{"id":"97e14b08-4e42-43cc-a308-e7ded33adde0","arxiv_id":"1908.08502","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The product of any key polynomial with a single-row Schur polynomial expands into key polynomials with coefficients in {-1,0,1}, via a weight-preserving bijection on Kohnert diagrams.","lead":"A new combinatorial rule describes how to multiply any key polynomial by a one-variable sum, expanding the product into key polynomials with coefficients only 1 or -1. It gives algebraic combinatorics a nonsymmetric analogue of the classical Pieri rule and a bijection that generalizes RSK insertion to Kohnert diagrams.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Imported thread-decomposition criterion (Lemma 2.3.8) is the load-bearing hinge; an unverified restriction there would invalidate the key Pieri formula.","rationale":"The reader identified exactly the right weak point: the paper imports Lemma 2.3.8 from Assaf-Searles without proof, and that lemma is used pervasively. My independent read of the manuscript confirms that the statement of the Pieri rule and the surrounding bijective machinery are internally coherent: the bottom insertion (Theorem 4.1.13), top insertion (Theorem 4.3.15), and stratum-map arguments (Section 5) all reduce membership of diagrams to the thread-weight criterion. If Lemma 2.3.8 is correct as stated for all generic Kohnert diagrams, the containment arguments and the reduction to k-addable cells are sound, and the inclusion–exclusion in Lemma 3.3.4 follows from that criterion. I found no internal inconsistency or circularity in the proof once the lemma is granted. The concern is therefore a verification risk rather than a demonstrated flaw. Since Lemma 2.3.8 is attributed to a published source, and the paper's own contributions beyond that point are presented in detail, the appropriate verdict remains the reader's ACCEPT with moderate confidence. A finite exhaustive check of the lemma for small n, together with a direct comparison to [5, Theorem 3.7], would settle whether the imported criterion is stated with exactly the needed hypotheses.","tokens_in":18,"tokens_out":9649,"duration_ms":217672,"concrete_test":"Verify Lemma 2.3.8 independently. Exhaustively, for all weak compositions a of length n≤4 with sum≤6: (i) generate KD(a) by Kohnert moves from key(a); (ii) generate all generic Kohnert diagrams T in the same bounding box using the column-weight criterion (2.3.1); (iii) compute θ(T) by the thread algorithm in Definition 2.3.6; (iv) check T∈KD(a) if and only if θ(T) lies weakly below a in left swap order. In parallel, locate the exact statement and proof of [5, Theorem 3.7] and confirm that it genuinely contains the full 'if and only if' with the left swap order, since the paper only describes the lemma as implicit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.3.6 rests on the weight-preserving bijection Theorem 3.1.4, and nearly every containment or membership step in the proof uses Lemma 2.3.8: for a generic Kohnert diagram T, T∈KD(a) if and only if θ(T)≼a. The lemma is imported as 'Implicit in [5, Theorem 3.7]' and is not proved. It underlies Proposition 2.3.9, Corollary 3.1.5, Lemma 3.2.5, Theorem 3.2.10, Lemma 3.3.4, and Theorem 4.1.8, which in turn supplies the matching-sequence criterion used for bottom insertion, top insertion, and the stratum maps. If the claimed 'if and only if' has a hidden genericity restriction—for instance if θ(T)≼a is necessary but not sufficient for T∈KD(a), or if it holds only after strengthening KD(a) to diagrams obtained without certain jumps—then the image inclusions in Lemmas 3.2.5 and 3.3.4, the maximality of k-addable terms in Theorem 3.2.10, and the inclusion–exclusion in Theorem 3.3.6 could all fail. The paper gives no independent derivation and no small worked test of the criterion beyond the motivating examples. This is not evidence of an error; it is the one place where the central claim's correctness depends on an imported statement whose exact hypotheses are not re-verified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a nonsymmetric generalization of the Pieri rule for key polynomials (Demazure characters of GL_n). The main bijective statement, Theorem 3.1.4, asserts a weight-preserving bijection between KD(a) × KD(e_k) and the union of KD(b + e_j) over b weakly below a in left swap order and 1 ≤ j ≤ k. From this, Theorem 3.3.6 derives a cancellation-free, multiplicity-free signed formula for κ_a · s_(1)(x_1, ..., x_k) as a sum over k-addable columns and nonempty subsets R of the corresponding row sets. Sections 4 and 5 prove the bijection through bottom insertion, top insertion via rectification, and a stratification by stratum maps; Section 6 extends the construction to horizontal strips and characterizes when the key expansion is nonnegative, with applications to Schubert polynomials.","tokens_in":65671,"tokens_out":7210,"duration_ms":72338,"significance":"If the result stands, it is a substantial contribution: it gives the first explicit nonsymmetric Pieri rule for key polynomials, generalizes the classical RSK-based proof of Pieri's rule, and provides a signed but cancellation-free expansion whose signs are fully understood. The paper is also notable for its detailed, example-rich combinatorial proof and for the clean reduction to classical Schur Pieri in the weakly increasing case. The positivity characterization in Section 6.3, including the vexillary case, is a further valuable dividend. The main caveat is that the proof leans on the imported thread-decomposition criterion of Lemma 2.3.8; this is a cited result rather than an internal error, but its exact statement should be made fully explicit given how many later statements depend on it.","major_comments":[],"minor_comments":[{"comment":"Lemma 2.3.8, stated as implicit in [5, Theorem 3.7], is the load-bearing criterion that T ∈ KD(a) if and only if θ(T) ≼ a. It is used repeatedly—for example in Proposition 2.3.9, Corollary 3.1.5, Lemma 3.2.5, Theorem 3.2.10, Lemma 3.3.4, and Theorem 4.1.8—and therefore underpins Theorem 3.3.6. Since the paper does not reproduce or prove this criterion, I ask the authors to provide either a self-contained proof or a precise quotation of [5, Theorem 3.7] together with an explanation of why the stated 'if and only if' follows for generic Kohnert diagrams. This is a completeness issue rather than an apparent error, but it should be resolved before publication.","section":"Section 2.3, Lemma 2.3.8"},{"comment":"In the displayed formula, the condition 'c_i−1 ∈ {a_1,...,a_n, c_i−1}' is tautological as written. Based on Example 6.3.2 and the surrounding text, it should read 'c_i−1 ∈ {a_1,...,a_n, c_{i−1}}' (with the second term being the previously chosen value, not the current value). Please correct this notation.","section":"Section 6.3, Corollary 6.3.1"},{"comment":"The notation 'ℓ = max_i{a_i > 0}' is ambiguous; it should be 'ℓ = max{i : a_i > 0}' or 'ℓ(a) = max{i : a_i > 0}', matching the use earlier in the paper. As written, the expression resembles a maximum over a set of inequalities rather than the largest index with a positive part.","section":"Section 4.3, Theorem 4.3.15 and text before it"},{"comment":"In the paragraph after Lemma 5.2.8, the sentence 'By Lemma 5.1.8, we have wt(M_{b+e_k}|_{U^-}) = wt(M_{b+e_k}|_{U^-})' is tautological and appears to contain a typo. Please check whether the intended equality involves θ(U^-) or the restriction of the Kohnert labeling, and revise accordingly.","section":"Section 5.2, proof of Theorem 5.1.13"}],"recommendation":"minor_revision","confidential_remarks":"The paper fits the journal's scope and the central argument appears sound. The only substantive concern is the reliance on the imported Lemma 2.3.8 without a proof or exact quotation; this should be fixed in revision. The remaining issues are local notational and typographical corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know about this paper if you care about key polynomials, Demazure characters, or bijective combinatorics. It proves a nonsymmetric Pieri rule for the product of an arbitrary key polynomial with a single-part key polynomial, with no restriction on the number of variables. The central bijection (Theorem 3.1.4) between KD(a) × KD(e_k) and a union of Kohnert diagram spaces is genuinely new, and the signed inclusion-exclusion formula (Theorem 3.3.6) with pairwise distinct terms is a clean statement that reduces to the classical Pieri rule when the composition is weakly increasing. I did not find a concrete error.\n\nWhat the paper does well: it builds the bijection from scratch, treating the k=1 case with bottom insertion, the k ≥ ℓ(a) case with top insertion via rectification, and the middle cases with stratification maps. The reduction to k-addable cells is natural, and the worked examples are extensive and consistent. The final applications to vexillary compositions and Schubert polynomials give real payoff, and the connection to Haglund–Luoto–Mason–van Willigenburg is handled honestly.\n\nWhere I am cautious: nearly every containment or membership step in the proof relies on Lemma 2.3.8, imported from Assaf–Searles and stated as implicit in their Theorem 3.7. That lemma says a generic Kohnert diagram T lies in KD(a) iff its thread weight is below a in left swap order. The stress-test note is right to identify this as load-bearing: if the imported criterion has a hidden genericity restriction, the bijection and formula could collapse. The paper does not reprove the lemma, and a few later arguments are compressed enough that I would not want to referee this without that lemma in front of me. But I have no evidence of an actual counterexample, and the authors clearly state the dependence.\n\nBottom line: this is a serious combinatorial advance, aimed at algebraic combinatorics and Schubert calculus readers. It deserves a careful peer review, not a desk reject. My recommendation: send it out, and ask the authors to either prove Lemma 2.3.8 in a self-contained appendix or give a precise statement with a more explicit reference. I would cite it if I worked in this area.","headline":"A substantial new bijective Pieri rule for key polynomials, with the main caveat being a load-bearing imported lemma that the authors do not reprove.","tokens_in":66241,"tokens_out":1814,"would_cite":true,"duration_ms":20943,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","14N10","14N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A signed Pieri rule for key polynomials, proved by box insertion into Kohnert diagrams.","keywords":["Demazure characters","key polynomials","Pieri rule","Kohnert diagrams","Robinson-Schensted-Knuth insertion","left swap order","Schubert polynomials","multiplicity-free expansion"],"falsifier":"Search for a generic Kohnert diagram $T$ and a weak composition $a$ with $\\theta(T)\\preceq a$ but $T\\notin KD(a)$; one such pair refutes the imported thread-weight criterion and, with it, Theorem 3.3.6. Because all objects are finite and algorithmically enumerable, this is a concrete brute-force check.","tokens_in":65195,"feed_emoji":"🧮","tokens_out":11717,"duration_ms":105836,"temperature":0.7,"pith_summary":"The paper establishes a nonsymmetric Pieri rule for key polynomials, the characters of Demazure modules for the general linear group. It proves that multiplying any key polynomial $\\kappa_a$ by a one-row Schur polynomial $s_{(1)}(x_1,\\ldots,x_k)$ expands in the key basis with no coefficients other than $+1$ and $-1$, and it gives an explicit indexing of the terms by $k$-addable cells and subsets of rows. The proof is bijective: a single-box version of Robinson--Schensted--Knuth insertion on Kohnert diagrams gives a weight-preserving bijection between the diagram set of $a$ times a single box and the union of diagram sets $KD(b+e_j)$ for $b$ below $a$ in left swap order and $j\\le k$, with signs arising only from overlaps in that union. If correct, the formula supplies the missing nonsymmetric analog of the classical Pieri rule and yields explicit positivity criteria, including a nonnegative expansion for products of Schubert polynomials with Grassmannian Schubert polynomials in the vexillary case.","feed_headline":"Key polynomials get a signed Pieri rule from Kohnert diagrams","feed_subtitle":"A new bijection extends RSK insertion and gives explicit ±1 expansions for products with one-row factors.","key_machinery":"The machinery is Kohnert diagrams together with the left swap order and thread decomposition. A Kohnert diagram is a finite set of unit cells obtained from the key diagram of a weak composition by repeatedly moving the rightmost cell of a row down to the first available position in its column, and key polynomials are the generating polynomials of these diagrams. The left swap order $b\\preceq a$ is the transitive closure of swapping a smaller part of $a$ with a larger part to its right; it controls containment, since $\\mathrm{key}_b\\in KD(a)$ exactly when $b\\preceq a$, and more generally a diagram $T$ lies in $KD(a)$ exactly when its thread weight $\\theta(T)$ lies below $a$ in this order. The paper's new insertion maps---bottom insertion, rectification, top insertion, and stratum maps---play the role of RSK insertion on tableaux and prove the bijection of Theorem 3.1.4. The drop composition $\\mathrm{drop}(c,R)_a$ is the weak composition obtained by lowering supporting cells from above the added cell down to the rows of $R$; it is the object that records the inclusion-exclusion data in the final formula.","core_discovery":"The central claim is Theorem 3.3.6: for a weak composition $a$ and positive integer $k$, $\\kappa_a s_{(1)}(x_1,\\ldots,x_k)$ equals the signed sum over $k$-addable columns $c$ of $a$ and nonempty subsets $R$ of the $k$-addable row set $\\mathrm{Row}^c_{a,k}$ of $(-1)^{|R|-1}\\kappa_{\\mathrm{drop}(c,R)_a+e_{\\min(R)}}$, and the terms are pairwise distinct. The indexing data are read directly from the key diagram of $a$: a $k$-addable cell is a position in a row at most $k$ to which a box can be added after supporting cells above row $k$ are dropped down through left swaps, and the drop composition records how far those supporting cells fall. The theorem is proved through the weight-preserving bijection of Theorem 3.1.4, $KD(a)\\times KD(e_k)\\leftrightarrow\\bigcup_{b\\preceq a,\\,1\\le j\\le k} KD(b+e_j)$, constructed by an insertion algorithm on Kohnert diagrams. The negative signs are not cancellations inside a single disjoint union; they come from genuine overlaps of the sets $KD(b+e_j)$, and the inclusion-exclusion in the formula removes exactly that redundancy.","pith_inferences":["The signs, which arise from overlaps of diagram sets rather than from a sign-reversing involution, suggest that the formula could be lifted to a K-theoretic or Euler-characteristic Pieri rule for Demazure characters; this is a testable extension, not a claim of the paper.","The same stratum machinery might yield a direct rule for products of a key polynomial with an arbitrary Schur polynomial in a fixed number of variables, with explicit sign data refining the earlier formula; the paper only treats the single-row case.","A brute-force verification on all weak compositions of a small fixed length would test both the formula and the imported thread-weight criterion, since the two are logically tied."],"forward_implications":["The product of any key polynomial with $x_1+\\cdots+x_k$ has a cancellation-free expansion in the key basis, with at most one positive term per $k$-addable column and signs recorded by subsets of rows.","For weakly increasing $a$ and $k=n$, the formula specializes to the classical Schur Pieri rule $s_\\lambda s_{(1)}=\\sum_{\\mu\\supset\\lambda,\\,|\\mu/\\lambda|=1} s_\\mu$.","For $k\\ge \\ell(a)$, the expansion is nonnegative and is indexed by choosing columns $c_1<\\cdots<c_m$ with each $c_i-1$ among the parts of the current composition, as stated in Corollary 6.3.3.","For weak compositions satisfying condition (2) of Macdonald's vexillary definition, the key expansion of $\\kappa_a s_{(m)}(x_1,\\ldots,x_k)$ is nonnegative for every $k$, giving an explicit formula for the Schubert polynomial product $S_wS_{v((m),k)}$ in Theorem 6.3.8.","Iterating the insertion gives a horizontal-strip bijection $KD(a)\\times KD(m e_k)\\leftrightarrow D^{(m)}(a,k)$, expanding the product $\\kappa_a\\kappa_{m e_k}$ by $k$-addable horizontal $m$-strips, as in Theorem 6.1.5 and Corollary 6.1.8."],"supporting_citations":[{"why":"Supplies the left swap order, thread decomposition, thread-weight criterion (Lemma 2.3.8), and Kohnert labelings that every containment argument in the proof uses.","marker":"[5]"},{"why":"Defines Kohnert diagrams and Kohnert moves, the model in which key polynomials are generated and in which the bijection is constructed.","marker":"[13]"},{"why":"Gives the RSK insertion bijection for the classical Pieri rule that the paper generalizes to diagrams.","marker":"[23]"},{"why":"States the classical Schur Pieri rule whose nonsymmetric analog is the paper's target.","marker":"[19]"},{"why":"Provides an earlier key-expansion formula for key times Schur in a fixed number of variables, the benchmark for the new formula's scope and extremal cases.","marker":"[10]"},{"why":"Defines vexillary weak compositions and links them to Schubert polynomials through Lehmer codes, used for the positivity and Schubert applications.","marker":"[16]"},{"why":"Introduces Schubert polynomials, the geometric objects whose products are given nonnegative expansions in the vexillary case.","marker":"[14]"}],"fun_headline_variants":["Signed Pieri rule for Demazure characters via Kohnert diagrams","Key polynomials: explicit signed Pieri rule from insertion","Kohnert insertion solves Demazure Pieri products","No cancellations: signed Pieri rule for Demazure characters","RSK generalized to Kohnert: signed Pieri for keys"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported criterion, used without proof, that a generic Kohnert diagram lies in $KD(a)$ exactly when its thread weight lies below $a$ in the left swap order; if that criterion has hidden restrictions, the bijection and the signed formula collapse.","fun_headline_variants_meta":{"raw":{"variants":["Signed Pieri rule for Demazure characters via Kohnert diagrams","Key polynomials: explicit signed Pieri rule from insertion","Kohnert insertion solves Demazure Pieri products","No cancellations: signed Pieri rule for Demazure characters","RSK generalized to Kohnert: signed Pieri for keys"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000777,"raw_usage":{"total_tokens":3432,"prompt_tokens":939,"completion_tokens":2493,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":2409}},"tokens_in":555,"tokens_out":2493,"duration_ms":19392,"temperature":1.0,"reasoning_tokens":2409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:37:22.656438+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search for a generic Kohnert diagram $T$ and a weak composition $a$ with $\\theta(T)\\preceq a$ but $T\\notin KD(a)$; one such pair refutes the imported thread-weight criterion and, with it, Theorem 3.3.6. Because all objects are finite and algorithmically enumerable, this is a concrete brute-force check.","supporting_citations":[{"cited_title":"Kohnert tableaux and a li fting of quasi-Schur functions","cited_arxiv_id":null,"evidence_quote":"Supplies the left swap order, thread decomposition, thread-weight criterion (Lemma 2.3.8), and Kohnert labelings that every containment argument in the proof uses."},{"cited_title":"W eintrauben, Polynome, Tableaux","cited_arxiv_id":null,"evidence_quote":"Defines Kohnert diagrams and Kohnert moves, the model in which key polynomials are generated and in which the bijection is constructed."},{"cited_title":"Schensted","cited_arxiv_id":null,"evidence_quote":"Gives the RSK insertion bijection for the classical Pieri rule that the paper generalizes to diagrams."},{"cited_title":"Sul problema degli spazi secanti","cited_arxiv_id":null,"evidence_quote":"States the classical Schur Pieri rule whose nonsymmetric analog is the paper's target."},{"cited_title":"Haglund, K","cited_arxiv_id":null,"evidence_quote":"Provides an earlier key-expansion formula for key times Schur in a fixed number of variables, the benchmark for the new formula's scope and extremal cases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines vexillary weak compositions and links them to Schubert polynomials through Lehmer codes, used for the positivity and Schubert applications."},{"cited_title":"Polyn ˆ omes de Schubert","cited_arxiv_id":null,"evidence_quote":"Introduces Schubert polynomials, the geometric objects whose products are given nonnegative expansions in the vexillary case."}],"review_version":1}