{"id":"c02b14b6-2cce-452c-a560-2a59381a53b3","arxiv_id":"2508.19079","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A survey of finite-dimensional Auslander-Gorenstein algebras, including classifications for monomial and incidence algebras and the identification of the Auslander-Reiten permutation with rowmotion, Ringel's homological permutation, and the Coxeter permutation.","lead":"This survey maps the finite-dimensional theory of Auslander-Gorenstein algebras, a class of algebras with strong symmetry in their injective and projective resolutions, and reports recent links between their Auslander-Reiten permutation and combinatorial bijections. A generalist might read it as a compact guide to an active area connecting homological algebra with posets, Dyck paths, and rowmotion.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.8.5, the advertised identification of the Coxeter permutation with Ringel's homological permutation for linear Nakayama algebras, is deferred to an in-preparation paper; this is the most load-bearing unverified claim in the survey.","rationale":"I read the paper in good faith as a survey. The central advertised claim is the existence of a canonical permutation on simples of Auslander-Gorenstein algebras that specializes to several known bijections. The fully proved Proposition 1.2.1 is correct: part (1) uses the finiteness of the number of simples to pass from finite projective dimension of each indecomposable injective to finite injective dimension of the regular module, and part (2) is a standard argument. The main theorems giving the coincidences with rowmotion and with the Coxeter permutation for Auslander regular algebras are referenced to available preprints ([50], [58]), so those parts are checkable. The single weakest spot is Theorem 1.8.5, which identifies the Coxeter permutation with Ringel's homological permutation for all linear Nakayama algebras. Its proof is deferred to [57] (in preparation), and even the auxiliary Lemma 1.8.1 used to compute the Coxeter permutation is deferred there. Since this theorem is a headline interaction in Section 1.8 and is not verifiable from the manuscript, the reader's CONDITIONAL verdict is appropriate. I would not move the verdict: the survey is honest about what is deferred, and the other advertised interactions are supported by available references. If anything, the paper should mark Theorem 1.8.5 and Lemma 1.8.1 more visibly as unproved announcements. The proposed computational test would settle whether the announced theorem holds for all small cases, providing evidence for or against the claim while the full proof is still forthcoming.","tokens_in":25795,"tokens_out":39222,"duration_ms":331238,"concrete_test":"Enumerate all linear Nakayama algebras with up to 10 simples (equivalently, all Dyck paths of semilength 10, totaling 16796 cases). For each algebra, compute (a) the Coxeter permutation by taking a Bruhat decomposition of the Cartan matrix and reading the row-permutation as in Lemma 1.8.1, and (b) Ringel's homological permutation h as defined in Section 1.8. Compare the two. If any mismatch occurs, Theorem 1.8.5 is false. For extra rigor, compute (a) also directly from a Bruhat decomposition of the Coxeter matrix -Phi^T Phi^{-1} to independently verify Lemma 1.8.1. The computation is elementary and should run in seconds to minutes in a computer algebra system.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The survey's central unification narrative is that the Auslander-Reiten permutation (equivalently, the grade permutation by Theorem 1.2.6) coincides with rowmotion, the Coxeter permutation, and Ringel's homological permutation. The rowmotion and Coxeter-permutation identifications for distributive lattices and Auslander regular algebras are backed by available references ([50], [58]). However, Theorem 1.8.5—asserting that for every linear Nakayama algebra the Coxeter permutation equals Ringel's homological permutation—is deferred entirely to the in-preparation reference [57]. The supporting Lemma 1.8.1, which lets one compute the Coxeter permutation from a Bruhat decomposition of the Cartan matrix rather than the Coxeter matrix, is also deferred to [57]. Thus the claimed interaction between the Coxeter permutation and Ringel's homological bijection, a headline result of Section 1.8, cannot be checked from the manuscript or from any publicly available proof. A counterexample here would not merely remove an example; it would break the thesis that the Coxeter permutation is the right generalisation of the Auslander-Reiten permutation for Nakayama algebras. Other deferred results (Theorem 1.7.6, Proposition 1.7.2, Theorem 1.4.7) are less central to the permutation-unification narrative. The paper is transparent about the deferral, but the load-bearing status of Theorem 1.8.5 remains a verification gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of Auslander-Gorenstein algebras, with emphasis on finite-dimensional algebras. It collects characterizations of the Auslander-Gorenstein property, discusses the Auslander–Reiten and grade bijections on simple modules, and reviews results in category O, higher Auslander algebras, monomial algebras, incidence algebras of posets, and Nakayama algebras. The paper's own contribution is Proposition 1.2.1, which places the Auslander–Reiten Conjecture between the Generalised Nakayama Conjecture and the Nakayama Conjecture; this is proved in the text. The advertised broader narrative is that the Auslander–Reiten permutation coincides with the grade permutation, with rowmotion in the distributive-lattice case, with the Coxeter permutation in admissible orderings, and with Ringel's homological permutation for linear Nakayama algebras. Several of these latter claims are deferred to in-preparation papers by the same authors.","tokens_in":26094,"tokens_out":9168,"duration_ms":85489,"significance":"If the stated results are correct, the survey is a useful and well-organized synthesis of a rapidly developing area, and Proposition 1.2.1 fills a small but real gap in the literature. The paper is also valuable for its explicit examples and for collecting many equivalent characterizations of Auslander-Gorenstein algebras. However, the central unification thesis--that the Coxeter permutation is the correct generalisation of the Auslander–Reiten permutation--depends on at least one theorem (Theorem 1.8.5) whose proof is not available to the reader. Several other advertised classification results are likewise deferred. This makes the survey's strongest claims unverifiable from the manuscript or from public sources, and these claims are load-bearing rather than peripheral.","major_comments":[{"comment":"The advertised unification of Section 1.8 rests on Theorem 1.8.5, which identifies the Coxeter permutation with Ringel's homological permutation for every linear Nakayama algebra. The proof is not given and is deferred entirely to the in-preparation reference [57]; Lemma 1.8.1 and Theorem 1.8.6 are also deferred to [57]. The reader cannot verify this central claim from the manuscript or from any public source. Since the section is explicitly framed as showing that the Coxeter permutation is the right generalisation of the Auslander-Reiten permutation, this is a load-bearing gap. Please either include proofs (an appendix would suffice), cite a public preprint, or downgrade these statements to conjectures with a clear caveat.","section":"§1.8, Theorem 1.8.5"},{"comment":"Several statements are presented as theorems/propositions but are deferred to in-preparation works: Theorem 1.7.6 (2-Gorenstein incidence algebras iff dissective posets), Proposition 1.7.2 (Conjecture 1.2.4 for lattices), and Theorem 1.4.7 (Koszul dual criterion). The manuscript explicitly states that [52] and [18] are in preparation. These results are part of the advertised classifications and interactions, so they are not merely expository. They should either be proved in the survey, supported by a publicly available preprint, or explicitly marked as unproved claims/conjectures.","section":"§1.7, Theorem 1.7.6 / Prop. 1.7.2; §1.4, Theorem 1.4.7"},{"comment":"The sentence 'Naturally, every semidistributive lattice is distributive, but the converse is not true' is mathematically false and is immediately contradicted by the displayed example of a non-distributive semidistributive lattice. The intended statement is presumably 'every distributive lattice is semidistributive, but the converse is not true.' This is not load-bearing for the main theorems, but it is a clear error in a prominent passage and should be corrected.","section":"§1.8, Theorem 1.8.3"}],"minor_comments":[{"comment":"There is a stray ']' after '[8,Proposition5.4]' in the displayed statement.","section":"§1.2, Theorem 1.2.4"},{"comment":"The phrase 'every semidistributive lattice is distributive' should be reversed: every distributive lattice is semidistributive. The example immediately following the assertion makes the typo obvious.","section":"§1.8, after Theorem 1.8.3"},{"comment":"The case descriptions in (i)-(iii) are garbled in the arXiv rendering; the diagrams and arrows for the three cases should be redrawn or described in text so that the rule for \\hat\\psi is unambiguous.","section":"§1.6, Theorem 1.6.5"},{"comment":"The display 'A=KQ_2/I' with 'Q_2=' is missing the actual quiver data; the arrows and relations are not readable. Please provide a complete display.","section":"§1.5, Example 1.5.1"},{"comment":"Several references lack publication years or full venue details, e.g. [6], [9], [25], [46], and [72]. Please standardize the reference list.","section":"References"},{"comment":"The opening line 'WegiveasurveyonAuslander-Gorensteinalgebras...' has missing spaces and appears to be a typesetting artifact. The abstract also uses 'bijection' for 'bijection'.","section":"Abstract and opening"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the authors' own unpublished and in-preparation works ([18], [52], [57]). This is not inherently improper, but the load-bearing status of these deferred claims is unusual for a survey. I would recommend requiring that the deferred results be made available in a public preprint or clearly downgraded to conjectures before acceptance. The survey is otherwise competent and well organized, and the proof of Proposition 1.2.1 is a useful addition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things up front. The survey is a genuinely useful map of the finite-dimensional Auslander-Gorenstein area, and it ties together classification results for monomial, gentle, incidence and Nakayama algebras through the Auslander-Reiten permutation. But the most load-bearing statement in that picture—Theorem 1.8.5, identifying the Coxeter permutation with Ringel's homological permutation for linear Nakayama algebras—is deferred to an in-preparation paper, along with the supporting Lemma 1.8.1. The advertised unification is currently an announcement, not a checkable theorem.\n\nThe genuinely new content is small: Proposition 1.2.1 places the Auslander-Reiten Conjecture between the Nakayama and Generalised Nakayama Conjectures, and there are several new conjectures and problems. The real contribution is the synthesis. References to published work are precise, examples like 1.8.3 genuinely help, and the paper is transparent about which statements are forthcoming.\n\nThe soft spots are proportional. First, the deferrals are not cosmetic: Theorems 1.8.5, 1.8.6, 1.7.6, 1.7.2 and 1.4.7 are announced without available proofs. If 1.8.5 breaks, the claimed connection between the Coxeter permutation and Ringel's bijection collapses, and that is a headline thread. Second, the proof of Proposition 1.2.1(1) skates from 'each indecomposable injective right module has finite projective dimension' to 'id A_A < ∞.' That inference needs the left-right symmetry of the Auslander condition (Theorem 1.2.2) and the fact D(A_A) is a finite direct sum of left indecomposable injectives; as written it is a gap. It is probably fixable, but a referee should ask for the line.\n\nThe published citations look consistent, and the paper is honest about what is unpublished. This is for a reader who wants a single source on the Auslander-Reiten permutation and its relatives, and it deserves a serious referee. I'd send it to review, with a request that the authors either make the promised papers available or recalibrate the wording so the deferred claims are not presented as established. I would cite it and take it to a reading group.","headline":"A useful, well-organised survey whose advertised unification rests on theorems still in preparation, plus a small new proof that needs a line repaired.","tokens_in":26627,"tokens_out":7906,"would_cite":true,"duration_ms":67012,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E10","16G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every Auslander-Gorenstein algebra has one canonical permutation on simple modules, unifying the grade, rowmotion, Nakayama, and Coxeter bijections.","keywords":["Auslander-Gorenstein algebras","Auslander-Reiten bijection","grade permutation","Coxeter permutation","rowmotion","Nakayama algebras","monomial algebras","incidence algebras"],"falsifier":"For the linear Nakayama algebra with Kupisch series [3,3,2,1] from Example 1.8.3, the survey lists the homological permutation as 1→4, 2→3, 3→1, 4→2; independently computing the Coxeter permutation from a Bruhat decomposition of its Coxeter matrix settles Theorem 1.8.5, since any difference is a counterexample.","tokens_in":25634,"feed_emoji":"🔁","tokens_out":13087,"duration_ms":105322,"temperature":0.7,"pith_summary":"This survey studies Auslander-Gorenstein algebras, finite-dimensional algebras whose minimal injective coresolution of the regular module has flat dimensions bounded by the index and finite total injective dimension. Its central claim is that every such algebra carries a canonical permutation of the simple modules, the Auslander-Reiten permutation, and that this permutation coincides with the grade bijection defined through first non-vanishing Ext-spaces. The survey then argues that this single permutation is a common ancestor of several bijections studied independently: rowmotion on distributive lattices, the Coxeter permutation of an Auslander regular algebra under an admissible ordering, and the homological permutation of linear Nakayama algebras. A reader should care because the claim converts combinatorial and matrix-theoretic bijections into one homological invariant, and it organises recent classification results for monomial algebras, incidence algebras, and blocks of category O.","feed_headline":"One canonical permutation unifies four known algebra bijections","feed_subtitle":"The Auslander-Reiten permutation equals the grade permutation and specializes to rowmotion and Coxeter maps.","key_machinery":"The load-bearing object is the Auslander-Reiten bijection ψ(I)=Ω^{pd I}(I) between indecomposable injective and projective modules, together with the permutation it induces on simple modules. The key identity is Theorem 1.2.6, which identifies this permutation with the grade bijection φ(S)=top(D Ext^{grade(S)}(S,A)). The third mechanism is the Coxeter permutation, defined by taking a Bruhat decomposition of the Coxeter matrix C_A = -Φ^T Φ^{-1}; it extends the Auslander-Reiten permutation to arbitrary finite-global-dimension algebras and is the tool that connects the survey to rowmotion and Nakayama combinatorics.","core_discovery":"The core discovery is that the Auslander-Reiten permutation—the bijection on simple modules induced by sending each indecomposable injective module I to the last non-zero term Ω^{pd I}(I) of its minimal projective resolution—is exactly the grade permutation, which sends a simple module S to the top of D(Ext^{g_S}(S,A)) where g_S is the smallest degree with non-vanishing Ext. This identity (Theorem 1.2.6) is the hinge of the survey. It lets the same permutation be recognised, in special classes, as rowmotion on the elements of a distributive lattice, as the Coxeter permutation obtained from a Bruhat decomposition of the Coxeter matrix, and as the homological permutation of linear Nakayama alg","pith_inferences":["Editorial inference: if the announced Bruhat-decomposition criterion for linear Nakayama algebras is correct, Auslander regularity of such an algebra becomes a finite computational check, making the open enumeration problem a calculation for each n.","Editorial inference: the survey's pattern suggests that any bijection on simples of a finite-dimensional algebra that agrees with the Auslander-Reiten permutation on all Auslander-Gorenstein examples is a candidate for a generalised rowmotion; the independence results for posets are the first test case.","Editorial inference: the same Bruhat-decomposition criterion could be tested on cyclic Nakayama algebras, where the reduction theorem for monomial algebras predicts the classification should also flow through Nakayama data."],"forward_implications":["The coincidence of the Auslander-Reiten and grade permutations gives a homological recipe for computing the canonical permutation: find the first non-vanishing Ext of each simple module and take the top of its dual.","For incidence algebras of distributive lattices, the Auslander-Reiten permutation is rowmotion, so a purely order-theoretic bijection is realised as a homological invariant.","For linear Nakayama algebras, the Coxeter permutation computed from the Cartan matrix's Bruhat decomposition coincides with the homological permutation, giving the requested elementary combinatorial description of that bijection.","For Auslander regular algebras with an admissible ordering of simple modules, the Coxeter permutation is a genuine generalisation of the Auslander-Reiten permutation.","If the proposed conjecture holds, the Auslander-Gorenstein property is equivalent to having a well-defined bijective Auslander-Reiten map; the survey reports this equivalence for monomial algebras and for incidence algebras of lattices."],"supporting_citations":[{"why":"Supplies the Auslander-Reiten bijection and the k-Gorenstein syzygy framework on which the survey's canonical permutation is built.","marker":"[8]"},{"why":"Supplies the grade bijection on simple modules whose coincidence with the Auslander-Reiten permutation is Theorem 1.2.6.","marker":"[49]"},{"why":"Proves the coincidence theorem and the identification of the Coxeter permutation with the Auslander-Reiten permutation under admissible orderings.","marker":"[58]"},{"why":"Provides the explicit projective and injective resolutions for incidence algebras of distributive lattices and the result from which the rowmotion identification follows.","marker":"[50]"},{"why":"Defines the homological permutation for Nakayama algebras that Theorem 1.8.5 identifies with the Coxeter permutation.","marker":"[70]"},{"why":"In-preparation reference asserted to prove the Nakayama/Coxeter identification and the Bruhat-decomposition criterion for Auslander regularity.","marker":"[57]"},{"why":"Supplies the monomial-algebra classification and the well-defined Auslander-Reiten map characterisation that motivate the general conjecture.","marker":"[56]"}],"fun_headline_variants":["AR permutation equals grade permutation","Four bijections reduce to one permutation","AR = grade unifies rowmotion and Coxeter","One bijection underlies four algebra maps","Survey: AR permutation is grade permutation"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The survey's unification depends on announced identifications whose proofs are not yet available—especially the equality of the Coxeter permutation with the Nakayama homological permutation—and on the cited theorem that the Auslander-Reiten and grade permutations coincide; if any of these statements fails, the surrounding web of identifications collapses.","fun_headline_variants_meta":{"raw":{"variants":["AR permutation equals grade permutation","Four bijections reduce to one permutation","AR = grade unifies rowmotion and Coxeter","One bijection underlies four algebra maps","Survey: AR permutation is grade permutation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000858,"raw_usage":{"total_tokens":3471,"prompt_tokens":566,"completion_tokens":2905,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":310,"completion_tokens_details":{"reasoning_tokens":2850}},"tokens_in":310,"tokens_out":2905,"duration_ms":22450,"temperature":1.0,"reasoning_tokens":2850,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:57:39.766503+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the linear Nakayama algebra with Kupisch series [3,3,2,1] from Example 1.8.3, the survey lists the homological permutation as 1→4, 2→3, 3→1, 4→2; independently computing the Coxeter permutation from a Bruhat decomposition of its Coxeter matrix settles Theorem 1.8.5, since any difference is a counterexample.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the grade bijection on simple modules whose coincidence with the Auslander-Reiten permutation is Theorem 1.2.6."},{"cited_title":"Iyama and R","cited_arxiv_id":null,"evidence_quote":"Provides the explicit projective and injective resolutions for incidence algebras of distributive lattices and the result from which the rowmotion identification follows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the homological permutation for Nakayama algebras that Theorem 1.8.5 identifies with the Coxeter permutation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"In-preparation reference asserted to prove the Nakayama/Coxeter identification and the Bruhat-decomposition criterion for Auslander regularity."},{"cited_title":"The Auslander-Gorenstein condition for monomial algebras","cited_arxiv_id":"2508.06957","evidence_quote":"Supplies the monomial-algebra classification and the well-defined Auslander-Reiten map characterisation that motivate the general conjecture."}],"review_version":1}