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REVIEW 4 major objections 5 minor 16 references

Demazure product and hopping in type D

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that the Demazure product (0-Hecke product) of two elements of a type D Coxeter group can be computed by a sequence of hopping moves on their one-line notation, with no need for reduced words, lengths, or Bruhat-order…

desk verdict The type-D hopping formula is essentially right, but the central theorem as printed has v and w swapped, so the statement is not what the proof and example actually establish. read the letter →

arxiv 2411.18584 v2 pith:VRW5CZ6T submitted 2024-11-27 math.CO

Darren Han , Michelle Huang , Benjamin Keller , Suho Oh , Jerry Zhang This is my paper · ORCID
classification math.CO MSC 05E1520F55
keywords Demazureproduct0-HecketypeDCoxetergrouphoppingoperatorevensignedpermutationsone-linenotationparabolicdecompositioncombinatorialunfolding
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 extends a formula for the Demazure product—also called the 0-Hecke product—from type A and type B Coxeter groups to type D. It proves that for any two even signed permutations $v, w \in D_n$, the product $v \star w$ is obtained by multiplying $v$ by $w$ in the usual way and then applying a sequence of hopping operators $h_{n-1,w{\uparrow} n-1} \cdots h_{1,w{\uparrow} 1}$. The lists $w{\uparrow} i$ are read directly from the one-line notation of $w$, so the computation never uses reduced words, element lengths, or Bruhat comparisons. This renders the type D Demazure product purely combinatorial and completes the picture for the classical finite Coxeter groups A, B, and D.

What carries the argument

The hopping operator $h_{i,L}$ is the workhorse: it scans the one-line notation of an even signed permutation to the right of $i$ and, as long as the list $L$ contains an element there larger than $i$, swaps $i$ with the rightmost such element. The paper pairs it with the lifting subword $w{\uparrow} i$, extracted from the interval $(i,-i)$ of the unfolding of $w$; this subword tells the hopping operator exactly which later entries are eligible. The maximal parabolic decomposition of $D_n$ into factors $Q_{n-1}\cdots Q_1$, each of one of four explicit forms, is what makes the proof tractable: for each form, Proposition 4.5 prescribes a list $L_i$ such that $Q_i \star w = h_{i,L_i}(Q_i w)$, and Proposition 4.10 shows that the lists recomputed after multiplication by the earlier factors coincide, up to a controlled equivalence, with $w{\uparrow} i$.

What would settle it

Run an exhaustive check over all pairs $v,w$ in $D_5$ (and, if that passes, $D_6$), computing $v \star w$ once by the reduced-word definition and once by the hopping formula of Theorem 4.11; the theorem is false if any pair differs, and the first differing pair would pinpoint the step that fails. The paper's worked examples all agree, so the search must go beyond them.

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Extended reading notes

Core claim

The central result, Theorem 4.11, states that for any $v, w \in D_n$, $$v \star w = h_{n-1, w{\uparrow} n-1} \cdots h_{1, w{\uparrow} 1}(v w),$$ where $w{\uparrow} i$ is the subword of the unfolding of $w$ consisting of entries strictly left of $i$ and lying between $i$ and $-i$ in the total order, and $h_{i,L}$ is the hopping operator that repeatedly swaps $i$ with the rightmost element of $L$ to its right that is larger than $i$. The proof decomposes $v$ into parabolic factors $Q_{n-1}\cdots Q_1$ according to the branch structure of the type $D$ Dynkin diagram, shows that each factor's Demazure action can be emulated by a single hopping operator (Proposition 4.5), and then slides all hoppings leftward past the factors using commutation identities. Along the way the paper shows why a naive unfolding trick from type $B$ fails in type $D$: combinatorial unfolding does not preserve the Demazure product for even signed permutations, so the argument must work directly in $D_n$. The resulting formula depends only on the one-line notation of $v w$ and the lifting subwords of $w$; no reduced-word data enters.

Load-bearing premise

The whole result depends on being able to interchange the order of a hopping step and a swap of two entries, and the paper proves this interchange directly only for swaps that do not flip signs, not for the sign-flipping swap or for strings of several swaps.

Editorial extensions

If this is right

  • The formula turns the Demazure product in type D into a direct algorithm: form $vw$, extract the lifting subwords of $w$, and perform $n-1$ scanning-and-swapping passes.
  • Because the lifting lists depend only on $w$, the same list can be reused for many different left factors $v$, which makes repeated computations of $v \star w$ more economical than recomputing reduced expressions.
  • The result extends the hopping description from the symmetric group and signed permutations to even signed permutations, showing that the one-line-only computation of the Demazure product is a common feature of the classical finite Coxeter groups A, B, and D.
  • The explicit treatment of the parabolic factors in Proposition 4.5 gives a local, blockwise description of how each factor acts under the Demazure product, which may be useful for understanding Bruhat intervals of type D elements.

Reading between the lines

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

  • The affine analogue the authors raise (Question 4.1) could be approached by extending the lifting subword to infinite one-line notation; the main obstacle is finding an interval $(i,-i)$ analogue when the negative side is unbounded.
  • The leaf-cutting parabolic decomposition is not type-D-specific in spirit: for other finite Coxeter diagrams with a single branch node, the same strategy might produce explicit lists for each quotient form, so testing $E_6$ would show whether the four-form classification is a general phenomenon.
  • Because the formula works entirely inside $D_n$ rather than through an embedding into $S_{2n}$, it suggests that the right way to transfer one-line thinking to type D is to keep evenness of signs as a built-in constraint; a monoid-level interpretation of the hoppings could connect the 0-Hecke monoid of type D to the one of type A in a way that unfolding cannot.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper claims an extension to Coxeter type D of the hopping-operator description of the Demazure (0-Hecke) product. After reviewing the type A and B results from [10], the authors introduce a right-quotient parabolic decomposition Q_{n−1}...Q_1 for elements of D_n, define lifting subwords w↑i from the combinatorial unfolding of even signed permutations, and prove local identities (Proposition 4.5) showing that each parabolic factor acts by a hopping operator. The main theorem (Theorem 4.11) is intended to say that the Demazure product of two elements is obtained by applying the hopping operators h_{i,w↑i}, in decreasing i order, to the ordinary product of the two elements, using only one-line notation. The proof proceeds by decomposing one factor, pulling hopping operators left past parabolic factors, and identifying the resulting lists with w↑i.

Significance. The intended result is a natural and useful type-D analogue of the Li–Oh–Richmond–Yan–You theorem: if correct, it gives a purely combinatorial, reduced-word-free algorithm for the Demazure product. The paper is honest about the failure of the type-B unfolding lemma for type D (Remark 3.2 and Example 3.4) and develops the parabolic-decomposition method directly; the final formula is explicit and falsifiable, and the worked examples support the intended statement. There are no fitted parameters and the type-D theorem is not assumed, so the argument is not circular. However, the written central theorem is misstated, and the proof contains several unproven commutation and induction steps; these must be repaired before the paper's main claim can be accepted.

major comments (4)
  1. [§4.3, Theorem 4.11 and Eq. (4.5)] The theorem as printed states v ⋆ w = h_{n−1,w↑n−1}...h_{1,w↑1}(v w), with the lifting lists taken from the second factor w. This is not the identity proved by the paper. The proof is built on Proposition 4.5, which is a left-factor identity Q_i ⋆ x = h_{i,L_i}(Q_i x); iterating it for a decomposition of the first factor yields W ⋆ V = h_{n−1,W↑n−1}...h_{1,W↑1}(W V) (with the lifts of the first factor). The displayed chain (4.5) and the final example are consistent only with the latter: the example applies h_{4,w↑4}...h_{1,w↑1} to w v and concludes w ⋆ v. The printed statement, with the roles of v and w interchanged, is not what is proved and, since the Demazure product is noncommutative, is very likely false in general. Please restate Theorem 4.11 as w ⋆ v = h_{n−1,w↑n−1}...h_{1,w↑1}(w v), and correct the proof's variables accordingly (the innermost term should be Q_1 v, and the final product w v).
  2. [Lemma 4.1 and the final chain of Theorem 4.11] The step 'Q_{n−1}...Q_{i+1}h_{i,L_i} = h_{i,Q_{n−1}...Q_{i+1}L_i}' is used to move hopping operators leftward, but Lemma 4.1 is proved only for a single simple generator s_j with j > i (j up to n−1), and its proof describes swapping adjacent values in one-line notation. The parabolic factors Q_k include form 2 and form 3 elements involving s_n, which swaps positions n−1 and n and flips their signs, and the identity is applied to products of such factors. No proof is given that the same commutation holds for s_n or for products. As stated, Lemma 4.1 therefore does not support the operator equality on which the proof of Theorem 4.11 depends. Please state and prove the required commutation identity for all generators, including s_n, and for products, and clarify the intended composition order in the displayed equality.
  3. [§4.2, Proposition 4.5] The proof of the second case (form 2, i ≤ n−2) is incomplete. Eq. (4.3) is asserted for s_i...s_j with j ≤ i, and the induction is left as 'the proof is omitted'; since Proposition 4.5 is the central tool that converts parabolic factors into hopping operators, this is load-bearing. The subsequent derivation of Eq. (4.4) also has unstated ingredients: the second line invokes Lemma 4.2 in a situation where the generator s_n (sign-flipping) is involved, and the third line invokes Lemma 4.4 in a context that goes beyond its statement. Please supply the full induction for Eq. (4.3) and verify each application in Eq. (4.4).
  4. [§4.2–4.3, Lemma 4.9 and Proposition 4.10] The proof of Lemma 4.9 asserts without proof that after a form-2 or form-3 right multiplication, the entries v(n) and v(−n) are adjacent in vQ_k ↑ i, which is what allows Lemma 4.7 to be applied. The proof of Proposition 4.10 then compresses the casework for the first claim into a single 'observation'. These steps are needed to replace the algorithmic lists L_i by the lifting subwords w↑i in the final theorem; please expand the arguments or at least give a precise statement of the adjacency claim.
minor comments (5)
  1. [Abstract] The word 'applicatio ns' should be 'applications'.
  2. [§1, Introduction] There is a duplicated article in 'the the form'; also, the notation [id,w] for a Bruhat interval in Proposition 1.2 should be clearly distinguished from the ordered-list notation [a1,...,ak] used later.
  3. [§2.2, Theorem 2.5] The displayed formula h_{n−1,w↑n−1}h_{2,w↑2}h_{1,w↑1}(wv) omits the intervening factors h_{n−2},...,h_3; presumably it should read h_{n−1,w↑n−1}...h_{2,w↑2}h_{1,w↑1}(wv).
  4. [§4.3, proof of Theorem 4.11] The displayed chain contains 'D_1 w' in one line, apparently a typo for 'Q_1 w' (or, after the correction in Major Comment 1, 'Q_1 v').
  5. [After Lemma 4.9] The sentence 'For each of them of them used in a hopping operator' contains a duplicated phrase and should be reworded.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.11 is derived from the hopping framework, not assumed; the cited lemmas are proved in the text.

full rationale

The claimed derivation is not circular. The central formula is obtained by combining Proposition 4.5, which proves Qi ⋆ w = h_{i,L_i}(Q_i w) by a five-case analysis from Lemma 4.3, with Proposition 4.10, which proves (Q_{n-1}...Q_{i+1})L_i ∼_i w↑i using the parabolic-form description and Lemma 4.7. Neither proposition assumes the final theorem. The lemmas labeled from [10] are restated with proofs in the present paper, so the proof does not reduce to an unverified self-citation chain, and [10] is about types A and B, not the type D statement proved here. There are no fitted parameters and no hidden use of the target result. Two non-circular defects were noticed: the proof of Theorem 4.11 invokes Lemma 4.1 to commute a whole product Q_{n-1}...Q_{i+1} past h_{i,L_i}, while Lemma 4.1 as stated proves only a single-generator commutation; and the statement of Theorem 4.11 writes v ⋆ w = h_{n-1,w↑n-1}...h_{1,w↑1}(vw), whereas the proof and the section 4.3 example actually compute w ⋆ v by applying the hopping chain to wv, suggesting a factor-order typo in the theorem statement. These would be correctness or proof-gap issues, not circularity: neither makes the conclusion equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Coxeter facts plus the paper's chosen type D notation and an unproved decomposition proposition. There are no fitted constants, no model parameters, and no new physical objects.

assumptions (5)
  • standard math Standard 0-Hecke/Coxeter monoid facts in Lemma 1.1: s1...sk <= s1 star ... star sk, with equality iff reduced, and s star w is either w or sw according to length.
    Imported from Norton [11]; the hopping algorithm is built to emulate this rule.
  • standard math Bruhat interval characterization [id, w star v] = {ab | a in [id,w], b in [id,v]}.
    Prop. 1.2, cited from He [6] and Kenney [7]; used as context for the Demazure product.
  • domain assumption Maximal parabolic decomposition w = Q_{n-1}...Q_1 with each Q_i in the stated quotient W^{{s_{i+1},...,s_n}}_{{s_i,...,s_n}}.
    Proposition 3.6 is stated without proof; it is the structural backbone for constructing the lists L_i and for Proposition 4.10.
  • domain assumption Type D one-line notation and simple transposition actions in Section 3.1, including s_n swapping positions n-1 and n and flipping their signs.
    All examples and proofs use this convention; it differs from the Bjorner-Brenti s0 convention.
  • domain assumption Combinatorial unfolding with total order 1 < 2 < ... < n < -n < ... < -2 < -1.
    Section 3.2; the hopping algorithm scans this unfolding and the lifting w lifting i is defined from it.

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Cite this review

Pith. "Pith review of Demazure product and hopping in type D." pith.science (2026). https://pith.science/paper/VRW5CZ6T

@misc{pith2026241118584,
  author       = {Pith},
  title        = {Pith review of: Demazure product and hopping in type D},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRW5CZ6T}},
  note         = {Machine review of arXiv:2411.18584}
}
read the original abstract

The Demazure product, also called the 0-Hecke product, is an associative operation on Coxeter groups with interesting properties and applications. In (Li et al 2024) it was shown that the Demazure product of two permutations can be described purely combinatorially: using only their one-line notation and not relying on reduced words. In this paper, we extend this to type D Coxeter groups.

Figures

Figures reproduced from arXiv: 2411.18584 by the authors.

Figure 1
Figure 1. The Coxeter diagram for Dn 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Bruhat order of W si+1,...,sn si,...,sn The elements of WJ S are one of the following four forms: • (form 0) id, • (form 1) si . . . sj where i ≤ j ≤ n − 1, • (form 2) si . . . sn−2snsn−1 . . . sj where i ≤ j ≤ n − 1, • (form 3) si . . . sn−2sn. We also classify the elements of W{sn−1,sn} using these forms: • (form 0) id, • (form 1) sn−1, • (form 2) snsn−1, • (form 3) sn. Throughout the paper, we will classify the e… view at source ↗

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Reference graph

Works this paper leans on

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