On the enumeration of double cosets and self-inverse double cosets

Ludovic Schwob

REVIEW 2 major objections 6 minor 17 references

On the enumeration of double cosets and self-inverse double cosets

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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).

desk verdict 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. read the letter →

arxiv 2506.04007 v1 pith:3E6SADJI submitted 2025-06-04 math.CO math.GR

Ludovic Schwob This is my paper

classification math.COmath.GR MSC 05A1505E1020C1520E4520G4020F55

keywords doublecosetsself-inverseconjugacyclassessquarerootsinfinitegroupscycleindexsymmetricgroupgenerallinearoverfieldCoxetertypeB

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $q^4+7q^3+32q^2+89q+117$.

Reading between the lines

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

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.

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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

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.

What 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.

The 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.

Bottom 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.

Referee Report

2 major / 6 minor

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.

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 (2)

[4.3, Eq. (4.10)] 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.
[3.2, Proposition 3.3] 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.

minor comments (6)

[2, Proof of Theorem 2.1] 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.
[3.2, Proposition 3.3] 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.
[4.6, Proof of Theorem 4.8] 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.
[Figure 1] 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.
[4.2, Polytope examples] 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.
[4.4, Example after Proposition 4.6] The set I in the numerical example contains the duplicate entry s_4; it should be s_3, s_4, s_6, ... .

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2.1 is proved from Frame's classical formula by explicit double counting and is shown equivalent to the character-theoretic expression; no fitted inputs or load-bearing self-citations appear.

full rationale

The paper's central claim, Theorem 2.1, is derived directly from Frame's fixed-point formula (2.6) by a transparent double-counting argument in (2.8)-(2.10), and the paper then explicitly verifies equivalence with the character-theoretic formula (2.3) via (2.11). No parameter is fitted to data and no input quantity is defined in terms of the target enumeration: Sq(C_i) is the standard number of square roots, and |H ∩ C_i| is computed from cycle-index data. The auxiliary Proposition 3.3 on Jordan blocks in even characteristic is a supporting computational lemma, not a premise of Theorem 2.1, and it is argued from matrix structure rather than assumed. Conjecture 4.10 is explicitly labeled a conjecture checked only up to n=8, not a derived prediction. The paper contains no self-citations, no imported uniqueness theorem, and no renamed known result used as if it were new support. The duplication of Theorem 1.1 and Theorem 2.1 is a presentation artifact only. Accordingly, the derivation chain is self-contained and none of the identified circularity patterns applies.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the only unproved inputs are the classical Frame formula and the Jordan block squaring fact. The conjecture about polynomiality is an empirical observation, not an assumption.

assumptions (4)

standard math Frame's formula (2.6): |Θ^G_H| = 1/|G| Σ_{g∈G} χ(g^2) where χ is the trace on G/H
Invoked in the proof of Theorem 2.1, from reference [5].
standard math Character decomposition of the permutation representation R_H (Eqs. 2.1-2.3) and the Frobenius-Schur indicator identity (Eqs. 2.4-2.5)
Used to show equivalence of (2.3) and (2.7).
domain assumption Squaring rule for Jordan blocks over even characteristic (Prop. 3.3): for q even, J_k(ρ)^2 splits as two Jordan blocks of sizes ceil(k/2) and floor(k/2)
Load-bearing for the GL_n(F_q) computation of Sq(C_f); the proof is only a displayed matrix, not a full derivation.
standard math Cycle index formulas for Z_n, D_n, S_λ, GL_λ(F_q), and W_I in type B (Eqs. 3.6-3.8, 3.16, 4.15)
Standard Pólya theory applied to the relevant permutation groups; used without proof.

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Pith. "Pith review of On the enumeration of double cosets and self-inverse double cosets." pith.science (2026). https://pith.science/paper/3E6SADJI

@misc{pith2026250604007,
  author       = {Pith},
  title        = {Pith review of: On the enumeration of double cosets and self-inverse double cosets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3E6SADJI}},
  note         = {Machine review of arXiv:2506.04007}
}

abstract

Double cosets appear in many contexts in combinatorics, for example in the enumeration of certain objects up to symmetries. Double cosets in a quotient of the form $H\backslash G / H$ have an inverse, and can be their own inverse. In this paper we present various formulas enumerating double cosets, and in particular self-inverse double cosets. We study double cosets in classical groups, especially the symmetric groups and the general linear groups, explaining how to obtain the informations on their conjugacy classes required to apply our formulas. We also consider double cosets of parabolic subgroups of Coxeter groups of type B.

Figures

Figures reproduced from arXiv: 2506.04007 by the authors.

**Figure 1.** Two mappings with their respective squares. The mapping f corresponds to a conjugacy class in GL14(F2), and g to a conjugacy class in GL7(F3). Similarly to the case of the symmetric group, the values of |H ∩ Cf | for H < GLn(Fq) and all Cf can be summarized in a cycle index (3.14) ZH((xϕ,k)ϕ∈Φq,k≥1) = 1 |H| X f |H ∩ Cf | · xf , where xf = Y ϕ∈Φq Y k∈f(ϕ) xϕ,k. If H1 < GLn(Fq) and H2 < GLm(Fq), their product H1 × H2 … view at source ↗

**Figure 2.** The cycle Cyc(513264) up to rotation and its inverse in Zn\Sn/Zn. The following results will be given without proof, since they are direct applications of Theorem 2.1 and Equations (3.6) and (3.7). Proposition 4.1. The number of self-inverse double cosets in Zn\Sn/Zn is (4.1) |Θ Sn Zn | = 1 n X d|n φ(d) · Sq(C[d n/d] ). n 2 3 4 5 6 7 8 9 10 11 12 |Zn\Sn/Zn| 1 2 3 8 24 108 640 4492 36336 329900 3326788 |Θ Sn Zn | 1 2… view at source ↗

**Figure 3.** Hexagons up to rotation and reflection. The ones without an arrow are self-inverse. We notice the following property, which follows from the fact that, when n ≡ 3 mod 4, reflections in Dn are odd permutations. Proposition 4.3. We have |Θ Sn Zn | = 2 · |Θ Sn Dn | when n ≡ 3 mod 4. Proof. Each self-inverse double coset in Zn\Sn/Zn is contained in a self-inverse double coset in Dn\Sn/Dn. Conversely, each double coset … view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Permutations of the vertices of a cube, up to isometry. The ones without an arrow are self-inverse. T(σ) on vertices {1, 2, ..., 2n} such that for all 1 ≤ i ≤ n, an edge connects vertices 2i − 1 and 2i, and another edge connects vertices σ(2i − 1) and σ(2i). Let λ be t…

**Figure 5.** Figure 5: ). These graphs have either 0 or 2 fixed points, i.e., vertices whose both coordinates are equal. There are k!/2 such graphs having 2 fixed points, and the number of those with no fixed point is equal to (k − 1)!/2 if k is odd and 0 otherwise. Since every orbit contain…

**Figure 6.** Figure 6: With n = 9 and I = {s2, s5, s6, s8, s9}, we get k = 2 and κ = (3, 2, 1, 1). We can use Proposition 4.6 to obtain the values of |WI∩Cλ,µ|, and efficiently compute |WI\Bn/WJ | and ΘBn WI . For example, by taking n = 42 and I = {s3, s4, s4, s6, s7, s8, s9, s12, s13, s15, …

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Reference graph

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Reviewed August 7, 2026 · model on record in the stance chip above.

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How large is the character degree sum compared to the character table sum for a finite group?

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Goulden and David M

Ian P. Goulden and David M. Jackson. Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials.Canadian Journal of Mathematics, 48:569–584, 1996

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