{"id":"ae5a7cb6-efe3-4be6-818b-6fd0a08f7478","arxiv_id":"1908.08310","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The geometric limit retraction and an algebraic lexicographic retraction on Weyl groups both equal the unique-closest-point retraction for Coxeter matroids.","lead":"This mathematics paper defines three different ways to send each element of a Coxeter group to a closest element in a special subset called a Coxeter matroid, and proves all three maps agree. It connects the geometry of torus orbits on flag varieties with combinatorial matroid theory.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4 is false for types B/C/D: the lex order (5.1) is not a linear extension of Bruhat order, so Theorem 5.7 fails on a two-element Coxeter matroid in B_2.","rationale":"The reader's (3.8) concern is real but cosmetic: replacing A^u_w by its closure in (3.8) and in the proof of Theorem 3.7 repairs the argument without changing the statement. The more serious problem is Theorem B. Under the standard signed-permutation convention used in the paper, Lemma 5.4 is false, and the two-element matroid {(\\bar{2},1),(1,\\bar{2})} gives a direct counterexample. This is not a disagreement with an external convention; it is an internal inconsistency with the paper's own definitions. Section 5 would need either a different order or a restriction to type A to be correct. Because Theorem B is announced in the abstract as a main result, the current version cannot be accepted as is.","tokens_in":15951,"tokens_out":42089,"duration_ms":405058,"concrete_test":"Enumerate W(B_2) with the paper's one-line notation and implement Definition 5.1 for M = {(\\bar{2},1),(1,\\bar{2})} and Definition (2.2) for R^m; print R^a_M(e) and R^m_M(e). The theorem is contradicted exactly if R^a_M(e) = (1,\\bar{2}) and R^m_M(e) = (\\bar{2},1). To rule out convention mismatches, also verify v^{-1}w = s1 and that v is a Bruhat subword of w.","verdict_should_be":"REJECT","load_bearing_attack":"Lemma 5.4 is false for the signed classical types. It asserts that v <_u w implies v ≺_u w, and the proof invokes the type-A criterion 'v ≤ w iff sorted d-subsets are coordinatewise ≤'. That criterion does not hold for B/C/D. Counterexample in W(B_2), with s0 changing the sign of coordinate 1 and s1 swapping coordinates: let v = s1 s0 = \\bar{2}1 and w = s1 s0 s1 = 1\\bar{2}. Then v^{-1}w = s1, so {v,w} is an edge of the W-permutohedron and hence a Coxeter matroid. Since v is a subword of w, v < w in Bruhat, so R^m_M(e) = v. But for u = e, the order (5.1) is 1 ≺ 2 ≺ \\bar{2} ≺ \\bar{1}, and comparing one-line words gives w = 1\\bar{2} ≺ v = \\bar{2}1; hence R^a_M(e) = w. Thus R^a_M(e) ≠ R^m_M(e), contradicting Theorem 5.7. The same pattern embeds into B_n and D_n (using an even number of bars in D_n), so the failure is not an artifact of rank 2.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies retractions of a finite Coxeter group W onto a Coxeter matroid M. It defines a matroid retraction R^m_M via the Bruhat-order Minimality Property, a geometric retraction R^g_Y for torus orbit closures Y in G/B, and an algebraic retraction R^a_M for Weyl groups of classical Lie type defined by a lexicographic order. The main claims are Theorem A (R^g_Y = R^m_{Y^T}) and Theorem B (R^a_M = R^m_M for every Coxeter matroid M in a product of classical Weyl groups). The paper also discusses representability of Coxeter matroids and gives a partial characterization of two-element Coxeter matroids in S_n.","tokens_in":16264,"tokens_out":7746,"duration_ms":68354,"significance":"The geometric result, if correct, gives a clean toric-geometric realization of the matroid retraction and would be a useful bridge between Coxeter matroid theory and torus orbit closures. The algebraic retraction is an appealing combinatorial construction, and its equality with the matroid retraction would provide an explicit algorithm for closest-point retractions in classical types. However, the central algebraic claim is false as stated: the lexicographic order in Lemma 5.4 does not refine Bruhat order outside type A, and the paper's Theorem B fails already in type B_2. The manuscript therefore cannot fulfill its main advertised contribution, although Theorem A appears salvageable after correcting a notational gap in equation (3.8).","major_comments":[{"comment":"Lemma 5.4 is false for W of type B_2, so its conclusion v <_u w implies v ≺_u w does not hold for signed classical types. Let s_0 be the sign change of coordinate 1 and s_1 the transposition swapping coordinates, and set v = s_1 s_0 = \\bar{2}1 and w = s_1 s_0 s_1 = 1\\bar{2}. Then v^{-1}w = s_1, so {v,w} is an edge of the W-permutohedron and hence a Coxeter matroid by the Gelfand–Serganova criterion. Since v is a subword of w, v < w in Bruhat order. However, for u = e the order (5.1) is 1 ≺ 2 ≺ \\bar{2} ≺ \\bar{1}, and comparing one-line words gives w = 1\\bar{2} ≺ v = \\bar{2}1. Thus v < w does not imply v ≺_e w. The proof of Lemma 5.4 relies on the type-A criterion 'v ≤ w iff sorted d-subsets are coordinatewise ≤', which is not valid for types B, C, and D.","section":"§5, Lemma 5.4"},{"comment":"Theorem 5.7 is false as stated. Using the same elements v = \\bar{2}1 and w = 1\\bar{2} in B_2, take M = {v,w}. This M is a Coxeter matroid because its two vertices are joined by an edge parallel to a root. For u = e, the unique Bruoth-minimal element of M is v, so R^m_M(e) = v. But with the lexicographic order (5.1), w is smaller than v, so R^a_M(e) = w. Hence R^a_M(e) ≠ R^m_M(e), contradicting Theorem 5.7. The proof of Theorem 5.7 goes through Lemma 5.5, whose proof depends on Lemma 5.4; the failure of Lemma 5.4 therefore invalidates the algebraic-retraction theorem outside type A. The same construction embeds into B_n and D_n by including extra coordinates, so the problem is not a rank-2 artifact.","section":"§5, Theorem 5.7 and Lemma 5.5"},{"comment":"Equation (3.8) is written as A^u_w = ⨆_{w ≤_u v ≤_u u w_0} A^u_v, and the following sentence asserts (A^u_w)^T = {v | w ≤_u v ≤_u u w_0}. As written this is false: the open Bruhat cell A^u_w is not equal to the union of cells in its closure, and its T-fixed points are not the entire interval. The correct statement is that \\overline{A^u_w} is the disjoint union of the A^u_v over that interval, and hence (\\overline{A^u_w})^T is the interval. The proof of Theorem 3.7 uses exactly the closure interpretation: from T·x ⊆ A^u_w the closure Y satisfies Y ⊆ \\overline{A^u_w}, so Y^T is contained in the interval. This is a load-bearing notational gap, but it is readily fixed by placing overlines in (3.8) and in the proof.","section":"§3, equation (3.8) and proof of Theorem 3.7"}],"minor_comments":[{"comment":"There is a typo: 'identitity' should be 'identity'.","section":"Remark 1.2"},{"comment":"In the sentence 'it tunrs out that R^a_M = R^m_M', 'tunrs' should be 'turns'.","section":"§5, before Definition 5.1"},{"comment":"Reference [1] is corrupted in the typeset text ('Bia/suppress lynicki Birula'); it should read A. Białynicki-Birula.","section":"References"}],"recommendation":"reject","confidential_remarks":"The counterexample in type B_2 is elementary and directly disproves Theorem 5.7, which is one of the paper's two central theorems. Since the algebraic retraction construction is a core contribution and the error cannot be repaired by a small modification within the paper's stated scope (it fails for classical signed types), I recommend rejection. Theorem A may be salvageable after correcting the open-cell/closure issue in equation (3.8), but the manuscript as a whole does not meet the bar for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Theorem A is a good result and probably correct once one fixes a closure/open-cell typo. Theorem B, as stated for all classical types, is false: Lemma 5.4 does not hold for B/C/D.\n\nThe geometric part deserves credit. Using limit points to define R^g_Y and then proving R^g_Y = R^m_{Y^T} via the B_u-Bruhat decomposition is a clean argument. The cone description in Corollary 3.3 is a nice by-product. The type-A representability discussion (including the Fano-plane non-representable example in S_7) is concrete and useful, and Proposition 6.1 is a careful small case.\n\nThe soft spots are real. Equation (3.8) as printed says the open cell A^u_w is the disjoint union of cells over the full Bruhat interval; that's only true for the closure. The T-fixed point computation in Theorem 3.7 uses the closure version, so the theorem is probably salvageable with that correction, but the manuscript needs to say it.\n\nThe bigger issue is Theorem B. Lemma 5.4 asserts that Bruhat order implies the lexicographic order (5.1), and the proof invokes the type-A characterization of Bruhat order by coordinatewise comparison of sorted d-subsets. That characterization is false for the signed types. Concrete counterexample in B_2, with s_0 the sign change of coordinate 1 and s_1 the transposition: v = s_1 s_0 = 2\\bar{1} and w = s_1 s_0 s_1 = 1\\bar{2}. Then v is a subword of w, so v < w in Bruhat, and v^{-1}w = s_1, so {v,w} is a Coxeter matroid. But for u = e the order (5.1) is 1 \\prec 2 \\prec \\bar{2} \\prec \\bar{1}, and lexicographically w = 1\\bar{2} \\prec v = 2\\bar{1}. Hence R^a_M(e) = w while R^m_M(e) = v, contradicting Theorem 5.7. The same pattern embeds in B_n and D_n, so this is not a rank-two artifact.\n\nRecommendation: the geometric half is solid and should survive peer review after the closure fix; the algebraic half needs to be restricted to type A or replaced with an ordering that genuinely extends Bruhat order. This is exactly the kind of paper a serious referee should see, because the flaw is subtle and the geometric contribution is real.","headline":"The geometric retraction result is worth keeping; the algebraic retraction theorem is false for B/C/D because Lemma 5.4 fails.","tokens_in":16772,"tokens_out":10105,"would_cite":false,"duration_ms":91462,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14M25","20F55","52B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Geometric limits of torus actions on flag varieties and the closest-point retraction of a Coxeter matroid define the same map on the Weyl group.","keywords":["Flag varieties","toric varieties","Coxeter matroids","retractions","Weyl groups","Bruhat order","torus orbit closures","Gelfand-Serganova polytopes"],"falsifier":"Compute the T-fixed points of the closure of A^u_w for a non-type-A group such as B_2 or G_2 and compare with the Bruhat interval from w to u w_0; finding any fixed point outside that interval is a concrete counterexample to the equality.","tokens_in":15786,"feed_emoji":"🔁","tokens_out":8580,"duration_ms":78415,"temperature":0.7,"pith_summary":"This paper proves that three ways of sending every element of a Weyl group to a prescribed subset—one combinatorial, one geometric, one algebraic—coincide whenever the subset is a Coxeter matroid. For any closure Y of a torus orbit in a flag variety G/B, the geometric retraction R^g_Y, which sends u to the limit point of a generic one-parameter torus action, equals the matroid retraction R^m_{Y^T}, the unique closest point to u in the Coxeter length metric. For products of classical Weyl groups, the algebraically defined retraction R^a_M, built from a lexicographic order, also equals R^m_M. The identification turns the abstract notion of a Coxeter matroid into a concrete geometric object and makes the fan of a torus orbit closure readable off the retraction.","feed_headline":"Torus geometry and Coxeter matroids define the same retraction","feed_subtitle":"The geometric limit map on a flag variety is the matroid's nearest-point projection.","key_machinery":"The load-bearing object is the Coxeter matroid retraction R^m_M, defined by the Minimality Property: for each u, exactly one element of M is minimal in the u-twisted Bruhat order ≤_u. The proof that R^g_Y=R^m_{Y^T} is carried by the Bruhat decomposition into cells A^u_w = uB^-$u^{{-1}}$wB/B: if x lies in A^u_w then the limit defining R^g_Y(u) is w, and the T-fixed points of the closure of A^u_w form the interval {v : w ≤_u v ≤_u u w_0}, so w is the unique ≤_u-minimal fixed point. The algebraic retraction R^a_M is carried by the u-lexicographic order ≺_u on one-line notation, which is compatible with Bruhat order in classical types; that compatibility plus left-invariance yields Lemma 5.5 and Theorem 5.7. All three constructions land in the same map because each picks the unique 'first' element of M when M is a Coxeter matroid.","core_discovery":"The central claim is that the abstract unique-closest-point retraction on a finite Coxeter group has two concrete incarnations. The paper defines the length metric d(v,w)=ℓ($v^{{-1}}$w), and for a Coxeter matroid M (a subset with a unique u-minimal element for each u) defines the matroid retraction R^m_M(u) as that unique minimal element. Theorem 3.7 states that for any T-orbit closure Y in G/B, the geometric retraction R^g_Y—defined by taking lim_{t→0} λ_u(t)·x for a generic λ_u in the chamber C(u)—equals R^m_{Y^T}. Theorem 5.7 states that for any Coxeter matroid M in a product of classical Weyl groups, the algebraic retraction R^a_M, given by the ≺_u-lexicographic minimum, equals R^m_M. Along the way the paper reformulates the geometric retraction through Bruhat cells A^u_w, uses the Gelfand–Serganova polytope criterion, shows every Bruhat interval is representable, and exhibits a non-representable Coxeter matroid in S_7 built from the Fano plane.","pith_inferences":["Since R^a_M is computed by a lexicographic scan, it gives a practical nearest-point oracle for Coxeter matroids that avoids explicit Bruhat comparisons; the paper does not draw this algorithmic consequence.","The S_7 example suggests the obstruction to representability is exactly the non-realizability of the underlying ordinary matroid; checking all small-rank Coxeter matroids against this condition would test that identification.","The equality R^g=R^m implies the fan of a torus orbit closure can be reconstructed purely combinatorially from the fixed-point matroid; one could try to compute toric invariants of Y from R^m without coordinates."],"forward_implications":["For any torus orbit closure Y, the limit that defines R^g_Y(u) is the unique T-fixed point of Y closest to u in the Coxeter length metric.","The maximal cone of the fan of Y attached to y∈Y^T is the union of the chambers C(u) over all u with R^g_Y(u)=y; equivalently, the geometric retraction encodes the fan.","In a product of classical Weyl groups, the algebraic retraction computes the closest-point retraction for every Coxeter matroid without comparing full Bruhat intervals.","Every Bruhat interval [v,w] in S_n occurs as the fixed-point set of a torus orbit closure, so Bruhat intervals are representable Coxeter matroids; however, the Fano-plane example gives a Coxeter matroid of S_7 that is not representable.","For two-element subsets of S_n, the two conditions (unique closest point for every u, and that closest point given by R^a_M) characterize Coxeter matroids; the same characterization is open for larger subsets."],"supporting_citations":[{"why":"Supplies the T-invariant affine cells S_w whose coordinate description makes the geometric limit well-defined.","marker":"[1]"},{"why":"Gives the one-line Bruhat criterion used in Lemma 5.4 to compare Bruhat order with the lexicographic order.","marker":"[2]"},{"why":"Is the source for Coxeter matroids, the Maximality Property, and the Gelfand–Serganova polytope characterization.","marker":"[3]"},{"why":"Establishes that T-fixed point sets of torus orbit closures are Coxeter matroids and gives the minor criterion used for representability.","marker":"[7]"},{"why":"Provides the tangent-space decomposition used to prove that limits in the Bruhat cell land on wB.","marker":"[8]"},{"why":"Supplies the earlier retraction idea and the fixed-point computation for Schubert varieties on which Definition 5.1 and Remark 4.3 build.","marker":"[11]"},{"why":"Supplies the non-realizability of the Fano plane that produces the non-representable Coxeter matroid in S_7.","marker":"[14]"}],"fun_headline_variants":["Geometric retraction equals matroid retraction on flag varieties","Torus orbit closures and Coxeter matroids share one retraction","Flag variety geometry yields the matroid's unique nearest point","Three definitions, one retraction: flag varieties and Coxeter matroids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the geometric–matroid equality assumes that the closure of each Bruhat cell A^u_w contains exactly the T-fixed points in the interval from w to u w_0; an extra fixed point anywhere would break the equality.","fun_headline_variants_meta":{"raw":{"variants":["Geometric retraction equals matroid retraction on flag varieties","Torus orbit closures and Coxeter matroids share one retraction","Flag variety geometry yields the matroid's unique nearest point","Three definitions, one retraction: flag varieties and Coxeter matroids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00039,"raw_usage":{"total_tokens":2135,"prompt_tokens":1111,"completion_tokens":1024,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":951}},"tokens_in":727,"tokens_out":1024,"duration_ms":9646,"temperature":1.0,"reasoning_tokens":951,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:43:22.195816+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the T-fixed points of the closure of A^u_w for a non-type-A group such as B_2 or G_2 and compare with the Bruhat interval from w to u w_0; finding any fixed point outside that interval is a concrete counterexample to the equality.","supporting_citations":[{"cited_title":"Bia/suppress lynicki Birula","cited_arxiv_id":null,"evidence_quote":"Supplies the T-invariant affine cells S_w whose coordinate description makes the geometric limit well-defined."},{"cited_title":"Billey and V","cited_arxiv_id":null,"evidence_quote":"Gives the one-line Bruhat criterion used in Lemma 5.4 to compare Bruhat order with the lexicographic order."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Is the source for Coxeter matroids, the Maximality Property, and the Gelfand–Serganova polytope characterization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that T-fixed point sets of torus orbit closures are Coxeter matroids and gives the minor criterion used for representability."},{"cited_title":"Guillemin, T","cited_arxiv_id":null,"evidence_quote":"Provides the tangent-space decomposition used to prove that limits in the Bruhat cell land on wB."},{"cited_title":"Lee and M","cited_arxiv_id":null,"evidence_quote":"Supplies the earlier retraction idea and the fixed-point computation for Schubert varieties on which Definition 5.1 and Remark 4.3 build."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the non-realizability of the Fano plane that produces the non-representable Coxeter matroid in S_7."}],"review_version":1}