A central limit theorem for the two-sided descent statistic on Coxeter groups

Benjamin Br\"uck; Frank R\"ottger

REVIEW 3 major objections 4 minor 17 references

A central limit theorem for the two-sided descent statistic on Coxeter groups

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For well-behaved finite Coxeter groups, the two-sided descent statistic satisfies the central limit theorem exactly when its variance diverges.

desk verdict Conditional CLT with a real hole: the well-behaved hypothesis is non-vacuous, and a factor-4 variance typo needs fixing, but the core proof architecture is sound and worth refereeing. read the letter →

arxiv 1908.07955 v2 pith:BRBIPDVR submitted 2019-08-21 math.CO math.GRmath.PR

Benjamin Brück , Frank Röttger This is my paper · ORCID

classification math.COmath.GRmath.PR MSC 05E1520F5560F0505A15

keywords Coxetergroupsdescentstatisticcentrallimittheoremtwo-sidedEulerianpolynomialdihedralLindebergconditionwell-behavedsequences

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

This paper establishes conditions under which the two-sided descent statistic—the number of descents of a random element of a finite Coxeter group plus the number of descents of its inverse—is asymptotically normal. For any "well-behaved" sequence of finite Coxeter groups whose ranks grow, the paper proves that the centred and scaled statistic converges to the standard Gaussian distribution exactly when its variance tends to infinity. Equivalently, normality holds precisely when the total rank of the non-dihedral irreducible components together with the sum of reciprocal dihedral parameters diverges. The result answers a question left open in earlier work on descents and extends previously known cases for symmetric and signed-permutation groups to arbitrary products of irreducible Coxeter groups.

What carries the argument

One central object is the two-sided Eulerian polynomial, the generating function $\sum_{w\in W} x^{\operatorname{des}(w)+\operatorname{des}(w^{-1})}$, which lacks the real-rooted factorization that makes the one-sided descent statistic easy to handle. To replace that structure, the paper derives recursive conditional-expectation formulas for the joint distribution of $(\operatorname{des}(w),\operatorname{des}(w^{-1}))$ in types $A_n$ and $B_n$, solves the recursions for fourth moments, and bounds the type-$D_n$ moment by comparison with $B_n$. Around this, the paper builds a triangular-array argument: the variance of $T_n$ adds over irreducible components, $\delta$-small components are separated from the rest, and a uniform tail-decay condition, called well-behaved, makes the Lindeberg condition hold.

What would settle it

Build a sequence of finite Coxeter groups with $\operatorname{rk}(W_n)\to\infty$ and $\operatorname{V}(T_n)\to\infty$ that fails the well-behaved tail condition (6.1)—for instance, by letting the ranks of small components grow just slowly enough to defeat uniform tail decay—and check whether the normalized $T_n$ still converges to $N(0,1)$; if it does not, Theorem 6.5 fails without the hypothesis.

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Extended reading notes

Core claim

The central claim is Theorem 6.5: for a well-behaved sequence $(W_n)$ of finite Coxeter groups with $\operatorname{rk}(W_n)\to\infty$, letting $T_n$ be the statistic $\operatorname{des}(w)+\operatorname{des}(w^{-1})$ on a uniform random element $w\in W_n$, the following are equivalent: $(T_n)$ satisfies the CLT; $\operatorname{V}(T_n)\to\infty$; and $\operatorname{rk}(G_n)+\sum_{i=1}^{\ell_n}1/m_{n,i}\to\infty$, where $G_n$ collects the non-dihedral irreducible components and $I_2(m_{n,i})$ are the dihedral factors. The paper proves the equivalence by splitting $T_n$ into independent per-component contributions, applying Lindeberg's theorem for triangular arrays through a weighted-sum criterion, and using fourth-moment bounds of order $n^2$ for the irreducible types $A_n$, $B_n$, and $D_n$.

Load-bearing premise

The equivalence is conditional on the sequence being "well-behaved"—a uniform decay condition on the tail variances of small irreducible components—and the paper does not prove that every sequence of Coxeter groups satisfies this condition.

Editorial extensions

If this is right

For any well-behaved sequence of finite Coxeter groups with growing rank, the two-sided descent statistic satisfies the CLT if and only if its variance diverges.
Equivalently, normality is controlled by the explicit quantity $\operatorname{rk}(G_n)+\sum_i 1/m_{n,i}$: once either the non-dihedral rank grows or the reciprocal dihedral parameters accumulate, the law of $T_n$ is Gaussian in the large-$n$ limit.
Sequences of products of dihedral groups obey the CLT exactly when $\sum_i 1/m_{n,i}\to\infty$, as in the harmonic-series example where the $n$-th group is the product of the first $n$ dihedral groups.
If the dihedral contributions do not accumulate, the variance stays bounded and no Gaussian limit is possible, so the CLT fails.

Reading between the lines

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

The authors' inability to construct a non-well-behaved sequence suggests the "well-behaved" hypothesis may be redundant; if so, Theorem 6.5 would hold for every sequence of finite Coxeter groups with growing rank, making the CLT criterion fully general.
The recursive conditional-expectation method for fourth moments is not tied to type $A$ or $B$; it could supply moment bounds for other Coxeter-invariant statistics or for higher moments, turning the CLT criterion into a finite-moment check.
Because $t$ counts geodesic-neighbour facets in the two-sided Coxeter complex, the theorem can be read probabilistically as a statement about the asymptotic size of the ball of radius one around a random chamber; the same variance condition might govern other local statistics on the complex.
The equivalence between CLT and variance divergence may hold more broadly for descent-like statistics on combinatorial groups, with the well-behaved condition serving as a technical bridge rather than a genuine restriction.

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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

Here is my take on arXiv:1908.07955. The new content is real: the weighted-sum CLT lemmas in Section 4 and the decomposition into dihedral/non-dihedral parts, plus the fourth-moment computations, go beyond Chatterjee–Diaconis, Özdemir, and Röttger. Theorem 6.5's equivalence between CLT, variance divergence, and rk(G_n)+Σ1/m_i diverging is a substantial step toward Kahle–Stump's question. The proof structure—split into irreducible components, check Lindeberg and fourth-moment bounds—is appropriate, and the honesty about using external CLTs from the literature is fine.

The soft spots are real but localized. First, the 'well-behaved' hypothesis is not as benign as the authors think. The stress-test construction W_R = ∏_{i=1}^R A_{⌊R/(i log R)⌋} violates Eq. (6.1) for every δ>0: the non-δ-small part has variance ∼δR/6, and the tail past any fixed k still ∼δR/6, so the ratio in (6.1) tends to 1, not 0. Yet this sequence satisfies the CLT. So the remark after Definition 6.2 claiming no non-well-behaved sequence exists is simply wrong, and Theorem 6.5 is only a conditional answer to Kahle–Stump. This doesn't invalidate the theorem, but it should be thoroughly qualified in revision.

Second, there is a factor-4 inconsistency between Theorem 2.3(2), which gives V(T)=Σ1/m_i for dihedral products, and Lemma 5.2's proof, which uses V(X_{n,i})=4/m_{n,i}. I suspect a typo, and the proof would survive with either constant, but it needs fixing.

Third, the fourth-moment formulas in Section 3 rest on unshown Sage/Mathematica output. Not a fatal flaw—the recursions are there—but a referee should ask for the scripts or at least the starting values to be documented.

Citations look appropriate; the self-citation to Röttger is a black-box use of a prior result, not self-serving.

Bottom line: this deserves a serious referee. I would accept it conditionally, asking for the well-behaved claim to be corrected, the variance typo fixed, and the moment computations made reproducible. The core architecture is sound and the result is a genuine advance.

Referee Report

3 major / 4 minor

Summary. The paper studies the statistic t(w)=des(w)+des(w^{-1}) on finite Coxeter groups and proves a central limit theorem for the associated random variable T_W under an additional 'well-behaved' hypothesis on the sequence of Coxeter groups. The main result, Theorem 6.5, states that for a well-behaved sequence of finite Coxeter groups with rank tending to infinity, the normalized T_n converges to N(0,1) if and only if its variance diverges, which is further equivalent to rk(G_n)+Σ 1/m_{n,i} → ∞, where G_n is the non-dihedral part and m_{n,i} are the dihedral parameters. The proof decomposes T_n into independent contributions from irreducible components, uses Lindeberg's theorem, and derives fourth-moment estimates for types A, B, and D via recursive conditional-expectation formulas solved with Sage and Mathematica.

Significance. If the result is correct, it provides a positive but conditional answer to a question of Kahle-Stump and generalizes prior CLTs for the two-sided descent statistic on symmetric groups and on irreducible types A, B, and D. The proof architecture is sensible, and the paper contains useful explicit mixed-moment computations in the appendix. However, the significance is limited by three issues: an internal factor-4 inconsistency in the variance formula for dihedral groups, a concrete natural sequence that violates the well-behaved hypothesis while still satisfying the CLT, and a reliance on unverified computer-algebra derivations for load-bearing fourth-moment bounds. These issues do not necessarily invalidate the theorem as stated, but they substantially weaken the claim that the paper answers Kahle-Stump's question in full generality.

major comments (3)

[Section 2.3, Theorem 2.3(2) and Lemma 5.2] Theorem 2.3(2) states that for a product of dihedral groups W=∏ I_2(m_i), the variance of T_W is Σ 1/m_i. However, Lemma 5.2 uses V(X_{n,i}) = 4/m_{n,i}, invoking Theorem 2.3 for this formula. A direct computation for I_2(m) gives V(T) = 4/m: the values of T are 0 for the identity, 2 for all non-identity elements except the longest element, and 4 for the longest element, yielding variance (4+4)/(2m) = 4/m. Thus the statement of Theorem 2.3(2) is incorrect and should be corrected to Σ 4/m_i. This factor affects the quantitative statement of Lemma 5.2 and the proof of Theorem 6.5, though the equivalence between variance divergence and rk(G_n)+Σ1/m_{n,i} → ∞ is preserved because of the constant factor 4.
[Section 6, Definition 6.2 and the remark after it] The remark after Definition 6.2 asserts that the authors failed to construct a sequence that is not well-behaved. This is contradicted by an explicit example. Let R→∞ and set W_R = ∏_{i=1}^R A_{⌊R/(i log R)⌋}. Then rk(W_R) ∼ R. For any fixed δ>0, an irreducible component is non-δ-small precisely when i ≲ R^δ/log R, so m_R ∼ R^δ/log R. The variance of the non-small part satisfies V(T_{M_R^δ}) ∼ (1/6)Σ_{i≤m_R} r_i ∼ δ R/6. For any fixed k, the tail Σ_{i=k}^{m_R} V(T_{R,i}) is also asymptotic to δ R/6 because the first k terms contribute o(R). Hence the ratio in Eq. (6.1) tends to 1 as R→∞ for every fixed k, so the sequence is not well-behaved for any δ>0. This sequence nevertheless satisfies the CLT: max_i s_{R,i}^2/s_R^2 ∼ 1/log R → 0 and Σ E(X_{R,i}^4)/s_R^4 = O(1/log^2 R) → 0, so the Lindeberg condition holds. Thus a natural family of Coxeter groups with growing rank satisfies the variance-divergence equivalence of Theorem 6.5 but lies outside the theorem's hypothesis. The paper should either prove the CLT under a weaker condition that includes such examples or explicitly state that the answer to Kahle-Stump's question is only conditional on well-behavedness.
[Section 3, Lemmas 3.3, 3.4, Propositions 3.6 and 3.9] The proofs of the fourth-moment formulas are not self-contained. The text repeatedly states that starting values were 'computed with Sage' and the recursion was 'solved with the RSolve command of Mathematica', but no code, no session transcript, and no detailed derivation of the intermediate recursions are provided. These formulas are load-bearing: Lemma 5.3 depends on the estimate E((T−E(T))^4)=O(n^2) from Theorem 3.1, and the type D case is handled only by an upper bound. To make the proof verifiable, the authors should include the Sage/Mathematica code (e.g., as an ancillary file) or give analytic derivations of the recursions and closed forms. This is essential for a rigorous proof of a central result.

minor comments (4)

[Section 4, proof of Lemma 4.4] In the estimate for the first summand, the expression Σ_{i=k+1}^{∞} a_{n,i} X_{n,i} should be the finite sum to k_n; as written, the index range is formally undefined. The intended argument is clear, but the notation should be fixed.
[Section 6, Definition 6.2] The definition of m_n := min {i ∈ N : W_{n,i+1} is δ-small} is unclear when all irreducible components are δ-small or when none are; a convention (e.g., m_n=0 or m_n=k_n) should be stated to make Eq. (6.1) well-defined in all cases.
[Section 6, text before Definition 6.2] The sentence 'if every W_{n,i} is of non-dihedral type and for some δ, one has lim_{n→∞} m_n = 0' appears to be a typo; m_n is a positive integer, so it cannot tend to 0. This should likely read that m_n is bounded or that the maximum condition holds.
[Throughout] There are several typographical errors, including 'random varibales' in the introduction, 'the the law of total expectation' in Section 3, and the rendered addresses 'F akult¨at' in the author affiliation lines. These should be corrected in a final revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a conditional synthesis of independent prior CLT, variance, and moment results.

full rationale

The derivation chain is not circular. Variance formulas (Theorem 2.3) and the integer-valued variance-divergence fact are imported from Kahle–Stump [8]; fourth-moment bounds for types A and B are derived from Özdemir's conditional-expectation recursions by explicit algebra and computer algebra, and the D-type bound is reduced to the B-type bound through a bounded coupling quoted from Röttger [13]. Section 4 proves the weighted-sum CLT directly from Lévy's theorem, and Section 5 verifies Lindeberg/maximum conditions rather than assuming the target CLT. Section 6 works under an explicitly stated 'well-behaved' hypothesis (Eq. (6.1)); this is a variance-concentration assumption, not a renamed version of the conclusion V(T_n)→∞, and no parameter is fitted to force the result. The self-citation [13] supplies base-case CLTs for B_n and D_n and a coupling bound; it is prior work with stated assumptions and is used transparently as external evidence, so it does not make the central claim circular. The authors' repeated remark that they could not construct a non-well-behaved sequence (Section 1 and immediately after Definition 6.2: 'the authors have failed to construct a sequence of finite Coxeter groups that is not well-behaved') is a scope/limitation claim, not a circular step; if the skeptical counterexample is correct, Theorem 6.5 is narrower than the authors suggest, but the proof remains conditional and non-circular. The reliance on unshown RSolve/Sage computations for the moment formulas in Section 3 and Appendix B is a reproducibility/correctness concern, not a definitional recycling of the conclusion.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim relies on standard probability theorems, prior variance and CLT results for irreducible types, Visontai's recurrence, the unproved well-behaved hypothesis, and the correctness of CAS computations. There are no fitted constants. The well-behaved condition is a structural hypothesis, not an invented entity.

assumptions (8)

standard math Lindeberg CLT for triangular arrays, Levy continuity theorem, Slutsky, Chebyshev, Minkowski, and smoothing theorem.
Invoked in Sections 4 and 5 to establish convergence in distribution and to verify Lindeberg-type conditions.
domain assumption Classification of finite irreducible Coxeter groups as A_n, B_n, D_n, I2(m), or exceptional types, with unique decomposition into irreducible components.
Used in Section 2.2 and throughout the decomposition arguments.
domain assumption Variance formulas for T from Kahle and Stump [8, Corollary 5.2], used in Section 5 in the form V(I2(m))=4/m and V(T)=Θ(rank) for non-dihedral irreducibles.
Theorem 2.3 and Lemma 5.2 rely on these formulas; note the printed Theorem 2.3(2) omits a factor 4 for the dihedral case.
domain assumption CLT for T on irreducible chains A_n, B_n, and D_n from Chatterjee-Diaconis [7] and Rottger [13].
Used in Proposition 6.4 to handle large irreducible components; [13] is by a coauthor but is an external prior result.
domain assumption Visontai's recursion for the type B_n two-sided Eulerian polynomial [17, Theorem 15].
Used to derive Lemma 3.7 and then Lemma 3.8.
domain assumption The two-dimensional conditional expectation formulas for (D_n,D'_n) from Ozdemir (Lemma 3.5) and the type B analogue (Lemma 3.8) are valid on the reachable support.
Used to derive mixed moment recursions in Propositions 3.6 and 3.9; entries P_i can be negative on some coordinates, so validity is only needed on the support.
ad hoc to paper Correctness of the Sage starting values and Mathematica RSolve solutions used for the explicit fourth moments in Lemmas 3.3, 3.4, Propositions 3.6, and 3.9.
No CAS code or intermediate derivation is shipped; the formulas are accepted on the authors' computational results.
ad hoc to paper The sequence (W_n) is well-behaved: there exists δ>0 such that Eq. (6.1) holds.
This is the extra hypothesis of Theorem 6.5; the authors state they could not construct a non-well-behaved sequence, so it is unproved.

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Cite this review

Pith. "Pith review of A central limit theorem for the two-sided descent statistic on Coxeter groups." pith.science (2026). https://pith.science/paper/BRBIPDVR

@misc{pith2026190807955,
  author       = {Pith},
  title        = {Pith review of: A central limit theorem for the two-sided descent statistic on Coxeter groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRBIPDVR}},
  note         = {Machine review of arXiv:1908.07955}
}

read the original abstract

We study the asymptotic behaviour of the statistic (des+ides) which assigns to an element w of a finite Coxeter group W the number of descents of w plus the number of descents of its inverse. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and generalises work of Chatterjee-Diaconis, \"Ozdemir and R\"ottger.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

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

[1] [1]

Abramenko and K

P. Abramenko and K. S. Brown. Buildings, volume 248 of Graduate Texts in Mathematics . Springer, New York, 2008

work page 2008

[2] [2]

Billingsley

P. Billingsley. Probability and Measure . Wiley Series in Probability and Statistics. Wiley, 1995

work page 1995

[3] [3]

Bj¨ orner and F

A. Bj¨ orner and F. Brenti. Combinatorics of Coxeter groups , volume 231 of Graduate Texts in Mathematics . Springer, New York, 2005

work page 2005

[4] [4]

F. Brenti. q-Eulerian polynomials arising from Coxeter groups. European Journal of Combinatorics , 15(5):417–441, 1994

work page 1994

[5] [5]

Carlitz, D

L. Carlitz, D. P. Roselle, and R. A. Scoville. Permutatio ns and sequences with repetitions by number of increases. Journal of Combinatorial Theory , 1:350–374, 1966

work page 1966

[6] [6]

Chatterjee

S. Chatterjee. A new method of normal approximation. Ann. Probab., 36(4):1584–1610, 2008

work page 2008

[7] [7]

Chatterjee and P

S. Chatterjee and P. Diaconis. A central limit theorem fo r a new statistic on permutations. Indian J. Pure Appl. Math., 48(4):561–573, 2017

work page 2017

[8] [8]

Counting inversions and descents of random elements in finite Coxeter groups

T. Kahle and C. Stump. Counting inversions and descents o f random elements in ﬁnite Coxeter groups. arXiv e-prints, page arXiv:1802.01389, Feb 2018

work page Pith review arXiv 2018