{"id":"e4cf3d39-c8ab-4942-8be7-efa70979464c","arxiv_id":"2411.18584","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Demazure product on type D Coxeter groups is computed by applying a sequence of hopping operators to the ordinary product, with each operator read off from the unfolding of the second factor.","lead":"This paper gives a purely combinatorial recipe for the Demazure (0-Hecke) product of two elements of a type D Coxeter group, using only one-line notation and a hopping algorithm. It extends a method known for types A and B, and it overcomes the failure of the naive unfolding approach in type D.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.11 as printed has the Demazure product factors swapped: it claims v*w = h_{w↑}(vw), but the proof and example establish w*v = h_{w↑}(wv). The stated central formula is not the one proved.","rationale":"The reader identified a genuine gap (Lemma 4.1 does not cover s_n), but my reading found a more fundamental issue: the theorem statement itself has a factor-order error. The proof and example show the correct identity is w⋆v = h_{...,w↑}(w v). The printed theorem states v⋆w = h_{...,w↑}(v w), which mixes the left factor (v) in the product with the right factor (w) in the lifting, an impossible combination. Since the Demazure product is not commutative, these are different statements. The proof's Eq. (4.5) also contains Q_1 w instead of Q_1 v and the final chain ends with v w instead of w v, confirming a systematic variable transposition. The paper's actual contribution—a hopping description of the Demazure product—is very likely correct once the statement is corrected to w⋆v = h_{w↑}(w v), and the Lemma 4.1 gap is repairable (s_n fixes i≤n−2 and preserves the set {x>i}, so the needed commutation holds). Therefore the verdict remains CONDITIONAL: require the authors to correct the statement/proof variable order and to supply the missing s_n/product commutation proof. I disagree with the reader's choice of weakest assumption because the factor-order error is more directly load-bearing for the central claim as written.","tokens_in":14537,"tokens_out":35802,"duration_ms":288737,"concrete_test":"Use the paper's own example in §4.3: w=[2,-4,-1,5,3], v=[-4,3,-5,-1,-2]. Compute v w (ordinary product), then apply h_{4,w↑4}h_{3,w↑3}h_{2,w↑2}h_{1,w↑1} to v w. Compare the result with the paper's claimed value of w⋆v ([-1,-3,-4,-2,5]) and with a direct computation of v⋆w using the correct left-factor formula (h_{v↑}(v w)) or the 0-Hecke recurrence from Lemma 1.1. If h_{w↑}(v w) does not equal v⋆w, the printed theorem is false; the correct statement must have the left factor's lifting and the product w v.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem as stated (Theorem 4.11: 'v ⋆ w = h_{n-1,w↑n-1}...h_{1,w↑1}(v w)') is inconsistent with the proof and the worked example. In the example at the end of §4.3, the authors apply h_{4,w↑4}...h_{1,w↑1} to w v and conclude that the result equals w ⋆ v. That is the correct pattern: in the type-A/B theorems (Theorem 2.2, 2.5) the lifting subwords are taken from the left factor of the Demazure product, and the hopping operators act on the ordinary product with the left factor on the left. Proposition 4.5 only gives Q_i ⋆ x = h_{i,L_i}(Q_i x), i.e., a left-factor computation. Iterating it for a decomposition w = Q_{n-1}...Q_1 yields w ⋆ v = h_{n-1,L_{n-1}}(Q_{n-1} h_{n-2,L_{n-2}}(... h_{1,L_1}(Q_1 v))), and Proposition 4.10 then replaces L_i by w↑i, giving w ⋆ v = h_{n-1,w↑n-1}...h_{1,w↑1}(w v). The printed Eq. (4.5) has Q_1 w (should be Q_1 v) and the final chain ends with v w (should be w v). Thus the formula stated in the theorem—with the list w↑ but the product v w and the product v⋆w—is not what is proved; as written it is likely false for generic v,w, since the Demazure product is noncommutative. This is a presentation-level error in the statement of the central claim, not merely a missing detail.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14956,"tokens_out":11952,"duration_ms":100696,"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":[{"comment":"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).","section":"§4.3, Theorem 4.11 and Eq. (4.5)"},{"comment":"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.","section":"Lemma 4.1 and the final chain of Theorem 4.11"},{"comment":"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).","section":"§4.2, Proposition 4.5"},{"comment":"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.","section":"§4.2–4.3, Lemma 4.9 and Proposition 4.10"}],"minor_comments":[{"comment":"The word 'applicatio ns' should be 'applications'.","section":"Abstract"},{"comment":"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.","section":"§1, Introduction"},{"comment":"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).","section":"§2.2, Theorem 2.5"},{"comment":"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').","section":"§4.3, proof of Theorem 4.11"},{"comment":"The sentence 'For each of them of them used in a hopping operator' contains a duplicated phrase and should be reworded.","section":"After Lemma 4.9"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on [10], which shares an author with this submission; the lemmas cited from [10] are appropriate, but the new contribution's boundary should be made explicit. The main theorem statement error must be fixed before publication, and the abstract and examples need to be consistent with the corrected left-factor version. The manuscript is suitable for math.CO once the proof gaps are closed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has a real result, but Theorem 4.11 as stated does not match the proof. The theorem claims v ⋆ w = h_{n-1,w↑n-1}...h_{1,w↑1}(vw). The proof, however, decomposes the left factor v into Q_{n-1}...Q_1, uses Proposition 4.5 to write v ⋆ w = h_{n-1,L_{n-1}}(Q_{n-1} h_{n-2,L_{n-2}}(... h_{1,L_1}(Q_1 w))), and then commutes the Q's past the hopping operators. The list that emerges at the end is the lifting of the left factor, i.e., v↑i, not w↑i. The worked example at the end of §4.3 confirms this: it takes w=[2,-4,-1,5,3], v=[-4,3,-5,-1,-2], computes h_{4,w↑4}...h_{1,w↑1}(wv), and concludes w ⋆ v. So the intended theorem is w ⋆ v = h_{n-1,w↑n-1}...h_{1,w↑1}(wv), matching the type A/B pattern. As printed, the commutativity of the Demazure product is not assumed, so the stated formula is likely false for generic noncommuting v and w. This is not a missing detail; it is a load-bearing swap in the central statement. It should be an easy fix, but it needs to be fixed.\n\nWhat is genuinely new: extending Li–Oh–Richmond–Yan–You's hopping description to type D is nontrivial. Section 3.2 honestly explains why the type B unfolding lemma fails for even signed permutations, and the maximal parabolic decomposition (Prop 3.6) together with the lifting equivalence (Prop 4.9, 4.10) is a real new ingredient. The paper is clearly written and the examples are helpful.\n\nThe soft spots beyond the swap: the proof of Eq. (4.3) in Proposition 4.5 says \"the proof is omitted\" for the induction; that is a genuine gap in the written argument. Lemma 4.1 is quoted from [10] and its one-line proof only addresses swapping adjacent values; the proof of Theorem 4.11 uses it to commute products of parabolic factors, including s_n actions, past hopping operators, and that step is not justified as written. These are all patchable, but they need to be supplied.\n\nWho is this for: people working on Coxeter monoids, 0-Hecke algebras, and combinatorial descriptions of Demazure products. It deserves a serious referee, and with the v/w swap corrected and the missing induction filled in, it would be a solid contribution. I would not cite it in its current form, but I would cite the corrected version.","headline":"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.","tokens_in":15470,"tokens_out":5756,"would_cite":false,"duration_ms":45037,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E15","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Demazure product","0-Hecke product","type D Coxeter group","hopping operator","even signed permutations","one-line notation","parabolic decomposition","combinatorial unfolding"],"falsifier":"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.","tokens_in":14352,"feed_emoji":"🔀","tokens_out":11924,"duration_ms":101913,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hopping operators compute Demazure product in type D","feed_subtitle":"No reduced words or lengths needed: one-line notation plus a scanning-and-swapping rule gives the product.","key_machinery":"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 $w{\\uparrow} i$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the hopping operator, the lifting subword notation, and the commutation lemmas on which the type D proof builds.","marker":"[10]"},{"why":"Defines the Demazure/0-Hecke product and the length-based rules that the paper computes with.","marker":"[11]"},{"why":"Provides the Coxeter-group background, reduced expressions, and Bruhat-order facts used throughout.","marker":"[2]"},{"why":"Fixes the type D simple-generator convention that determines the one-line notation and the action of $s_n$.","marker":"[1]"},{"why":"Introduced the Demazure product, the object the paper gives a combinatorial formula for.","marker":"[4]"}],"fun_headline_variants":["Type D Demazure product from one-line notation and scanning swaps","Demazure product in type D via hopping, no reduced words","Hopping operators give type D Demazure product without reduced words","Scanning-swapping rule yields Demazure product in type D"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Type D Demazure product from one-line notation and scanning swaps","Demazure product in type D via hopping, no reduced words","Hopping operators give type D Demazure product without reduced words","Scanning-swapping rule yields Demazure product in type D"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000731,"raw_usage":{"total_tokens":3241,"prompt_tokens":883,"completion_tokens":2358,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":2299}},"tokens_in":499,"tokens_out":2358,"duration_ms":14923,"temperature":1.0,"reasoning_tokens":2299,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:05:05.354998+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the hopping operator, the lifting subword notation, and the commutation lemmas on which the type D proof builds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Demazure/0-Hecke product and the length-based rules that the paper computes with."},{"cited_title":"Björner and F","cited_arxiv_id":null,"evidence_quote":"Provides the Coxeter-group background, reduced expressions, and Bruhat-order facts used throughout."},{"cited_title":"Billey and V","cited_arxiv_id":null,"evidence_quote":"Fixes the type D simple-generator convention that determines the one-line notation and the action of $s_n$."},{"cited_title":"Demazure , Dèsingularisation des variètès de Schubert gènèralisèes , Ann","cited_arxiv_id":null,"evidence_quote":"Introduced the Demazure product, the object the paper gives a combinatorial formula for."}],"review_version":1}