}_q[X]$-module decompositions, on which the GL_n parametrization rests."},{"cited_title":"Statistical enumeration of groups by double cosets.Journal of Algebra, 607A:214–246, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the bijection between parabolic double cosets in $S_n$ and contingency matrices with prescribed row and column sums, one of the paper's main applications."}],"review_version":1}