{"id":"7b9d0b5b-8dd3-44d8-b557-4cbf35138a8a","arxiv_id":"2501.09447","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Auslander regular algebras, the grade bijection equals the permutation matrix in the Bruhat factorization of the Coxeter matrix, with applications to permanents and distributive lattices.","lead":"This paper proves that two bijections attached to Auslander-Gorenstein algebras, Iyama's grade bijection and the Auslander-Reiten bijection, are the same, and that for Auslander regular algebras this bijection is encoded by the permutation matrix in the Bruhat decomposition of the Coxeter matrix. The new linear algebra description makes the bijection quick to compute and yields a test for Auslander regularity plus a new characterization of distributive lattices.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.13 rests on two structural facts about Ext^r(S,A) quoted from inside the proof of Iyama's Theorem 2.10; until these are verified, the upper-triangularity of φ^{-1}M and the ±1 diagonal of U2 are not fully supported.","rationale":"I read the paper as establishing a genuinely useful bridge between the homological grade bijection and the Bruhat permutation of the Coxeter matrix. The proof of Theorem 3.13 is short and elegant, and once Proposition 3.9 is granted, the upper-triangularity argument with the admissible ordering and the identity pdim I(j)=grade S(j), idim P(φ(i))=grade S(i) is sound. I checked the main internal steps: the equivalence in Theorem 3.3, the use of Lemma 3.1, the sign identification M=-C, and the permanent corollary. I found no circularity or counterexample. The only genuinely load-bearing dependency is the unproved quotation from the proof of Iyama's Theorem 2.10 inside Proposition 3.9. This is exactly the reader's weakest assumption, so I agree with the conditional verdict: the paper is likely correct but should make that dependency explicit by stating and proving (or precisely referencing) the two structural facts. The missing citation of [MTY] and the terseness of the rowmotion pivot argument are secondary and do not affect the central claim.","tokens_in":16118,"tokens_out":27680,"duration_ms":286976,"concrete_test":"Read the proof of [I, Theorem 2.10] and extract the exact assertions corresponding to the two facts used in Proposition 3.9: that in Ext^r_A(S,A) all composition factors except the socle Dφ(S) have grade greater than r, and that Dφ(S) appears with multiplicity one. Record the relevant sentence(s). If both are literally in [I], the proof of Theorem 3.13 is supported; if either is absent, Proposition 3.9 needs an independent proof or Theorem 3.13 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result (Theorem 3.13) reduces to Proposition 3.9. There, the set P of projectives P for which I(S) appears in degree r in the injective coresolution of P is identified with the support of Ext^r_A(S,A). The proof then cites 'part (ii) of the proof of [I, Theorem 2.10]' for two facts: (1) every composition factor of Ext^r_A(S,A) other than the socle Dφ(S) has grade > r, and (2) the simple Dφ(S) occurs with multiplicity one. These facts are not stated as a theorem, not proved in this paper, and the second is essential: if Dφ(S) had multiplicity m>1, then the (φ(i),i) entry of M would be ±m and the upper triangular factor U2 in C=P U2 would have diagonal m, not ±1. Fact (1) is equally essential for the inequality grade S(j)<grade S(i) that makes φ^{-1}M upper triangular. The rest of the proof of Theorem 3.13 is a clean consequence of Proposition 3.9, so this citation is the single load-bearing point. This is a dependency on the literature, not an internal contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes that for Auslander-Gorenstein algebras, Iyama's grade bijection coincides with the Auslander-Reiten bijection, and for Auslander regular algebras with an admissible ordering of simples, the Coxeter matrix admits a Bruhat decomposition C = P U2 in which P is the permutation matrix of the grade bijection and U2 is upper triangular with diagonal entries ±1. From this it derives that the permanent of the Coxeter matrix is ±1. In a second part, it gives a characterisation of finite distributive lattices in terms of the Bruhat decomposition of their incidence algebra's Coxeter matrix, and applies the results to blocks of category O.","tokens_in":16420,"tokens_out":37873,"duration_ms":337088,"significance":"The linear-algebraic reinterpretation of the grade bijection is elegant and has immediate computational value, and the permanent corollary gives a fast numerical necessary condition for Auslander regularity. The lattice characterisation is striking and connects rowmotion to Coxeter matrices. The proof strategy is mostly clean: Theorem 3.3's equivalence and the AR/grades bijection proof are well structured, and the matrix argument for Theorem 3.13 is transparent once its hypotheses are granted. The main caveat is that the proof of Theorem 3.13 delegates a structurally essential point to a fact quoted from inside the proof of Iyama's Theorem 2.10 rather than to a stated theorem.","major_comments":[{"comment":"The proof of Theorem 3.13 rests on Proposition 3.9, which invokes \"part (ii) of the proof of Theorem 2.10 in [I]\" for two facts: all composition factors of Ext^r_A(S,A) other than the socle Dφ(S) have grade > r, and Dφ(S) occurs with multiplicity one. These facts are not stated in [I, Theorem 2.10] (as cited) and not proved here. The upper triangularity of φ^{-1}M in the proof of Theorem 3.13 depends on the first fact, and the assertion that the diagonal entries of U2 are ±1 depends on the second: if the multiplicity of Dφ(S) were m > 1, the corresponding diagonal entry would be ±m. Please provide a self-contained proof or a direct reference to a stated theorem containing these properties.","section":"Proposition 3.9 / Theorem 3.13"},{"comment":"The dot-product rule in the proof of Lemma 4.1 reads \"C_{x·} · [P(y)] is −1 if y ≥ x and 0 otherwise,\" but no convention for the dimension vectors of projectives and injectives in the incidence algebra is specified. In particular, with Cartan entries e_{i,j} = dim e_j A e_i, the identity C·dim P(y)=-dim I(y) determines the direction of the inequality, and the subsequent equivalence \"for y ∈ I, this is the same as −1 if y ≥ x ∨ z\" depends on that direction. Since Corollary 4.2 and Theorem 4.9 rely on Lemma 4.1, please state the convention explicitly and verify the proof of Lemma 4.1 under that convention.","section":"Lemma 4.1"},{"comment":"The verification of the Coxeter matrix formula is given as a few dot-product assertions (\"the dot product is therefore zero, as desired\"). This is load-bearing for Theorems 4.7 and 4.9, which identify leftmost nonzero entries of rows via Δ(Y) and χ_red(Δ(Y)). Please expand the proof to show explicitly how the entries follow from the definition C = -ω^T ω^{-1} and the Möbius inversion for the incidence algebra, and state the precise support conventions for projective and injective dimension vectors.","section":"Lemma 4.5"},{"comment":"The statement \"the leftmost non-zero entry in row Y is in column ⟨min Y^c⟩ = row^{-1}(Y)\" needs clarification: the standard rowmotion formula row(I) = ⟨min(P \\ I)⟩ gives ⟨min Y^c⟩ = row(Y), not row^{-1}(Y). The inverse appears because Y is a set of meet-irreducibles and the correspondence with R reverses order, but this is not spelled out. Please state the convention explicitly and define the rowmotion used on R and on order ideals of meet-irreducibles, so that the conclusion that the Coxeter permutation is the grade bijection/rowmotion is verifiable.","section":"Theorem 4.7"}],"minor_comments":[{"comment":"The running title contains a typo: \"ma trices\" should be \"matrices\".","section":"Title page"},{"comment":"The phrase \"where is acts as ν\" should read \"where it acts as ν\".","section":"Section 2.3.4"},{"comment":"The matrix M is introduced as containing \"multiplicity (with sign)\"; please specify the sign convention, since the diagonal entries of U2 depend on it.","section":"Proof of Theorem 3.13"},{"comment":"The poset example would be easier to verify if a Hasse diagram were included.","section":"Example 3.15"},{"comment":"The notation \"idim\" is used without definition; it should be defined where first used, parallel to \"pdim\".","section":"Section 2.1"}],"recommendation":"major_revision","confidential_remarks":"The main theorem's dependence on a fact quoted from the proof of Iyama's Theorem 2.10 should be verified against [I] by the editor or a second referee. If the facts in Proposition 3.9 are not in the published statement, the authors must add a proof. The conventions in Section 4 also need careful checking, particularly the direction of the inequality in Lemma 4.1 and the rowmotion identification in Theorem 4.7."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one before sending it back. The headline claim is real: for Auslander regular algebras, the permutation part of the Bruhat decomposition of the Coxeter matrix is exactly Iyama's grade bijection, and this also matches the Auslander-Reiten bijection. That is a new and useful way to compute a homological invariant from a matrix factorization.\n\nThe paper does a lot right. Theorem 3.3, the characterization of Auslander-Gorenstein by grade S = pdim I(S), is clean and self-contained. The proof of Theorem 3.13 is short: once you accept Proposition 3.9, the upper-triangularity argument via admissible orderings is elegant. The permanent corollary is a genuinely fast numerical obstruction, and the example with permanent -1501 is nice. The distributive lattice section gives a new characterization (U1 = id in a Bruhat decomposition) and connects to rowmotion, building on [IM]. I also like the category O application.\n\nThe soft spot is exactly the one the stress-test note highlights. Proposition 3.9 cites 'part (ii) of the proof of [I, Theorem 2.10]' for two structural facts: all composition factors of Ext^r(S,A) except the socle have grade > r, and the socle occurs with multiplicity one. The second fact is essential for the ±1 diagonal of U2; without it you could get other diagonal entries. The first is essential for upper triangularity. These facts may well be true and proved by Iyama, but citing a step inside a proof rather than a stated theorem is fragile, and the authors should either state and prove them as a lemma or point to a theorem number that covers them. This is addressable, not fatal.\n\nSection 4 is more compressed. Lemma 4.5's formula for the Coxeter matrix of a lattice with all proper upper intervals distributive is plausible but only sketched; Lemma 4.8's induction is quick. The rowmotion identification in Theorem 4.7 is dense, and I'd want a referee to check the edge cases around 'min Y^c' and the inverse rowmotion convention. Also, the relation to the authors' earlier [MTY] is only a one-line remark; they should say a bit more about what is genuinely new beyond that paper. (The reader's note that [MTY] is never cited is wrong; it appears in Section 2.3.4.)\n\nAll in all, this paper deserves a serious referee. It is a genuine contribution, the main arguments are sound on a first pass, and the issues are fixable with a revision. I would cite it and would bring it to a reading group that cares about homological algebra or lattice combinatorics.","headline":"A genuinely new and mostly clean paper identifying the grade bijection with the Bruhat permutation of the Coxeter matrix, with one load-bearing citation to a proof-part of Iyama that should be promoted to a lemma.","tokens_in":16908,"tokens_out":3561,"would_cite":true,"duration_ms":33560,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","16E10","06A11"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that the grade bijection of an Auslander regular algebra coincides with the permutation matrix in the Bruhat decomposition of its Coxeter matrix, and derives consequences for permanents, distributive lattices, and category…","keywords":["Partially ordered sets","Distributive lattices","Auslander regular algebras","Coxeter matrix","Rowmotion bijection","grade bijection","Bruhat decomposition"],"falsifier":"Compute the permanent of the Coxeter matrix of any finite-dimensional algebra of finite global dimension; if an Auslander regular algebra has a Coxeter matrix with permanent other than 1 or −1, the central theorem fails, and since the permanent is independent of the ordering of simples, a single such example suffices.","tokens_in":15962,"feed_emoji":"🔢","tokens_out":7355,"duration_ms":65367,"temperature":0.7,"pith_summary":"The paper establishes that two apparently different ways of associating a permutation to an Auslander regular algebra—one homological (the grade bijection on simple modules), one linear algebraic (the permutation part of a Bruhat factorization of the Coxeter matrix)—always give the same answer. This makes the grade bijection computable by Gaussian elimination instead of by injective resolutions. As consequences, the permanent of any Auslander regular Coxeter matrix is 1 or −1, giving a quick numerical way to disprove Auslander regularity; and a finite lattice is distributive exactly when the first nonzero entry of each row of its Coxeter matrix lies in a distinct column. The paper also shows that the grade bijection agrees with the classical injective-to-projective syzygy bijection, and applies the results to blocks of category O.","feed_headline":"Grade bijection matches Coxeter permutation","feed_subtitle":"Permutation in a Bruhat factorization is a homological map, giving fast computations and a ±1 permanent test.","key_machinery":"The load-bearing identity is the Bruhat factorization $C = P U_2$. The proof identifies the Coxeter matrix $C = -\\omega^T \\omega^{-1}$ with the matrix $M$ recording, with signs, which indecomposable injectives appear in the minimal injective coresolutions of indecomposable projectives; the row/column dimension-vector relation then gives $M = -C$. The admissible ordering of simples (grades nonincreasing along the order) makes the matrix $\\varphi^{-1}M$ upper triangular, using the key property that the grade bijection $\\varphi$ sends grade to cograde. The permutation matrix $P$ in the Bruhat decomposition is thus forced to be $\\varphi$.","core_discovery":"The central claim is that for a finite-dimensional Auslander regular algebra, the grade bijection—the map sending a simple module to the top of the socle of its first nonzero Ext against the algebra—is encoded verbatim by linear algebra. When the simple modules are ordered so that grades increase with index, the Coxeter matrix factorizes as $C = P U$, where $P$ is the permutation matrix of the grade bijection and $U$ is upper triangular with diagonal entries 1 or −1. Equivalently, the grade bijection is the unique permutation part of the Bruhat decomposition of the Coxeter matrix. Along the way the paper proves that the grade bijection coincides with the classical bijection between indecomposable injective and projective modules given by syzygies, and uses the factorization to show that the permanent of any Auslander regular Coxeter matrix is 1 or −1, and to characterize distributive lattices by the pivot pattern of their Coxeter matrix.","pith_inferences":["Beyond the paper: because a Bruhat permutation is computed by pivoting, the result should make grade bijections computable in polynomial time for algebras where injective resolutions are slow; an implementation on quiver algebras would be a direct test.","Beyond the paper: the ±1 permanent constraint is a necessary condition that may be much easier to verify than Auslander regularity; applying it to families of algebras of finite global dimension with unknown homological dimension could identify new candidates for Auslander regular algebras.","Beyond the paper: the lattice theorem suggests a purely matrix-theoretic route to rowmotion on distributive lattices; one could test whether the entries of $U_2$ in the factorization carry additional order information about the lattice."],"forward_implications":["The permanent of the Coxeter matrix of any Auslander regular algebra is either 1 or −1, providing a fast necessary test: a permanent outside this set shows the algebra is not Auslander regular.","For a finite lattice ordered by a linear extension, distributivity is equivalent to the condition that the first nonzero entry of each row of the Coxeter matrix lies in a different column; in that case the Coxeter permutation is the rowmotion bijection.","For blocks of category O, the set of grades of modules equals twice the values of Lusztig's a-function.","For an Auslander regular algebra with a simple-preserving duality, the Cartan matrix is symmetric and the grade bijection is the identity.","The grade bijection of an Auslander-Gorenstein algebra equals the classical injective-to-projective syzygy bijection, so the two historical definitions describe the same map."],"supporting_citations":[{"why":"Supplies the definition and structural properties of the grade bijection, including the fact that all but one composition factor and top have higher grade or cograde, on which the upper-triangularity argument rests.","marker":"[I]"},{"why":"Introduces the classical injective-to-projective syzygy bijection and the r-Gorenstein properties used to identify final terms in resolutions.","marker":"[AR]"},{"why":"Establishes that for incidence algebras of distributive lattices the grade bijection is the rowmotion bijection, which the lattice characterization builds on.","marker":"[IM]"},{"why":"Shows that blocks of category O are Auslander regular, the input for the application relating grades to Lusztig's a-function.","marker":"[KMM]"},{"why":"Provides the theorem that projective dimensions of injectives in blocks of category O are twice Lusztig's a-function values.","marker":"[Maz]"},{"why":"Supplies the standard fact that the Coxeter matrix times the dimension vector of a projective equals minus the dimension vector of the corresponding injective, used to identify the Coxeter matrix with the coresolution matrix.","marker":"[ASS]"}],"fun_headline_variants":["Grade bijection equals Bruhat permutation","Coxeter factorization pinpoints grade bijection","Permanent of Auslander Coxeter matrix is ±1","Distributive lattices from Coxeter pivot pattern","Bruhat map is the grade bijection"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes a known structural property of the grade bijection, cited from the literature rather than proved here: in the module $\\mathrm{Ext}^r_A(S,A)$, every simple composition factor other than the socle has grade greater than $r$, and dually every top except one has cograde greater than $r$.","fun_headline_variants_meta":{"raw":{"variants":["Grade bijection equals Bruhat permutation","Coxeter factorization pinpoints grade bijection","Permanent of Auslander Coxeter matrix is ±1","Distributive lattices from Coxeter pivot pattern","Bruhat map is the grade bijection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000317,"raw_usage":{"total_tokens":1780,"prompt_tokens":922,"completion_tokens":858,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":786}},"tokens_in":538,"tokens_out":858,"duration_ms":6818,"temperature":1.0,"reasoning_tokens":786,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:02:41.326392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the permanent of the Coxeter matrix of any finite-dimensional algebra of finite global dimension; if an Auslander regular algebra has a Coxeter matrix with permanent other than 1 or −1, the central theorem fails, and since the permanent is independent of the ordering of simples, a single such example suffices.","supporting_citations":[],"review_version":1}