{"id":"9f018bee-47aa-4fea-89ed-e5023cc4615b","arxiv_id":"2508.07857","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The canonical filtration of a right-angled Iwahori-Hecke algebra satisfies the Ozawa-Rieffel Haagerup condition iff the graph of commuting generators has no induced square, yielding compact quantum metric spaces and propinquity continuity at q=1.","lead":"For right-angled Coxeter groups, the natural length filtration of their q-deformed Iwahori-Hecke algebra gives a compact quantum metric space exactly when the commuting graph has no induced square, and the deformed spaces converge to the group algebra as q tends to 1. This connects Coxeter combinatorics with noncommutative metric geometry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.9 relies on a Lip-norm that is not lower semicontinuous, so the propinquity claim is not established as stated.","rationale":"The Reader's weakest_assumption targeted Lemma 3.7. On reading Lemma 3.7, the hyperbolicity inequality appears legitimate: with g_1=v_1, g_2=v_1', g_3=e, g_4=x, the four-point inequality gives exactly |v_1^{-1}v_1'|+|x| ≤ max(|v_1|+|x^{-1}v_1'|, |v_1'|+|x^{-1}v_1|)+δ, and the prefix assumptions then yield the claimed bound. The counting argument is terse but not apparently wrong. However, the paper contains an explicit and more serious gap in Section 4: the Lip-norm used for the propinquity theorem is not lower semicontinuous. This is not a matter of consensus versus convention; it is internal to the definitions quoted in the paper. Definition 2.4(2) demands lower semicontinuity, and the construction in (3.4) assigns ∞ outside the dense algebraic subspace C_q[W]. In any infinite-dimensional situation, such a seminorm is typically not lsc, and D_∞ provides a concrete instance. Since Theorem 4.9 is one of the headline results, the paper should either prove lsc for a modified/closed Lip-norm or explicitly state that all propinquity statements are for the lower semicontinuous envelopes. The core characterization Theorem 3.6 and compact quantum metric space Theorem 3.11 are not affected, so the verdict should be CONDITIONAL rather than REJECT: the main structural results stand, but the continuity theorem needs a correction or clarification before acceptance as stated.","tokens_in":25821,"tokens_out":30662,"duration_ms":384554,"concrete_test":"Settle the lower-semicontinuity claim by an explicit counterexample in W=D_∞. Identify C*_r(D_∞) with a continuous-function algebra on the circle and C[D_∞] with finite Fourier polynomials. Let f(t)=|t| on [-π,π], a non-polynomial 1-Lipschitz function, and let p_n be Fejér means of f. Compute the reduced C*-algebra norm error ||p_n-f|| (which tends to 0) and compute ||[D,p_n]|| (which stays bounded by a constant independent of n). If both hold, then p_n belongs to {L≤C}, p_n→f, but f∉C[D_∞], so L(f)=∞ and L is not lower semicontinuous. This directly tests the claim that (C*_{r,q}(W),L_S^{(q)}) is a Leibniz quantum compact metric space for q=1; the same mechanism applies for q near 1.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The continuity theorem (Theorem 4.9) is formulated for Latrémolière's propinquity, which is defined only for Leibniz quantum compact metric spaces. Definition 2.4(2) requires the Lip-norm to be lower semicontinuous. The paper defines L_S^{(q)}(x)=||[D_S,x]|| for x in the dense algebraic subspace C_q[W], and +∞ otherwise. Since C_q[W] is not norm-closed in C*_{r,q}(W), the sublevel sets {x: L_S^{(q)}(x)≤r} are not generally closed. Concretely, take W=D_∞, the infinite dihedral group, which is right-angled with Γ empty (no induced square). For q=1, C*_r(D_∞) can be realized as a C*_r-algebra of continuous functions on the circle, and C[D_∞] corresponds to finite Fourier polynomials; L is a Lipschitz-type seminorm. A sequence of polynomials p_n with ||p_n'||_∞≤1 converging uniformly to a non-polynomial Lipschitz function f satisfies L(p_n)≤C but L(f)=∞, so {L≤C} is not closed. The paper asserts immediately before Question 4.8 that lower semicontinuity is 'straightforward to verify'; this appears false. The proof of Theorem 4.9 could likely be repaired by replacing L_S^{(q)} with its lower semicontinuous envelope, but as written the objects are not in the category for propinquity. This does not invalidate Theorem 3.6 or Theorem 3.11, but it is load-bearing for Theorem C.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies quantum metric structures on Iwahori–Hecke algebras C_q[W] attached to finite-rank right-angled Coxeter systems. After recalling Ozawa–Rieffel's Haagerup-type condition and a generator decomposition from Caspers–Klisse–Larsen, it proves Theorem 3.6: the canonical word-length filtration satisfies the Haagerup-type estimate with constant C_q if and only if the commuting graph Γ contains no induced square. The positive direction uses a counting estimate (Lemma 3.7) together with word hyperbolicity via Moussong's theorem; the negative direction is an explicit induced-square counterexample. Theorem 3.11 then produces compact quantum metric spaces (C*_{r,q}(W), L_S^{(q)}) for no-induced-square systems. In Section 4 the authors construct bridges using Schur multipliers m_κ coming from the positive definite functions κ^{|·|}, and prove Theorem 4.9: as q→1 these spaces converge in Latrémolière's propinquity to the reduced group C*-algebra with its word-length Lip-norm. The final section discusses open questions and possible extensions.","tokens_in":26171,"tokens_out":9094,"duration_ms":114870,"significance":"If correct, Theorem 3.6 gives a sharp combinatorial characterization of the Haagerup-type condition in the right-angled Hecke setting and provides a systematic class of compact quantum metric structures for q-deformations beyond group algebras. The proof strategy is modular: the geometric counting input is isolated from the algebraic generator decomposition, and the negative direction is an explicit, checkable counterexample. The reliance on Proposition 3.9 from [15] and on Ozawa–Rieffel machinery is not circular; those are independent published results. Theorem 4.9, once the category-theoretic defect noted below is repaired, would be a valuable continuity result in propinquity that avoids finite-dimensional approximations. The paper is clearly organized and appropriately situated in the literature.","major_comments":[{"comment":"The assertion that L_S^{(q)} is 'straightforward to verify' lower semi-continuous is false as stated. Definition 2.4(2) requires the sublevel sets {L ≤ r} to be norm-closed in sa(C*_{r,q}(W)). But L_S^{(q)} takes the value +∞ outside the algebraic subspace C_q[W], which is not norm-closed in C*_{r,q}(W); hence the sublevel sets are not closed. Concretely, for W = D_∞ (right-angled with Γ empty), C_1[W] corresponds to finite trigonometric polynomials in a commutative realization, and a sequence of polynomials with uniformly bounded derivative can converge uniformly to a non-polynomial Lipschitz function, for which the defined Lip-norm is ∞. Thus (C*_{r,q}(W), L_S^{(q)}) is not, as written, a Leibniz quantum compact metric space, and Theorem 4.9 is not a statement about Latrémolière's propinquity. This is load-bearing for Theorem C, though it does not affect Theorems 3.6 or 3.11. The repai","section":"Lemma 3.7"},{"comment":"The proof of Lemma 3.7 is too compressed for a result on which Theorem 3.6(1) depends. Starting from Moussong's hyperbolicity inequality, the manuscript jumps to bounds such as |v_1^{-1}v_1'| ≤ #S + δ and then asserts that 'using u = w_1 w_2 y' yields R_{x,y}(i) ≤ (#Cliq(Γ))^2 R^4. It is not shown in detail why the word-length constraints force the relevant geodesic quadruples to satisfy the hyperbolicity estimate with these particular quantities, nor why the product of four ball-counts controls all components of the tuple independently of i, x, y. Since the constant K in Lemma 3.7 is precisely the uniform constant used in the Cauchy–Schwarz estimate following (3.6), this omission is load-bearing. Please expand the proof with the intermediate steps and the counting argument.","section":"Proposition 4.11"},{"comment":"The proof uses the step 'By Proposition 3.12 and the structure of the Schur multipliers, we may choose 0 < κ < 1 such that sup_{x∈B} ||x − m_κ(x)|| < ε/2', where B is the union of totally bounded sets over a compact parameter set. This uniformity is not automatic from pointwise convergence m_κ(x)→x. It requires a finite ε-net argument together with the contractivity of the completely positive maps m_κ, and should be written out. The same applies to the subsequent choice of i0, which depends on the explicit decay of F(q_i,1). These compactness steps are load-bearing for the uniform estimate in Proposition 4.11 and hence for the convergence claim in Theorem 4.9.","section":"Proposition 4.11"}],"minor_comments":[{"comment":"The notation oscillates between C_q[W] and C_q[W], and the running title reads 'IW AHORI–HECKE ALGEBRAS' with an extra space. These should be standardized.","section":"Throughout"},{"comment":"The symbol x is used both for an element of C_q[W] and for a group element in the reindexing ('X_{x∈W:|x|=n}'). This makes the displayed sums hard to parse; please use a different letter for the group element.","section":"Proof of Theorem 3.6, around (3.6)"},{"comment":"In the displayed formula, the subscripts K0, K1, K2 should be Γ0, Γ1, Γ2. As typeset they are undefined and obscure the otherwise clear argument.","section":"Lemma 4.6"},{"comment":"Part (2) states the failure with K∥xδ_e∥_2, while part (1) and the abstract include the factor C_q. This is not an error, but the asymmetry in the formulation may confuse readers; a brief sentence explaining that the lower bound in the counterexample is independent of q would help.","section":"Abstract and Theorem 3.6(2)"}],"recommendation":"major_revision","confidential_remarks":"The lower semicontinuity defect in Section 4.2 is local and repairable, but as written it removes Theorem 4.9 from the category in which propinquity is defined. I recommend major revision rather than rejection: Theorems 3.6 and 3.11 appear sound, and the continuity result can likely be fixed by using the lower semicontinuous envelope of the Lip-norm and rechecking the bridge estimates."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the quick take. The main theorem (3.6) is genuine: an iff characterization of the Ozawa–Rieffel Haagerup condition for right-angled Hecke algebras, with the no-induced-square hypothesis exactly matching hyperbolicity. The proof is traceable: Proposition 3.9 from [15] gives the generator decomposition, Lemma 3.7 gives the counting bound via Moussong, and the q-dependent coefficients are handled cleanly. The induced-square counterexample is explicit. Theorem 3.11 follows from Ozawa–Rieffel without trouble. That part is solid and worth having.\n\nThe soft spot is Theorem 4.9. The objects are presented as Leibniz quantum compact metric spaces, but L_S^(q) is infinite off the dense subalgebra C_q[W], so the sublevel sets aren't norm-closed. The paper says lower semicontinuity is 'straightforward to verify' just before Question 4.8. That's false: take W=D_infty, q=1; C[D_infty] is the finite Fourier polynomials, L is the Lipschitz seminorm, and bounded-derivative polynomials converge uniformly to a non-polynomial Lipschitz function. So the propinquity isn't defined for these spaces as stated. This is load-bearing for Theorem C. The fix is likely to take the lower semicontinuous envelope of the seminorm; the proof should carry through, but as written the theorem needs revision.\n\nTwo smaller issues. Lemma 3.7 is telegraphic; the word-hyperbolicity reduction is real but written so tightly that a referee should ask for details. Proposition 4.11 has some implicit compactness/uniformity steps in the q→1 argument; not wrong as far as I can tell, but a referee should check the limit exchanges. Also, the authors cite [15] for the generator decomposition, so the overlap is transparent; a sentence on exactly which inequalities are new relative to [15] would help readers.\n\nNet: the core result (Theorem 3.6/3.11) deserves publication, and the paper should go to a serious referee. The referee should focus on the lsc gap and the uniform estimates. With those addressed, I'd be happy to see it in print.","headline":"Solid Haagerup characterization on Hecke algebras, but the propinquity theorem as stated uses a Lip-norm that isn't lower semicontinuous—fixable, but the paper needs revision.","tokens_in":26685,"tokens_out":4175,"would_cite":true,"duration_ms":45789,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C08","46L87","20F55","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For right-angled Coxeter systems, the word-length filtration satisfies the Haagerup-type condition exactly when the generator graph has no induced square, yielding compact quantum metric spaces and convergence as q→1.","keywords":["Iwahori–Hecke algebras","compact quantum metric spaces","Haagerup-type condition","right-angled Coxeter groups","word-hyperbolic Coxeter groups","quantum Gromov–Hausdorff propinquity","Lip-norms","Schur multipliers"],"falsifier":"Compute the counting function $R_{x,y}(i)$ from Lemma 3.7 for a square-free right-angled Coxeter system and find a family with $R_{x,y}(i)\\to\\infty$ as $i\\to\\infty$; that would falsify the uniform bound and Theorem 3.6(1). Alternatively, exhibit a square-free right-angled system whose canonical Lip-norm $L_S^{(q)}$ fails to metrizes the weak-$*$ topology on the state space, directly contradicting Theorem 3.11.","tokens_in":25694,"feed_emoji":"📏","tokens_out":12280,"duration_ms":117153,"temperature":0.7,"pith_summary":"This paper asks when the Iwahori–Hecke algebra of a Coxeter group can be viewed as a noncommutative metric space, with distance measured through the word-length operator $D_S=\\sum_{n\\in\\mathbb{N}} nP_n$. For finite-rank right-angled Coxeter systems, the answer is governed by one combinatorial feature: the graph connecting commuting generators must contain no induced square. When it does not, the canonical filtration satisfies the Haagerup-type estimate and the algebra becomes a compact quantum metric space; when it does, no uniform estimate of that kind exists. The paper also proves that, as the deformation parameter $q$ approaches $1$, these deformed metric spaces converge to the reduced group $C^*$-algebra with its word-length metric in the quantum Gromov–Hausdorff propinquity.","feed_headline":"Square-free Coxeter groups yield quantum metric Hecke algebras","feed_subtitle":"No four-cycle in the generator graph makes every Hecke C*-algebra a compact quantum metric space","key_machinery":"The load-bearing structure is the canonical word-length filtration: finite-dimensional subspaces $C_q^{(n)}[W]$ spanned by $\\{T_w^{(q)}: |w|\\le n\\}$, the associated projections $\\chi_n$ and $P_n$ onto length-$n$ vectors in $\\ell^2(W)$, and the Dirac operator $D_S=\\sum_n nP_n$. The central estimate is the Haagerup-type bound $\\|P_i x P_j\\|\\le K C_q\\|x\\delta_e\\|_2$ for $x\\in\\chi_n(C_q[W])$. Its proof rests on two items: a decomposition of each basis element into creation, annihilation, and diagonal operators indexed by cliques (Proposition 3.9), and a uniform bound on a counting function $R_{x,y}(i)$ that counts such decompositions (Lemma 3.7), obtained from word-hyperbolicity of the Coxeter g","core_discovery":"The central result (Theorem 3.6) is an equivalence. Let $(W,S)$ be a finite-rank right-angled Coxeter system and let $\\Gamma$ be the graph with vertex set $S$ and edges between commuting generators. The canonical word-length filtration of $C_q[W]$ satisfies the Haagerup-type condition — there is $K$ with $\\|P_i x P_j\\| \\le K C_q \\|x\\delta_e\\|_2$ for all $i,j,n\\in\\mathbb{N}$ and $x\\in\\chi_n(C_q[W])$ — if and only if $\\Gamma$ contains no induced square. In the square-free case, the same estimate feeds the criterion of [56] to produce a compact quantum metric space structure on $C^*_{r,q}(W)$ with Lip-norm $L_S^{(q)}(x)=\\|[D_S,x]\\|$ (Theorem 3.11). Theorem 4.9 then shows that $(C^*_{r,q}(W),L_S","pith_inferences":["The iff characterization suggests a broader principle the authors leave implicit: for general Coxeter systems the canonical filtration should satisfy the Haagerup-type condition precisely when the Coxeter group is word-hyperbolic; the right-angled no-induced-square case is the sharp instance of that heuristic.","Since right-angled Hecke C*-algebras are amalgamated free products of subalgebras, the failure for induced squares gives an explicit counterexample to preservation of the Haagerup-type condition under arbitrary amalgamated free products, while preservation may hold when amalgamating over finite-dimensional subalgebras.","One can test whether the no-go for induced squares is a feature of the word-length filtration rather than of the algebra: alternative filtrations or weighted Dirac operators might still make $C^*_{r,q}(W)$ a compact quantum metric space in the square-containing case.","The Schur-multiplier strategy for convergence at $q=1$ does not directly transfer to convergence between two non-trivial deformations, because the multipliers do not preserve $C^*_{r,q}(W)$ for $q\\ne 1$; modifying this family is a concrete next step."],"forward_implications":["For every finite-rank right-angled Coxeter system whose generator graph has no induced square, $C^*_{r,q}(W)$ is a compact quantum metric space for every $q>0$ (Theorem 3.11).","For graph containing an induced square, the canonical filtration fails the Haagerup-type condition for all multi-parameters $q$, so this particular metric construction is impossible (Theorem 3.6(2)).","The continuous deformation statement is explicit: $(C^*_{r,q}(W), L_S^{(q)}) \\to (C^*_r(W), L_S^{(1)})$ in propinquity as $q \\to 1$, with no finite-dimensional approximation device (Theorem 4.9).","The uniform constant depends on $q$ only through the clique products $C_q$, and the proof establishes total boundedness uniformly over compact parameter sets (Proposition 3.12), which is what makes the convergence proof work.","Word-hyperbolicity of the underlying Coxeter group is exactly the geometric input that supplies the uniform counting bound, so the square-free condition is what converts the filtration into a genuine metric structure."],"supporting_citations":[{"why":"Provides the Haagerup-type condition and the criterion converting it into a compact quantum metric space; used in Theorem 3.11 and throughout Section 3.","marker":"[56]"},{"why":"Supplies the generator decomposition (Proposition 3.9) into creation, annihilation, and diagonal operators on which the proof of Theorem 3.6 and Lemma 4.6 rests.","marker":"[15]"},{"why":"Gives the word-hyperbolicity characterization of Coxeter groups used in Lemma 3.7 to bound the counting function R_{x,y}(i).","marker":"[53]"},{"why":"Defines compact quantum metric spaces and the quantum Gromov–Hausdorff distance; provides the general framework the paper instantiates.","marker":"[64]"},{"why":"Introduces the quantum Gromov–Hausdorff propinquity and the Leibniz Lip-norm setting used in Theorem 4.9.","marker":"[46]"},{"why":"Establishes conditional negative definiteness of the Coxeter word length, which yields the positive definite functions used for the Schur multipliers in the convergence proof.","marker":"[10]"}],"fun_headline_variants":["No four-cycles: Hecke algebras gain quantum metric","Square-free Coxeter graphs turn Hecke algebras into quantum metrics","As q→1, Hecke algebras converge in quantum propinquity","Coxeter graphs without 4-cycles make Hecke C*-algebras metric"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The whole result depends on the uniform combinatorial bound of Lemma 3.7: in a square-free right-angled Coxeter group, the number of ways a pair of group elements can be decomposed at any length scale is bounded by a constant independent of the pair and the scale; if that bound fails, the Haagerup-type estimate and the compact quantum metric structure collapse.","fun_headline_variants_meta":{"raw":{"variants":["No four-cycles: Hecke algebras gain quantum metric","Square-free Coxeter graphs turn Hecke algebras into quantum metrics","As q→1, Hecke algebras converge in quantum propinquity","Coxeter graphs without 4-cycles make Hecke C*-algebras metric"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000834,"raw_usage":{"total_tokens":3477,"prompt_tokens":749,"completion_tokens":2728,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":2651}},"tokens_in":493,"tokens_out":2728,"duration_ms":22008,"temperature":1.0,"reasoning_tokens":2651,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T21:50:26.973373+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the counting function $R_{x,y}(i)$ from Lemma 3.7 for a square-free right-angled Coxeter system and find a family with $R_{x,y}(i)\\to\\infty$ as $i\\to\\infty$; that would falsify the uniform bound and Theorem 3.6(1). Alternatively, exhibit a square-free right-angled system whose canonical Lip-norm $L_S^{(q)}$ fails to metrizes the weak-$*$ topology on the state space, directly contradicting Theorem 3.11.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Haagerup-type condition and the criterion converting it into a compact quantum metric space; used in Theorem 3.11 and throughout Section 3."},{"cited_title":"Caspers, M","cited_arxiv_id":null,"evidence_quote":"Supplies the generator decomposition (Proposition 3.9) into creation, annihilation, and diagonal operators on which the proof of Theorem 3.6 and Lemma 4.6 rests."},{"cited_title":"Moussong, Hyperbolic Coxeter groups, Thesis (Ph.D.)-The Ohio State University","cited_arxiv_id":null,"evidence_quote":"Gives the word-hyperbolicity characterization of Coxeter groups used in Lemma 3.7 to bound the counting function R_{x,y}(i)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines compact quantum metric spaces and the quantum Gromov–Hausdorff distance; provides the general framework the paper instantiates."},{"cited_title":"Kumar, Kac-Moody groups, their flag varieties and representation theory , Progr","cited_arxiv_id":null,"evidence_quote":"Introduces the quantum Gromov–Hausdorff propinquity and the Leibniz Lip-norm setting used in Theorem 4.9."},{"cited_title":"Bo˙ zejko, T","cited_arxiv_id":null,"evidence_quote":"Establishes conditional negative definiteness of the Coxeter word length, which yields the positive definite functions used for the Schur multipliers in the convergence proof."}],"review_version":1}