{"id":"c8307ac8-321c-43b1-b529-9754cbc32e56","arxiv_id":"2505.07623","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Order polytopes of graded posets are gamma-effective, meaning the equivariant gamma-polynomial of any graded poset has nonnegative integer character coefficients.","lead":"This paper proves that order polytopes of graded posets satisfy an equivariant version of gamma-positivity: when a finite group acts on the poset, the gamma-polynomial has coefficients that are honest group characters. The result provides a large new class of polytopes where Stapledon's effectiveness conjecture holds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Load-bearing dependence on unpublished G-invariant triangulation lemma; linear-extension triangulation should be verified directly.","rationale":"I read the proof of Theorem 4.5 carefully. The saturation formula (Theorem 3.6) is proven by character evaluation and depends only on Brändén's non-equivariant saturation theorem plus the quotient lemma (2.12); the algebra leading to the gamma-decomposition (13) is valid because each shifted cube h*-summand is palindromic with the same center, so the gamma-map is linear on that subspace. The effectiveness of induced/restricted products of cube gamma-polynomials is preserved under induction, restriction, and tensor product. The only truly load-bearing external input is Lemma 4.1, specifically the existence of a G-invariant lattice triangulation. This is cited to an unpublished preprint, and no proof or reference to a published source is supplied. In fact, the standard triangulation by linear extensions is G-invariant and unimodular, so the statement is true and easily checked; therefore the concern is about rigor and provenance rather than mathematical falsehood. Because the paper leaves this key ingredient to an unpublished citation, the reader's CONDITIONAL verdict is appropriate, and my stress-test does not change it.","tokens_in":25085,"tokens_out":41130,"duration_ms":364085,"concrete_test":"Verify [6, Lemma 6.4] for order polytopes directly: show that the standard triangulation of O(P) into the simplices {1 ≥ x_{σ(1)} ≥ ... ≥ x_{σ(n)} ≥ 0} indexed by the linear extensions σ of P is invariant under Aut(P). Concretely, for g ∈ Aut(P), the simplex indexed by σ maps under the coordinate permutation g to the simplex indexed by gσ, so the set of simplices is permuted. Checking that each such simplex is unimodular (vertices are 0-1 vectors) completes the proof and makes Lemma 4.1 self-contained, removing the dependence on the unpublished preprint.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.5 rests on Lemma 4.1, which asserts that h*_P^G(t) is a palindromic polynomial with effective coefficients. The proof of Lemma 4.1 cites [6, Lemma 6.4] for the existence of a G-invariant lattice triangulation of O(P) for every finite G ≤ Aut(P), and then applies [17, Theorem 1.4] to obtain polynomiality and effectiveness. This is the step that makes gamma-effectiveness well-defined: without it, the equivariant h*-series might not be a polynomial, and the gamma-polynomial could not be formed. The cited lemma comes from an unpublished preprint, and the paper gives no proof or indication of why such a triangulation exists. If the triangulation result required extra hypotheses, or if the standard triangulation were only invariant up to refinement (not as a simplicial complex), the central claim would lose its foundation. This is a rigor gap rather than an observed contradiction, but it is genuinely load-bearing because the main theorem depends on it directly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an equivariant version of Brändén's theorem on gamma-nonnegativity for order polytopes of graded posets. For a graded poset P and a finite subgroup G of Aut(P), the authors define the equivariant h*-polynomial h*_P^G(t) and its gamma-polynomial, and prove that the coefficients of the equivariant gamma-polynomial are effective G-characters (Theorem 4.5, Theorem B). The proof introduces an equivariant theory of order polytopes of sign-graded posets, including a non-closed convex set O(P,ε), and proves a structural formula expressing the equivariant h*-series of O(P,ε) as a sum of induced representations indexed by saturations (Theorem 3.6, Theorem A). The paper then specializes to the parity-grading of a graded poset, uses known gamma-effectiveness for cubes, and obtains the main result. A detailed example with the dihedral group D4 is worked out in Section 5.","tokens_in":25202,"tokens_out":4633,"duration_ms":48898,"significance":"If the main theorem is fully justified, it is a substantial contribution: it extends Brändén's gamma-nonnegativity from the ordinary h*-polynomial to the equivariant setting, giving effective characters rather than nonnegative integers, and it provides a large new class of polytopes for which Stapledon's effectiveness conjecture holds. Theorem 3.6 is a new structural decomposition that appears independently useful, and the explicit computation in Section 5 is a valuable illustration of the machinery. The writing is careful and the main line of argument is coherent. However, the central statement rests on an unpublished result about G-invariant lattice triangulations, and the treatment of the non-closed polytope O(P,ε) is somewhat informal; these points affect the well-definedness of the main objects and should be addressed before the result can be considered fully established.","major_comments":[{"comment":"The proof of Lemma 4.1 depends on [6, Lemma 6.4] for the existence of a G-invariant lattice triangulation of O(P) for every finite subgroup G of Aut(P), and on [17, Theorem 1.4] to conclude that the equivariant h*-series is a polynomial with effective coefficients. Both [6] and [17] are preprints, and no proof or verification of the triangulation claim is given in the present paper. This is load-bearing: without a G-invariant lattice triangulation, h*_P^G(t) may not be a polynomial, so the gamma-polynomial in Definition 4.2 and the statement of Theorem 4.5 are not even well-defined. Please either include a proof of the triangulation statement (the standard linear-extension triangulation of an order polytope is a natural candidate and should be checked for G-invariance and unimodularity directly) or state the missing result as a lemma with a proof.","section":"§4, Lemma 4.1"},{"comment":"The paper applies the standard equivariant Ehrhart determinant identity to the non-closed convex set O(P,ε). Lemma 3.3 asserts an identity involving h*(O(P,ε)^g;t) and a factor det(Id - η_G(g)t)|_{M_g^⊥}, and Theorem 3.6 freely uses evaluations of h*G_{P,ε}(t) at group elements. Since O(P,ε) is not closed, the interpretation of the formal Ehrhart series and of the fixed-point set O(P,ε)^g needs a precise justification; in particular, the identity Ehr(O(P,ε),η_G;t)(g)=Ehr(O(P,ε)^g;t) and the subsequent determinant factorization should be proved for these non-closed sets rather than invoked as if they were closed lattice polytopes. This gap affects Theorem 3.6 and therefore the proof of the main theorem, so it should be closed by an explicit formal argument.","section":"§3, Definition 3.1 and Lemma 3.3"}],"minor_comments":[{"comment":"The proof of Lemma 3.2 leaves the unimodularity check to the reader; please spell out the lattice basis change between ⊕_{O∈P/G} Z·e^*_O and ⊕_{O∈P/G} Z·(∑_{p∈O} e^*_p), since this is needed for the Ehrhart-theoretic identifications in the rest of the paper.","section":"§3.1, Lemma 3.2"},{"comment":"The set \\SYT_d introduced before equation (10) is not defined explicitly; it is only described as 'all standard Young tableaux with d boxes having neither a double descent nor a final descent'. Please clarify the notation and define 'higher row' precisely in the descent definition.","section":"§4.1"},{"comment":"In the example, the computations for Q_3 and Q_4 use specific choices of stabilizer subgroups ('{e,τσ}' and '{e,τ}'); it would help the reader if the text stated explicitly why the resulting induced characters are independent of the choice of representative in each G-orbit.","section":"§5"},{"comment":"Reference [9] is missing the year of publication; please add it to match the journal style.","section":"References"},{"comment":"The phrase 'h*G_P(t) is γ-effective' in Theorem B is slightly imprecise: γ-effectiveness is a property of the poset P together with the G-action, not of a polynomial. Consider restating the theorem as 'the equivariant gamma-polynomial of O(P) has effective coefficients' or define the phrase explicitly in Definition 4.2.","section":"§1.2, Theorem B wording"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is likely correct in its main idea, but the proof of the main theorem depends essentially on [6, Lemma 6.4] and [17, Theorem 1.4], both unpublished preprints. In particular, the first of these is by the first author, and the triangulation statement is exactly what makes the equivariant h*-polynomial well-defined. I would ask the authors to either prove the triangulation lemma or state it as a standalone result with a full proof. Given the importance of this dependency, I recommend major revision rather than rejection, as the gap appears repairable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. It proves that order polytopes of graded posets are gamma-effective under any automorphism group: the equivariant h*-polynomial is palindromic and its gamma-coefficients are honest characters. That is a natural equivariant strengthening of Br\\\"and\\'en's theorem, and it is genuinely new. The main technical work is a saturation formula (Theorem 3.6) expressing the equivariant h*-series as a sum of induced characters from stabilizers of saturations, then reducing gamma-effectiveness to the already-known gamma-effectiveness of cubes. The proof is detailed, and I checked the key steps: the character evaluation argument in Theorem 3.6 is coherent, and the decomposition into ordinal sums of antichains works.\n\nThe result is as advertised. In the trivial group case it recovers Br\\\"and\\'en, which is the correct baseline. The authors also do something useful beyond that: they develop order polytopes of sign-graded posets, including the non-closed objects O(P,epsilon), and show how automorphisms interact with saturations. The example in Section 5 is worked out carefully and helps.\n\nSoft spots are minor. The one real issue flagged by stress-test is Lemma 4.1, which relies on [6, Lemma 6.4] for existence of a G-invariant lattice triangulation of order polytopes, from an unpublished preprint. But that fact is standard and easily proved: the linear-extension triangulation of O(P) is permuted by Aut(P), so it is G-invariant. The citation to [6] is unnecessary for that point. So this is a rigor gap in presentation, not in the mathematics. The treatment of h* for the non-closed O(P,epsilon) is also a bit informal, but the generating function is defined by the same Ehrhart-series mechanism, and the evaluation-at-elements argument justifies the formulas. A few verifications are left to the reader (notably unimodularity in Lemma 3.2), but they are routine.\n\nNo circularity: the proof uses Br\\\"and\\'en's saturation theorem and Stapledon's equivariant framework, and the reliance on [17] is for a known theorem about invariant triangulations, not the target result. The citation pattern looks fine; self-citation of [6] is load-bearing only via that triangulation fact, which is independent.\n\nBottom line: this is a solid, publishable paper in equivariant Ehrhart theory. A serious referee could reasonably ask for a one-line proof of the G-invariant triangulation and for a slightly more careful treatment of the non-closed polytopes, but I would not put any obstacle in front of publication. I'd recommend accepting it for peer review, with minor revisions at most.","headline":"A new equivariant gamma-effectiveness theorem for order polytopes; the proof is sound and the main soft spot is just an unnecessary citation to an unpublished preprint for a standard triangulation fact.","tokens_in":25839,"tokens_out":2617,"would_cite":true,"duration_ms":22555,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E18","52B20","06A07","52B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every graded poset and every subgroup of its automorphism group, the order polytope's equivariant h*-polynomial is gamma-effective: it expands in the gamma basis with coefficients that are genuine group characters.","keywords":["equivariant Ehrhart series","order polytope","graded poset","sign-graded poset","gamma-effectiveness","lattice polytope","group action"],"falsifier":"One can test the theorem by computing the equivariant gamma-polynomial of a graded poset with a nontrivial automorphism group using the paper's saturation formula — the dihedral-group example in Section 5 is the worked template — and checking that every coefficient decomposes into irreducibles with nonnegative multiplicities; a single negative multiplicity would disprove the claim. A sharper target is the triangulation lemma itself: exhibiting a graded poset $P$ and a subgroup $G\\le \\mathrm{Aut}(P)$ for which $O(P)$ has no $G$-invariant lattice triangulation would destroy the proof's foundation, even though the theorem might still be true by another route.","tokens_in":24799,"feed_emoji":"📐","tokens_out":17155,"duration_ms":139865,"temperature":0.7,"pith_summary":"A classical 2004 result [4] shows that the numerator of the Ehrhart series of the order polytope of a graded poset is gamma-nonnegative: it expands in the basis $t^i(1+t)^{s-2i}$ with nonnegative real numbers. This paper proves the equivariant analogue: when a finite group $G$ acts on the poset by automorphisms, the equivariant $h^*$-series — the character-valued count of lattice points fixed by each group element in every dilation — expands in the same gamma basis with coefficients that are genuine $G$-characters, not just virtual ones. The statement is proved for every graded poset and every subgroup of its automorphism group, so it is a uniform, large class of examples where the effectiveness conjecture [16] holds along with the stronger gamma structure. The proof is constructive: it gives an explicit formula for the gamma-characters in terms of the group action on the saturations of the poset, each saturation being a stack of antichains whose gamma-data is the known equivariant gamma-expansion of a cube.","feed_headline":"Graded order polytopes stay gamma-effective under symmetry","feed_subtitle":"Every equivariant gamma coefficient is a genuine group character, not just a virtual one.","key_machinery":"The central mechanism is the equivariant Ehrhart series of the order polytope $O(P) = \\{f: P\\to[0,1] : p\\le q \\Rightarrow f(p)\\ge f(q)\\}$, whose numerator $h^{*G}_P(t)$ records, for each group element $g$, the Ehrhart series of the fixed-point polytope. Two tools carry the argument. First, the sign-graded posets introduced in [4]: an edge labeling $\\varepsilon$ makes $P$ $\\varepsilon$-consistent when every principal order ideal has a well-defined rank function, and a saturation of $(P,\\varepsilon)$ is a maximal extension of the order that preserves this rank and makes elements at rank distance one comparable; for parity-graded posets all saturations are ordinal sums of antichains. Second, Theorem 3.6 (proved by evaluating at each $g$ and using orbit-stabilizer and induction) expresses the equivariant $h^*$-series of $O(P,\\varepsilon)$ as a sum of induced-and-restricted equivariant $h^*$-series of the saturations, and Lemma 3.5 shifts between the grading $\\varepsilon$ and the canonical parity grading $\\varepsilon_{\\mathrm{par}}$ by a monomial. The gamma-effectiveness then reduces to the antichain/cube case, where the equivariant gamma expansion is known explicitly through standard Young tableaux with no double or final descent (formula (10) in the paper).","core_discovery":"Let $P$ be a finite poset with all maximal chains of the same length and let $G$ be any subgroup of its automorphism group. The paper proves that the equivariant $h^*$-polynomial $h^{*G}_P(t)$ is palindromic and gamma-effective: in the expansion $h^{*G}_P(t) = \\sum_{i=0}^{\\lfloor s/2\\rfloor} \\gamma_i t^i(1+t)^{s-2i}$, each $\\gamma_i$ is an effective character of $G$. Equivalently, the equivariant gamma-polynomial is a polynomial in $t$ whose coefficients are sums of irreducible representations with nonnegative multiplicities, so it records the symmetries without introducing negative multiplicities. The key formula is $\\gamma^G_P(t) = \\sum_i \\mathrm{Ind}^{G}_{\\mathrm{stab}_G(Q_i,\\varepsilon_{\\mathrm{par}})} \\mathrm{Res}^{\\mathrm{Aut}(Q_i)}_{\\mathrm{stab}_G(Q_i,\\varepsilon_{\\mathrm{par}})} \\left( t^{(r_{Q_i}(1)-r_P(1))/2} \\cdot \\gamma^{\\mathrm{Aut}(Q_i)}_{Q_i}(t)\\right)$, where the sum ranges over representatives of the $G$-orbits of saturations of the parity-graded structure; each $Q_i$ is an ordinal sum of antichains, and the cube case supplies effective characters by the explicit standard-Young-tableaux formula [11].","pith_inferences":["The saturation-plus-induction structure suggests a general recipe for proving equivariant gamma-effectiveness of any polytope whose gamma-vector factors into cube contributions via $G$-invariant lattice triangulations; the formula may transfer to other families such as alcoved polytopes or $G$-parking-function polytopes.","The explicit character formula for cubes (via standard Young tableaux with no double or final descent) hints that the gamma-characters of a graded poset could be modeled as sums of irreducible symmetric-group modules indexed by tableaux on each antichain block, yielding a purely combinatorial interpretation of the equivariant gamma-vector.","A natural next test concerns the gamma-positivity conjecture for flag spheres: the quotient polytopes $P/\\langle g\\rangle$ are themselves order polytopes of graded posets (by Lemma 2.5), so the machinery gives effective gamma-vectors for the fixed-point polytopes along every cyclic subgroup; comparing these with known flag-sphere constructions might reveal whether gamma-effectiveness is inherited "],"forward_implications":["For every graded poset $P$ and every $G\\le \\mathrm{Aut}(P)$, the equivariant $h^*$-polynomial is palindromic, effective, and gamma-effective; in particular the effectiveness conjecture [16] holds for the whole class of order polytopes of graded posets.","The explicit formula (13) expresses the equivariant gamma-characters as induced/restricted products of the symmetric-group characters of cubes, so the gamma-vector of any graded poset action can be computed from the $G$-orbits of its saturations.","Evaluating the equivariant gamma-polynomial at the identity recovers the classical gamma-nonnegativity of the ordinary $h^*$-polynomial, so the result is a genuine extension of the prior theorem [4].","Because induction and restriction send effective characters to effective characters, and products of effective characters are effective, the conclusion survives passing to any subgroup of $\\mathrm{Aut}(P)$; no condition on the group action beyond preservation of the order is needed."],"supporting_citations":[{"why":"It introduces sign-graded posets and proves gamma-nonnegativity of the h*-polynomial of order polytopes of graded posets, supplying the saturation construction used in Theorem 3.6.","marker":"[4]"},{"why":"It supplies the G-invariant lattice triangulation of order polytopes of graded posets on which Lemma 4.1 depends.","marker":"[6]"},{"why":"It proves that the h*-polynomial of an order polytope of a graded poset is palindromic, the non-equivariant base of the argument.","marker":"[10]"},{"why":"It gives the explicit formula for the equivariant gamma-characters of cubes via standard Young tableaux, used for the antichain factors in the proof of Theorem 4.5.","marker":"[11]"},{"why":"It establishes gamma-effectiveness of the equivariant h*-polynomial of the cube under the symmetric group.","marker":"[13]"},{"why":"It provides the foundation of equivariant Ehrhart theory, including the definition of the equivariant h*-series and the relations between effectiveness, polynomiality, and palindromy.","marker":"[16]"},{"why":"It proves that a G-invariant lattice triangulation produces a polynomial equivariant h*-series with effective coefficients.","marker":"[17]"}],"fun_headline_variants":["Graded poset polytopes: gamma-effective under any symmetry","Symmetry preserves gamma-effectiveness for graded order polytopes","Equivariant gamma coefficients are genuine characters for graded posets","Order polytopes of graded posets: gamma-effective with symmetries","Graded order polytopes: equivariant gamma stays effective"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the cited result that the order polytope of every graded poset admits a $G$-invariant lattice triangulation for every finite subgroup $G$ of its automorphism group; if that triangulation were missing for some pair $(P,G)$, the equivariant $h^*$-series could stop being a polynomial, and gamma-effectiveness would not even be defined.","fun_headline_variants_meta":{"raw":{"variants":["Graded poset polytopes: gamma-effective under any symmetry","Symmetry preserves gamma-effectiveness for graded order polytopes","Equivariant gamma coefficients are genuine characters for graded posets","Order polytopes of graded posets: gamma-effective with symmetries","Graded order polytopes: equivariant gamma stays effective"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000852,"raw_usage":{"total_tokens":3765,"prompt_tokens":1065,"completion_tokens":2700,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":681,"completion_tokens_details":{"reasoning_tokens":2612}},"tokens_in":681,"tokens_out":2700,"duration_ms":19031,"temperature":1.0,"reasoning_tokens":2612,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:13:29.628564+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One can test the theorem by computing the equivariant gamma-polynomial of a graded poset with a nontrivial automorphism group using the paper's saturation formula — the dihedral-group example in Section 5 is the worked template — and checking that every coefficient decomposes into irreducibles with nonnegative multiplicities; a single negative multiplicity would disprove the claim. A sharper target is the triangulation lemma itself: exhibiting a graded poset $P$ and a subgroup $G\\le \\mathrm{Aut}(P)$ for which $O(P)$ has no $G$-invariant lattice triangulation would destroy the proof's foundation, even though the theorem might still be true by another route.","supporting_citations":[{"cited_title":"Sign-graded posets, unimodality ofW-polynomials and the Charney-Davis conjecture","cited_arxiv_id":null,"evidence_quote":"It introduces sign-graded posets and proves gamma-nonnegativity of the h*-polynomial of order polytopes of graded posets, supplying the saturation construction used in Theorem 3.6."},{"cited_title":"Equivariant Hilbert and Ehrhart series under translative group actions, 2024","cited_arxiv_id":null,"evidence_quote":"It supplies the G-invariant lattice triangulation of order polytopes of graded posets on which Lemma 4.1 depends."},{"cited_title":"Distributive lattices, affine semigroup rings and algebras with straightening laws","cited_arxiv_id":null,"evidence_quote":"It proves that the h*-polynomial of an order polytope of a graded poset is palindromic, the non-equivariant base of the argument."},{"cited_title":"Gamma vectors of partitioned permutohedra","cited_arxiv_id":"2405.09242","evidence_quote":"It gives the explicit formula for the equivariant gamma-characters of cubes via standard Young tableaux, used for the antichain factors in the proof of Theorem 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes gamma-effectiveness of the equivariant h*-polynomial of the cube under the symmetric group."},{"cited_title":"Equivariant Ehrhart theory","cited_arxiv_id":null,"evidence_quote":"It provides the foundation of equivariant Ehrhart theory, including the definition of the equivariant h*-series and the relations between effectiveness, polynomiality, and palindromy."}],"review_version":1}