{"id":"79e04c8f-a33c-4534-945e-3960934c345f","arxiv_id":"2506.22730","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Toric double determinantal rings are Hibi rings of a product of three chains, and this identification gives closed formulas for all main numerical invariants and a Gorenstein characterization.","lead":"This paper proves that toric double determinantal rings, defined by the 2x2 minors of stacked generic matrices, are Hibi rings of a simple three-chain poset. That identification yields closed formulas for multiplicity, regularity, Hilbert function, and a Gorenstein criterion, answering open questions of Li in the toric case.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the inclusion asserted in Theorem 3.4 is routine and correct, and the regularity formula applies in the non-pure cases needed here.","rationale":"The reader's weakest assumption correctly identifies the one place where the proof is abbreviated, namely the containment I^r_mn ⊆ I_{A_mnr} in Theorem 3.4. However, this is not a genuine load-bearing risk: the containment is immediate from the definition of phi, since each 2-minor specializes to a zero difference of identical monomials. I checked all cases: minors within one X_k, minors across blocks in the horizontal concatenation, and minors across blocks in the vertical concatenation; in every case the two terms of the determinant have the same image under phi. The converse inclusion is explicit and correct. The second concern raised by the reader concerns the use of the regularity formula reg(K[J(P)]) = |P| - rank(P) - 1 for non-pure posets. For the poset P_mnr this formula gives m+n+r-2-max{m,n,r}, which agrees with the degree of the multiset Eulerian h-polynomial derived independently in Corollary 4.7 from the Hilbert function and MacMahon's identity. Thus the formula is valid in the non-pure situations appearing here. The remaining results (minimal number of generators, multiplicity, Hilbert function, a-invariant, Gorenstein criterion) follow formally once the Hibi-ring isomorphism is established, and I found no circularity or unsupported hypothesis. The paper would benefit from spelling out the small check in Theorem 3.4, but its absence does not threaten correctness. Hence no adjustment to the reader's conditional verdict is required.","tokens_in":21838,"tokens_out":28531,"duration_ms":282513,"concrete_test":"Verify the skipped inclusion by writing phi on the three elementary types of 2-minors (same block, horizontal cross-block, vertical cross-block) and expanding: in each case the two determinant terms become identical monomials. This can be done by hand or in a CAS in a few lines; if any case produced different monomials, Theorem 3.4 would fail and the Hibi-ring identification would collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The one step the reader flags is the unstated verification of I^r_mn ⊆ ker phi in Theorem 3.4. That inclusion is immediate: under phi(x^k_ij)=x_i y_j z_k, every 2-minor of H and V has its two terms mapped to the same monomial. For a vertical minor with rows (i1,k1),(i2,k2) and columns j1,j2, both terms map to x_{i1}x_{i2}y_{j1}y_{j2}z_{k1}z_{k2}; for a horizontal cross-block minor the two terms map to the same monomial because y_{j1} and y_{j2} commute; for a same-block minor the two terms map to x_{i1}x_{i2}y_{j1}y_{j2}z_k^2. Thus the inclusion is not merely plausible but easily verified, and the converse direction is supplied by explicitly expressing each sorting relation as a minor or a sum of minors. The second flagged point, the Hibi regularity formula |P|-rank(P)-1, is also safe here: the h-polynomial computed in Corollary 4.7 is the multiset Eulerian polynomial of {1^{m-1},2^{n-1},3^{r-1}}, whose degree is m+n+r-2-max{m,n,r}, exactly the value obtained from the cited formula in both pure and non-pure cases. The Hibi-ring realization, the invariant formulas, and the Gorenstein criterion are internally consistent and check out.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the quotient R/I^r_mn of the polynomial ring R = K[x^k_{ij}] by the ideal generated by all 2-minors of the horizontal concatenation (X_1 ... X_r) and the vertical concatenation of r generic m x n matrices. The main structural result is Theorem 3.4 and Corollary 3.5: R/I^r_mn is isomorphic to the Hibi ring K[L_mnr] of the distributive lattice L_mnr = J(P_mnr), where P_mnr is the disjoint union of three chains of lengths m-1, n-1, and r-1. From this identification the paper derives explicit formulas for the minimal number of generators, dimension, multiplicity, regularity, a-invariant, Hilbert function, and h-polynomial, proves a symmetry isomorphism under permutations of m, n, r, characterizes when the ring is Gorenstein, and describes the facets of the Stanley-Reisner complex of the diagonal initial ideal.","tokens_in":22050,"tokens_out":14511,"duration_ms":157227,"significance":"If the Hibi-ring identification is correct, it provides a clean conceptual explanation of the previously known normality and Cohen-Macaulayness of toric double determinantal rings and yields closed, parameter-free formulas for their basic invariants, answering Li's questions in the toric case. The paper is largely self-contained, makes careful use of the sortable-set presentation of Hibi rings, and includes several internal consistency checks, such as Remark 4.4(1) and Remark 4.8. I regard the central claim as sound: the one potentially load-bearing step, the inclusion I^r_mn subset of I_A in Theorem 3.4, is true and easily verified, so the omitted verification is an exposition issue rather than a mathematical gap.","major_comments":[],"minor_comments":[{"comment":"The inclusion I^r_mn subset of I_A is asserted with the words \"it is not so hard to directly check that any minor belongs to ker phi\" and no verification is displayed. Because this inclusion is load-bearing for the identification with the Hibi ring, please add the two-line check: under phi(x^k_ij) = x_i y_j z_k, the two terms of every 2-minor of H and V map to the same monomial (for a same-block minor both terms map to x_{i_1}x_{i_2}y_{j_1}y_{j_2}z_k^2, and for a cross-block minor both map to x_{i_1}x_{i_2}y_{j_1}y_{j_2}z_{k_1}z_{k_2}).","section":"Theorem 3.4"},{"comment":"There is a typo in the Hilbert-function formula: the factor should be binom(n-1+d, d), not binom(n-1+d, t).","section":"Theorem 4.3(6)"},{"comment":"The parenthetical \"see Section 4 for the definition\" is inaccurate: diagonal term orders are defined in Section 5, not Section 4.","section":"Remark 4.2(2)"},{"comment":"The notation \"(g_1,h_1) - -> (g_s,h_s)\" should presumably be \"(g_1,h_1) - -> (g_t,h_t)\", since the path has t points.","section":"Definition 5.2"},{"comment":"The symbol R is used both for the polynomial ring and for one of the four generating sets in Theorem 3.3; using a different letter for the set (for example S or \\mathcal{R}) would avoid confusion.","section":"Theorem 3.3 and Section 3.1"},{"comment":"There is a typo in \"the the facets\" in the paragraph preceding Question 5.10.","section":"Section 5, before Question 5.10"},{"comment":"The application of [9, Theorem 1.1] to the non-pure poset P_mnr is a point the authors could clarify with one sentence; I see no correctness issue, since the resulting formula agrees with the degree of the h-polynomial obtained in Corollary 4.7.","section":"Section 4.3 (regularity)"}],"recommendation":"minor_revision","confidential_remarks":"For the editor: the paper is well within the scope of the journal and is a solid contribution. The only substantive request is to expand the omitted verification in Theorem 3.4; once that is added, I expect the paper to be acceptable. I do not view the citation of [10] as a circularity concern, since the result is published independently and the proof is outlined in the present text."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The main new thing is the identification (Cor. 3.5) of R/I^r_mn with the Hibi ring K[L_mnr], where L_mnr = J(P_mnr) and P_mnr is the disjoint union of chains of lengths m-1, n-1, r-1. That's a real structural insight, not a repackaging. It converts a class of ideals defined by minors into the well-developed Hibi ring machinery, and the authors use it to answer Li's questions in the toric case: closed formulas for the number of minimal generators, multiplicity, regularity, a-invariant, Hilbert function, h-polynomial, and the Gorenstein criterion. The symmetry isomorphism Theorem 4.1 is a nice surprise, proved cleanly via the poset.\n\nThe paper does what it claims. The explicit minimal generating set M ∪ N ∪ R ∪ T in Theorem 3.3 is new and correctly proved. The equality I^r_mn = I_A is load-bearing; the harder inclusion I_A ⊆ I^r_mn is handled by writing each sorting relation as a minor or sum of minors, and that part is detailed. The other inclusion, I^r_mn ⊆ I_A, is only asserted as 'not so hard to directly check.' The stress-test note confirms it's genuinely immediate—under phi(x^k_ij)=x_i y_j z_k, both terms of every 2-minor of H and V map to the same monomial, which is exactly the kind of thing the authors could have shown in one line. So this is an exposition gap, not a correctness issue. Similarly, the regularity formula from Ene-Herzog-Madani is quoted without discussing the non-pure case, but the computed h-polynomial checks out, so it's fine.\n\nThe facet description in Section 5 generalizes the r=2 result, and the bijection with multiset words is neat. The paper is self-contained apart from standard references; no code or data is involved, which is appropriate.\n\nIn short: the central claim holds up, the soft spots are minor and easily fixed. This is a good subfield paper that deserves a serious referee. I would accept it for review and suggest the authors spell out the one inclusion and add a sentence on the Hibi regularity hypotheses.","headline":"Solid Hibi-ring realization of toric double determinantal rings with closed formulas; worth refereeing, only minor exposition gaps.","tokens_in":22692,"tokens_out":2178,"would_cite":true,"duration_ms":21219,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E40","13F65","14M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the quotient $R/I^r_{mn}$ by all $2$-minors of the horizontal and vertical concatenations of $r$ generic $m\\times n$ matrices is isomorphic to the Hibi ring of the distributive lattice…","keywords":["Hibi rings","double determinantal ideals","toric rings","sorting relations","Hilbert functions","Gorenstein rings","Stanley-Reisner complex","regularity"],"falsifier":"Compute the kernel of $\\phi(x^k_{ij})=x_i y_j z_k$ for a small instance such as $m=n=r=3$ and compare it with the ideal generated by all 2-minors of $H$ and $V$: the central claim is true exactly if the two ideals coincide, and any 2-minor with nonzero image under $\\phi$ would refute it.","tokens_in":21579,"feed_emoji":"🧮","tokens_out":8696,"duration_ms":77146,"temperature":0.7,"pith_summary":"The paper studies the toric double determinantal ring $R/I^r_{mn}$, where $R$ is the polynomial ring in the entries of $r$ generic $m\\times n$ matrices and $I^r_{mn}$ is generated by all $2$-minors of both the horizontal concatenation $(X_1 \\cdots X_r)$ and the vertical concatenation $(X_1;\\dots;X_r)$. It proves that this ring is isomorphic to the Hibi ring $K[L_{mnr}]$ of a very simple distributive lattice: the lattice of order ideals of three disjoint chains of sizes $m-1$, $n-1$, $r-1$. Because Hibi rings have a well-developed invariant theory, this identification turns previously hard invariants of $R/I^r_{mn}$ into lattice counts: minimal generator number, multiplicity, Castelnuovo--Mumford regularity, $a$-invariant, Hilbert function, and $h$-polynomial all get closed formulas. It also characterizes when the ring is Gorenstein, answering a question from the bipartite determinantal literature, and describes the facets of the Stanley--Reisner complex of the initial ideal with respect to a diagonal term order.","feed_headline":"Toric double determinantal rings are Hibi rings","feed_subtitle":"The identification yields closed formulas for generators, multiplicity, regularity, and Gorensteinness.","key_machinery":"The central object is the Hibi ring $K[L]$ of a finite distributive lattice $L$, together with its realization as a toric ring generated by sortable monomials. For $L_{mnr}=\\mathcal{J}(P_{mnr})$, the monomial model is $\\{x_i y_j z_k:1\\le i\\le m,\\,1\\le j\\le n,\\,1\\le k\\le r\\}$, with the sorting map sending an unsorted pair to the meet and join of the corresponding lattice elements. The load-bearing mechanism is the equality of two descriptions of the same ideal: the $2$-minors of the two concatenated matrices generate the same kernel as the sorting relations, which are the Hibi relations of $L_{mnr}$. Once this equality is in place, invariants of the double determinantal ring become lattice-theoretic counts of elements, chains, linear extensions, and order-preserving maps.","core_discovery":"On the paper's own terms, the discovery is that the quotient ring $R/I^r_{mn}$ is exactly the Hibi ring $K[L_{mnr}]$, so every invariant of the ring can be read off from the three-chain poset $P_{mnr}$. Concretely, the map sending the variable $x^k_{ij}$ to the monomial $x_i y_j z_k$ has kernel equal to $I^r_{mn}$; the sorting relations of the monomial set $\\{x_i y_j z_k\\}$ coincide with the Hibi relations of the lattice of order ideals of $P_{mnr}$. The paper then reads off: $\\mu(I^r_{mn})=\\binom{mnr+1}{2}-\\binom{m+1}{2}\\binom{n+1}{2}\\binom{r+1}{2}$ minimal generators; dimension $m+n+r-2$; multiplicity $\\frac{(m+n+r-3)!}{(m-1)!(n-1)!(r-1)!}$; regularity $m+n+r-2-\\max\\{m,n,r\\}$; $a$-invariant $-\\max\\{m,n,r\\}$; Hilbert function $\\binom{m-1+d}{d}\\binom{n-1+d}{d}\\binom{r-1+d}{d}$; and Gorensteinness exactly when $\\{m,n,r\\}\\subseteq\\{1,\\max\\{m,n,r\\}\\}$.","pith_inferences":["Beyond the paper, the same sortable-monomial model suggests that the 2-minor construction for three concatenated chains is one instance of a general identification with Hibi rings of products of chains; replacing three chains by $s$ chains should give analogous formulas with $s$ parameters.","Beyond the paper, the failure of the isomorphism for minors of size at least $3$, noted in the paper's Remark 4.2, indicates that the Hibi-ring dictionary is specific to the 2-minor case and that different tools would be needed for larger minors.","Beyond the paper, the word encoding of facets raises the possibility that ordering facets by adjacent-letter exchanges gives a shelling order, which would yield a purely combinatorial proof that the quotient by the initial ideal is Cohen--Macaulay; the paper leaves this as an open question."],"forward_implications":["The symmetry $R/I^r_{mn}\\cong R/I^{\\sigma(r)}_{\\sigma(m)\\sigma(n)}$ for any permutation $\\sigma$ of $\\{m,n,r\\}$ means all computed invariants are unchanged by permuting the three parameters.","Since Hibi rings of distributive lattices are normal Cohen--Macaulay domains, the ring $R/I^r_{mn}$ is a normal Cohen--Macaulay domain of dimension $m+n+r-2$ for all positive integers $m,n,r$.","The Hilbert function is the product $\\binom{m-1+d}{d}\\binom{n-1+d}{d}\\binom{r-1+d}{d}$, so the $h$-polynomial is the descent polynomial of multiset permutations of $\\{1^{m-1},2^{n-1},3^{r-1}\\}$.","The ring $R/I^r_{mn}$ is Gorenstein exactly when $\\{m,n,r\\}\\subseteq\\{1,\\max\\{m,n,r\\}\\}$; equivalently, at most one of the three parameters exceeds $1$.","The facets of the Stanley--Reisner complex of the initial ideal with respect to a diagonal term order are in bijection with words on $\\{M^{m-1},N^{n-1},R^{r-1}\\}$, making that complex pure of dimension $m+n+r-3$."],"supporting_citations":[{"why":"Supplies the theorem that a sortable set of monomials has a presentation ideal generated by sorting relations with a Gröbner basis, used to identify $I_{A_{mnr}}$.","marker":"[25]"},{"why":"Gives the realization of a Hibi ring as a toric ring generated by sortable monomials, with sorting matching meet and join; this is the bridge to $K[L_{mnr}]$.","marker":"[10]"},{"why":"Establishes that Hibi rings of distributive lattices are toric, normal, Cohen--Macaulay domains, and gives the dimension and Gorenstein criteria used in Section 4.","marker":"[14]"},{"why":"Provides formulas for multiplicity, Hilbert function, and Hilbert series of Hibi rings in terms of linear extensions and order-preserving maps, used in Theorem 4.3 and Corollary 4.5.","marker":"[15]"},{"why":"Supplies the regularity formula for Hibi rings of $\\mathcal{J}(P)$, used in Theorem 4.3(4).","marker":"[9]"},{"why":"Gives the Gröbner basis theorem for Hibi relations and the Stanley--Reisner facts used to identify facets.","marker":"[13]"},{"why":"Proves Gröbner bases and the Cohen--Macaulay property for double determinantal ideals, supplying the diagonal Gröbner basis used in Section 5.","marker":"[17]"},{"why":"Introduces bipartite determinantal ideals, gives path-count formulas for $h$-vectors and multiplicity, and poses the Gorenstein question answered here.","marker":"[19]"}],"fun_headline_variants":["Toric double determinantal rings proven to be Hibi","All invariants of toric double determinantal now closed","Hibi ring equivalence yields full invariant list for toric double determinantal","Gorenstein toric double determinantal characterized","Toric double determinantal dimension via Hibi rings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on the unexpanded assertion that every 2-minor of the two concatenated matrices lies in the kernel of the map $x^k_{ij}\\mapsto x_i y_j z_k$; if that inclusion failed for even one minor, the identification with the Hibi ring and all formulas built on it would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Toric double determinantal rings proven to be Hibi","All invariants of toric double determinantal now closed","Hibi ring equivalence yields full invariant list for toric double determinantal","Gorenstein toric double determinantal characterized","Toric double determinantal dimension via Hibi rings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000719,"raw_usage":{"total_tokens":3248,"prompt_tokens":981,"completion_tokens":2267,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":2186}},"tokens_in":597,"tokens_out":2267,"duration_ms":19211,"temperature":1.0,"reasoning_tokens":2186,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:59:29.493251+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the kernel of $\\phi(x^k_{ij})=x_i y_j z_k$ for a small instance such as $m=n=r=3$ and compare it with the ideal generated by all 2-minors of $H$ and $V$: the central claim is true exactly if the two ideals coincide, and any 2-minor with nonzero image under $\\phi$ would refute it.","supporting_citations":[{"cited_title":"Sturmfels, Gr¨ obner bases and convex polytopes, University Lecture Series, 9, American Mathe- matical Soc., 1996","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that a sortable set of monomials has a presentation ideal generated by sorting relations with a Gröbner basis, used to identify $I_{A_{mnr}}$."},{"cited_title":"Gasanova, L","cited_arxiv_id":null,"evidence_quote":"Gives the realization of a Hibi ring as a toric ring generated by sortable monomials, with sorting matching meet and join; this is the bridge to $K[L_{mnr}]$."},{"cited_title":"Com- mutative Algebra and Combinatorics","cited_arxiv_id":null,"evidence_quote":"Establishes that Hibi rings of distributive lattices are toric, normal, Cohen--Macaulay domains, and gives the dimension and Gorenstein criteria used in Section 4."},{"cited_title":"Hibi, Hilbert functions of Cohen-Macaulay integral domains and chain conditions of finite partially ordered sets, J","cited_arxiv_id":null,"evidence_quote":"Provides formulas for multiplicity, Hilbert function, and Hilbert series of Hibi rings in terms of linear extensions and order-preserving maps, used in Theorem 4.3 and Corollary 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the regularity formula for Hibi rings of $\\mathcal{J}(P)$, used in Theorem 4.3(4)."},{"cited_title":"Herzog, T","cited_arxiv_id":null,"evidence_quote":"Gives the Gröbner basis theorem for Hibi relations and the Stanley--Reisner facts used to identify facets."},{"cited_title":"Klein, N","cited_arxiv_id":null,"evidence_quote":"Proves Gröbner bases and the Cohen--Macaulay property for double determinantal ideals, supplying the diagonal Gröbner basis used in Section 5."},{"cited_title":"Li, Bipartite determinantal ideals and concurrent vertex maps, J","cited_arxiv_id":null,"evidence_quote":"Introduces bipartite determinantal ideals, gives path-count formulas for $h$-vectors and multiplicity, and poses the Gorenstein question answered here."}],"review_version":1}