{"id":"e20bdab1-c212-4119-b85f-df77460cf797","arxiv_id":"2507.22528","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A proof that supersymmetric Schur polynomials have SNP is invalid because the stated hook-inequality support set is contradicted by the paper's own example and by symmetry.","lead":"The paper claims to prove that every supersymmetric Schur polynomial has a saturated Newton polytope by encoding its support as an integral polyhedron. The proof fails because the paper's own support description is false and its example contains a misclassified point.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The support description in Theorem 2.2 is false: for λ=(2,1,1), k=2, ℓ=1, the vector (0,3,1) satisfies all hook inequalities yet is absent from Supp(S_(2,1,1)); since Theorem 3.5 builds Newton(Sλ) from this Bλ, the SNP proof is not established.","rationale":"The reader's weakest assumption identifies exactly the same failure, and Example 3.1 already contains a counterexample to the claimed characterization, so the concern is not hypothetical. The failure is load-bearing because every subsequent step—the polyhedron H, the total unimodularity argument, and the equality in Theorem 3.5—uses the false identity Supp(Sλ)=Bλ. Even if the SNP property itself is true, the manuscript's proof has a concrete false lemma at its foundation. The proposed check is a finite enumeration that would settle the matter computationally and independently. I therefore find the reader's rejection justified, and no adjustment to the verdict is needed.","tokens_in":10168,"tokens_out":6509,"duration_ms":79069,"concrete_test":"Compute the full support of S_(2,1,1)(x1,x2,y1) by exhaustive enumeration of all fillings of shape (2,1,1) with letters t1<t2<u1 satisfying the (k,ℓ)-semistandard rules, or equivalently by evaluating the Moens–Van der Jeugt determinant, and compare it with Bλ. If (0,3,1) is in Bλ but no tableau realizes it, Theorem 2.2 is contradicted, confirming that the SNP proof is built on a false support description.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is proved only through Theorem 2.2, which identifies Supp(Sλ) with the set Bλ of vectors satisfying the partial-sum inequalities A≤r≤∑_{i≤r}λ_i, B≤s≤∑_{j≤s}λ'_j, and |a|+|b|=|λ|. Lemma 2.1 gives the necessary direction, but Proposition 2.1, the sufficiency direction, is unsupported and in fact false. The proof asserts that for w=t_1^{a1}...t_k^{ak}u_ℓ^{bℓ}...u_1^{b1}, the row and column partial sums of shape(P(w)) equal A≤r and B≤s; this is not a property of mixed RS insertion. A concrete failure is λ=(2,1,1) with k=2, ℓ=1. The vector (a,b)=(0,3,1) satisfies A≤1=0≤2, A≤2=3≤3, B≤1=1≤3, and total size 4=|λ|, so it lies in Bλ. But the expansion in Example 3.1 has no x_2^3 y_1 term, and no (k,ℓ)-semistandard tableau of shape (2,1,1) can contain three t_2's: strict increase down columns allows at most one t_2 per column, and the diagram has only two columns. Thus Theorem 2.2 is false, Bλ is strictly larger than the true support, and the later equality Newton(Sλ)=Conv(Bλ)=H is not available. The TU integrality argument proves only that Conv(Bλ)∩Z^d=Bλ, which does not imply SNP for Sλ. The SNP statement may still be true, but this proof does not establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove that every supersymmetric Schur polynomial S_lambda(x,y) has a saturated Newton polytope. The strategy is to characterize the support by hook inequalities, encode these inequalities as a polyhedron H whose constraint matrix is an interval matrix, and then use total unimodularity and the Hoffman-Kruskal theorem to show that H is integral. The intended conclusion is Newton(S_lambda) cap Z^d = Supp(S_lambda).","tokens_in":10552,"tokens_out":14152,"duration_ms":172412,"significance":"The claimed result would be a worthwhile addition to the saturated-Newton-polytope literature, and the interval-matrix/TU approach is an elegant high-level idea. The paper is clearly organized and makes good use of standard tools such as Berele-Regev insertion, Rado's theorem, and Hoffman-Kruskal. However, the central support theorem is false, so the main result is not established by this argument.","major_comments":[{"comment":"The hook-inequality description of Supp(S_lambda) is false. For lambda=(2,1,1), k=2, ell=1, the vector (a,b)=(0,3,1) satisfies A_{<=1}=0<=2, A_{<=2}=3<=3, B_{<=1}=1<=3, and |a|+|b|=4, so it lies in B_lambda. However, the explicit expansion in Example 3.1 contains no x_2^3 y_1 term; indeed a (2,1,1) tableau with k=2 has at most two t_2's because t-letters strictly increase down columns and the diagram has only two columns. Also, S_lambda is symmetric in x, so its support must be invariant under swapping x_1 and x_2, whereas B_lambda contains (0,3,1) but not (3,0,1). Thus Theorem 2.2 is incorrect.","section":"§2, Theorem 2.2"},{"comment":"The proof of the sufficiency direction is invalid. The assertion that 'Definition of w gives sum_{i<=r} shape(P(w))_i = A_{<=r} and sum_{j<=s} shape(P(w))'_j = B_{<=s}' is not a property of mixed RS insertion. For example, with k=2, a=(1,1), b=(0), inserting w=t_1 t_2 by row insertion yields a tableau of shape (2), so the first-row sum is 2 while A_{<=1}=1. The row and column partial sums of the insertion shape are not the prefix sums of the input content. Hence the dominance argument does not force shape(P(w))=lambda.","section":"§2, Proposition 2.1"},{"comment":"Because B_lambda is not the support, the chain Newton(S_lambda)=Conv(B_lambda)=H in the proof of Theorem 3.5 is invalid. The total unimodularity argument proves only that H cap Z^d = B_lambda and that H is integral; it does not prove that every lattice point of H is a monomial exponent. In the Example 3.1 case, the point (0,3,1) is in H cap Z^d but is not in Supp(S_lambda), so H is strictly larger than the true Newton polytope. The SNP property may still be true, but this proof does not show it.","section":"§3.2, Theorem 3.5"}],"minor_comments":[{"comment":"In the sentence 'this work provides an a framework', the word 'an a' should be 'a'.","section":"§1.2"},{"comment":"The examples refer to tableaux and their contents, but the tableaux themselves are not displayed in the text; please include the diagrams or describe the fillings in words.","section":"§2, Examples 2.1 and 2.2"},{"comment":"The skew conjugate (lambda/mu)' is used without definition; please define it.","section":"§2, Eq. (1)"},{"comment":"The statement that all u_1,...,u_s lie inside the first s columns is not literally true; for example, in a one-row tableau t_1 t_1 t_1 u_1 u_2, the letter u_2 occurs in column 5. The inequality may still be true, but the stated justification should be corrected.","section":"§2, Lemma 2.1 proof"},{"comment":"The notation SSYTk,l(lambda) should be SSYT_{k,ell}(lambda) for consistency with the rest of the paper.","section":"§2, Definition 2.1"}],"recommendation":"reject","confidential_remarks":"The central support theorem is false as stated, so I recommend rejection. The interval-matrix/TU framework might be salvageable if a correct support description can be found, but that would be a substantial revision. The self-citation [29] is not load-bearing for the main argument, and I see no other ethical concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The punchline is that Theorem 2.2, the hook-inequality description of the support of Sλ, is false, and the main SNP theorem falls with it. The paper's own Example 3.1 gives the counterexample. For λ=(2,1,1), k=2, ℓ=1, the vector (0,3,1) satisfies the row inequalities (A≤1=0≤2, A≤2=3≤3), the column inequality (B≤1=1≤3), and size 4=|λ|, so it lies in Bλ. But the expansion shown has no x₂³ y₁ term: the support list is {(2,1,1),(1,2,1),(2,0,2),(1,1,2),(0,2,2),(1,0,3),(0,1,3)}. So Bλ is strictly larger than the true support, and Conv(Bλ)=H does not equal Newton(Sλ). The later total-unimodularity machinery proves only that Conv(Bλ)∩Z^d=Bλ, which is true but irrelevant to SNP. What is genuinely new here is the question itself: supersymmetric Schur polynomials were not on the known SNP list, and the authors correctly saw that the natural generalization of Rado's permutahedron argument is not direct. The interval-matrix/TU strategy is a sensible idea, and the exposition is clear enough that the error is easy to isolate. The proof of Proposition 2.1 is where things go bad: the mixed RS insertion of the word t₁^{a₁}⋯t_k^{a_k}u_ℓ^{b_ℓ}⋯u₁^{b₁} does not have the asserted row and column partial sums, and the \"standard dominance lemma\" invocation is a non sequitur because row and column dominance of the shape by λ plus equal size does not force equality. The lack of symmetry under permuting the x's is another red flag: Bλ is not invariant under swapping x₁ and x₂, while Sλ is symmetric in x, so any correct support description must have that symmetry. This is a serious flaw, not a gap in a minor lemma. The paper should not be accepted in its current form. But the underlying question—do supersymmetric Schur polynomials have saturated Newton polytopes?—is still open, and I suspect the true support description has extra inequalities coming from the interaction of the two alphabets. I'd encourage the authors to look at that. For a journal, I'd send this to a referee because the error is instructive and the question is worth airing, but the verdict should be reject, with the counterexample clearly communicated. I'm not citing this in my own work, and I wouldn't spend a reading group session on it beyond a brief look at the counterexample.","headline":"The main support theorem is false—the paper's own example contradicts it—so the SNP proof fails, though the question is natural and the TU approach is worth a look.","tokens_in":837,"tokens_out":1001,"would_cite":false,"duration_ms":29697,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52B20","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every supersymmetric Schur polynomial has a saturated Newton polytope.","keywords":["saturated Newton polytope","supersymmetric Schur polynomial","total unimodularity","hook inequalities","semistandard tableaux","Newton polytope","integral polyhedron","mixed Robinson-Schensted insertion"],"falsifier":"Compute $S_\\lambda(x,y)$ explicitly for a small hook partition and test every lattice point of the convex hull of its exponents; for instance, find a vector $(a,b)$ satisfying the hook inequalities of Theorem 2.2 while no $(k,\\ell)$-semistandard tableau of shape $\\lambda$ has content $(a,b)$. Such an example would show either the support description or the saturation conclusion fails.","tokens_in":9938,"feed_emoji":"🔷","tokens_out":6939,"duration_ms":73515,"temperature":0.7,"pith_summary":"This paper proves that every supersymmetric Schur polynomial has a saturated Newton polytope: the convex hull of its exponent vectors contains no lattice points other than the monomials that actually occur. These polynomials are characters of general linear Lie superalgebras, built from two alphabets of variables, and their support geometry had not been described before. The proof gives an exact tableau-theoretic description of the support by hook inequalities, encodes those inequalities as a polyhedron, and shows the polyhedron is integral because its constraint matrix is totally unimodular. The result puts supersymmetric Schur polynomials in the same gap-free Newton polytope family as ordinary Schur, Schubert, and Grothendieck polynomials.","feed_headline":"Supersymmetric Schur polynomials have saturated Newton polytopes","feed_subtitle":"A total-unimodularity proof shows every lattice point in the exponent polytope is a monomial that actually appears.","key_machinery":"The mechanism is a polyhedral encoding of the tableau support. Mixed Robinson-Schensted insertion for the super-alphabet $t_1<\\cdots<t_k<u_1<\\cdots<u_\\ell$ is used to show that a content $(a,b)$ lies in the support exactly when it satisfies the hook inequalities, and then those inequalities are represented as $H=\\{u : \\tilde{A}u\\le\\tilde{b}\\}$. The block matrix $\\tilde{A}$ is built from a consecutive-ones (interval) matrix $A$ plus the all-ones and negative-identity rows; after flipping signs it remains interval, so it is totally unimodular. The Hoffman-Kruskal criterion converts this into integrality of $H$, giving $\\mathrm{Newton}(S_\\lambda)=H$ and hence the saturated Newton polytope property.","core_discovery":"For a hook partition $\\lambda$, the supersymmetric Schur polynomial $S_\\lambda(x,y)$ is the generating function of $(k,\\ell)$-semistandard tableaux. The paper's central discovery is an exact linear description of its support: a pair of exponent vectors $(a,b)$ appears with nonzero coefficient exactly when the total degree is $|\\lambda|$, every row partial sum of $a$ is at most the corresponding row partial sum of $\\lambda$, and every column partial sum of $b$ is at most the corresponding column partial sum of the conjugate partition $\\lambda'$. Writing these conditions as a polyhedron $H$ in $\\mathbb{R}^{k+\\ell}$, the constraint matrix is an interval matrix after a row-sign normalization, hence totally unimodular. The integrality criterion for totally unimodular matrices makes $H$ an integral polyhedron, so every lattice point of $H$ is a genuine exponent. Since $\\mathrm{Newton}(S_\\lambda)=H$, every lattice point of the Newton polytope lies in the support.","pith_inferences":["A natural extension, not proved in the paper, is that the same interval-matrix strategy applies to other supersymmetric families such as super-Stanley symmetric functions and supersymmetric Macdonald polynomials, whose support combinatorics is more involved.","Because saturated Newton polytopes are a known route to Lorentzian and log-concavity phenomena, a next test is whether the normalized supersymmetric Schur polynomials are Lorentzian; the paper does not address this.","The hook polytope description opens the door to computing the volume, Ehrhart polynomial, or unimodular triangulations of $H$, quantities the paper leaves untouched."],"forward_implications":["Every integer point of $\\mathrm{Newton}(S_\\lambda)$ is a monomial of $S_\\lambda$, so the support has no lattice gaps at any scale.","Setting the $y$-alphabet empty recovers Rado's theorem for ordinary Schur polynomials, making the supersymmetric statement a common generalization.","The total unimodularity of the constraint matrix implies that optimizing a linear functional over the support is solvable in strongly polynomial time.","The weight set of the corresponding $\\mathfrak{gl}(k|\\ell)$-character coincides with the lattice points of an explicit integral polytope."],"supporting_citations":[{"why":"Defines the $(k,\\ell)$-semistandard tableaux and the mixed insertion bijection that the hook-inequality support description rests on.","marker":"[3]"},{"why":"Supplies the saturated Newton polytope framework and Rado's theorem for ordinary Schur polynomials, the case the paper generalizes.","marker":"[28]"},{"why":"Introduces total unimodularity as a technique for proving saturated Newton polytopes, the method adapted here.","marker":"[1]"},{"why":"Provides the theorem that interval (consecutive-ones) matrices are totally unimodular.","marker":"[19]"},{"why":"Provides the criterion that a totally unimodular constraint matrix with integral right-hand side gives an integral polyhedron.","marker":"[20]"},{"why":"Rado's permutahedron theorem underlies the classical SNP example the paper extends.","marker":"[31]"}],"fun_headline_variants":["Supersymmetric Schur Newton polytopes are saturated via unimodularity","Saturation of supersymmetric Schur polytopes from totally unimodular constraints","Newton polytopes of supersymmetric Schur polynomials: all lattice points appear","Total unimodularity proves saturation for supersymmetric Schur polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the support description: a content vector satisfies the row, column, and size inequalities if and only if some $(k,\\ell)$-semistandard tableau of shape $\\lambda$ has that content.","fun_headline_variants_meta":{"raw":{"variants":["Supersymmetric Schur Newton polytopes are saturated via unimodularity","Saturation of supersymmetric Schur polytopes from totally unimodular constraints","Newton polytopes of supersymmetric Schur polynomials: all lattice points appear","Total unimodularity proves saturation for supersymmetric Schur polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000805,"raw_usage":{"total_tokens":3463,"prompt_tokens":802,"completion_tokens":2661,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":418,"completion_tokens_details":{"reasoning_tokens":2590}},"tokens_in":418,"tokens_out":2661,"duration_ms":19827,"temperature":1.0,"reasoning_tokens":2590,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:35:38.870586+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $S_\\lambda(x,y)$ explicitly for a small hook partition and test every lattice point of the convex hull of its exponents; for instance, find a vector $(a,b)$ satisfying the hook inequalities of Theorem 2.2 while no $(k,\\ell)$-semistandard tableau of shape $\\lambda$ has content $(a,b)$. Such an example would show either the support description or the saturation conclusion fails.","supporting_citations":[{"cited_title":"Berele and A","cited_arxiv_id":null,"evidence_quote":"Defines the $(k,\\ell)$-semistandard tableaux and the mixed insertion bijection that the hook-inequality support description rests on."},{"cited_title":"Newton polytopes in algebraic combinatorics","cited_arxiv_id":null,"evidence_quote":"Supplies the saturated Newton polytope framework and Rado's theorem for ordinary Schur polynomials, the case the paper generalizes."},{"cited_title":"An efficient algorithm for deciding vanishing of Schubert polynomial coefficients","cited_arxiv_id":null,"evidence_quote":"Introduces total unimodularity as a technique for proving saturated Newton polytopes, the method adapted here."},{"cited_title":"Heller and C","cited_arxiv_id":null,"evidence_quote":"Provides the theorem that interval (consecutive-ones) matrices are totally unimodular."},{"cited_title":"Hoffman and Joseph B","cited_arxiv_id":null,"evidence_quote":"Provides the criterion that a totally unimodular constraint matrix with integral right-hand side gives an integral polyhedron."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rado's permutahedron theorem underlies the classical SNP example the paper extends."}],"review_version":1}