{"id":"93c73c03-4829-49bc-b43f-5e0bd073e342","arxiv_id":"2502.05618","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A lowering-operator formula expresses closed K-k-Schur functions as sums of K-k-Schur functions in the Bruhat order, yielding a new proof of a theorem by Ikeda, Iwao and Naito.","lead":"This paper studies lowering operators, a tool that changes symmetric functions by removing boxes, acting on K-k-Schur functions from K-theoretic Schubert calculus. It obtains a new operator formula for closed K-k-Schur functions and uses it to give a combinatorial proof of a theorem previously proved by Ikeda, Iwao and Naito.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Claim 4.6 collapses lowering-operator orbit sets to the full Bruhat interval using only the forward inclusion of Proposition 4.4; the required reverse inclusion is never proved.","rationale":"The reader’s weakest assumption pinpoints a genuine gap in the proof of the central theorem. The paper proves, in Proposition 4.4, that every partition produced by the lowering-operator straightening process lies in the Bruhat interval below λ. The proof of Claim 4.6, however, repeatedly uses the converse identification: the union of all Ω-sets over n is asserted to be exactly the full Bruhat interval. The cited Proposition 4.4 cannot justify this converse. The step appears in equation (96) and again in the base case of equation (98), where the sum over n of Ω_{ν,{d}^n} is replaced by the interval {μ : wμ ≤ wν}. That identification is load-bearing because Claim 4.6 is the engine of Theorem 4.5, and Theorem 4.5 in turn supplies the combinatorial proof of Theorem 1.2. I find no other issue that is equally central: the lowering-operator calculations in Section 3 are detailed and checkable, and Theorem 1.2’s reduction to Theorem 4.5 is straightforward once Theorem 4.5 is available. The gap is internal rather than a disagreement with external consensus, and it is plausibly repairable: one could try to prove the missing surjectivity by induction on Bruhat length using strong covers, or by showing directly that every μ ≤ λ is reachable by a sequence of allowable lowering operators. Until such a lemma is supplied, the paper should remain conditional. The reader’s verdict is therefore unchanged.","tokens_in":43511,"tokens_out":6982,"duration_ms":71573,"concrete_test":"Implement definitions (15)–(16) and Definition 3.16 in SageMath for a small case, e.g. k=2, ℓ=3, λ=(2,1,0) and k=3, ℓ=4, λ=(3,2,1,0). Enumerate Ω_{λ,{z}^n} for all z ∈ [ℓ] and n up to |P^k_ℓ|, compute wμ via the bijections of Section 4.1, and compare ∪_{n≥0} Ω_{ν,{z}^n} with {μ ∈ P^k_ℓ : wμ ≤ wν}. In particular, test the base case of (98) with z = bot_{Δ_k(ν)}(z). If any interval element is missing, equality (88) fails and Theorem 4.5 is not established; if none is missing, check larger λ to confirm the collapse is general.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Claim 4.6, after applying Theorem 3.18(a), sums over ν ∈ Ω_{ν0,{z}^n} are replaced by the full set {μ ∈ P^k_ℓ : wμ ≤ wν0} with the citation “by Proposition 4.4” (see equations (96) and the base case of (98)). Proposition 4.4 establishes only the forward direction: if ν ∈ Ω_{λ,z}, then wν < wλ. It does not show that every μ with wμ ≤ wλ lies in ∪_{n≥0} Ω_{λ,{z}^n} for the relevant z, nor that every μ in the interval is reachable through the iterative Γ process. This surjectivity is exactly what is needed to make equality (88) true, and it is also used in the induction proving (98). Without it, Theorem 4.5 may only assert that the lowering-operator sum equals a sub-sum over the Bruhat interval, and the subsequent application to Theorem 1.2 would not cover the full closed K-k-Schur function. The gap may be fillable from strong-cover properties such as Lemma 4.2, but no such statement or proof appears in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a systematic study of lowering operators L_z acting on K-k-Schur functions g^{(k)}_λ, for λ in P^k_ℓ. The main technical result is Theorem 3.18, a recursive formula expressing L_z^n g^{(k)}_λ as a sum of K-k-Schur functions indexed by sets Ω_{λ,z}, together with a telescoping refinement in Corollary 3.21. These tools are then used to prove Theorem 4.5 (Theorem 1.1): the sum of lowering operators ∏_{z∈S} L_z applied to g^{(k)}_λ over all multisets S with support in [ℓ]_λ equals the closed K-k-Schur function, i.e., the sum of g^{(k)}_μ over μ in the Bruhat interval w_μ ≤ w_λ. As an application, the paper gives a new proof of Theorem 1.2, originally due to Ikeda, Iwao and Naito, expressing the closed k-Schur Katalan function as (1 - G_1^⊥) applied to that Bruhat-interval sum. The proof of Theorem 4.5 is long and is mostly built from the cited Mirror Lemmas of Blasiak–Morse–Seelinger, but the key step Claim 4.6 contains a load-bearing surjectivity assertion that is not proved.","tokens_in":43700,"tokens_out":3509,"duration_ms":36099,"significance":"If the proof can be completed, the result would be valuable: it gives an explicit lowering-operator formula for closed K-k-Schur functions and a combinatorial route to a theorem previously proved by different methods in [8]. The paper contains no fitted parameters and relies on established Katalan-function machinery; the main claimed identity is a genuine new statement. The proof strategy is plausible and the combinatorial structures (Ω_{λ,z}, Γ sets, Bruhat intervals) are natural. However, the gap identified in Claim 4.6 affects the central theorem, so the significance is conditional on repairing that argument.","major_comments":[{"comment":"The proof of Claim 4.6 replaces sums over the sets Ω_{ν(0),{z}^n} by sums over the full Bruhat interval {μ ∈ P^k_ℓ : w_μ ≤ w_{ν(0)}} with the citation \"by Proposition 4.4\". Proposition 4.4 proves only the forward inclusion: if ν ∈ Ω_{λ,z}, then w_ν < w_λ. It does not prove the reverse inclusion that every μ with w_μ ≤ w_λ lies in the union over n of Ω_{λ,{z}^n}, nor that the iterative Γ process reaches every element of the Bruhat interval. This surjectivity is exactly what is needed to justify equation (96) and the base case of (98), and without it equation (84) may assert equality only with a proper sub-sum of the Bruhat interval. The gap is load-bearing for Theorem 4.5 and, through it, Theorem 1.2. A proof of the reverse inclusion, perhaps from strong-cover properties such as Lemma 4.2, is required.","section":"§4.2, Claim 4.6, equations (96) and (98)"},{"comment":"The final containment step of Proposition 4.4 reads \"c(ν)[1,z-1] ⊊ c(λ)[1,z-1] ⇒ c(ν) ⊊ c(λ) (by c(ν)[z,ℓ] ⊊ c(λ)[z,ℓ])\". The second containment c(ν)[z,ℓ] ⊊ c(λ)[z,ℓ] is asserted without proof; the preceding sentence says it follows from w_{ν'} < w_λ, (74), and the effect of -ǫ[z,z_a-1], but the argument is not written out and is not transparent. Since Proposition 4.4 is the only bridge between the combinatorially defined sets Ω and the Bruhat order, this step needs a rigorous justification.","section":"§4.1, proof of Proposition 4.4"},{"comment":"The definition of the sets Γ^i_{ν^i_{(j)}} depends on choices of z ∈ Z_{i,j} and of ν^i_{(j)} ∈ Ω_{ν^i_{(j-1)}, z}; it is not proved that the choices can be made so that the process runs for all i ∈ [0,r] and yields nonempty Γ^i at each step. The proof of (83) only examines the initial Γ^i_{ν^i_{(0)}}; for the updated sets after applying (79), the cardinality statement is argued in Case 2 via (91), but the case analysis is not fully exhaustive because the proof of (87) assumes the specific identity (91) without treating possible coincidences or empty intersections in the interval [1, down_{Δ_k(ν(0))}(z)-1]. These details need to be supplied for Claim 4.6 to be complete.","section":"§4.2, equations (79) and (83)"}],"minor_comments":[{"comment":"There are numerous OCR-type typographical errors, e.g., \"nelement\" for \"∉\" in Definition 2.7(d) and elsewhere, \"gk\" for \"g^{(k)}\" in several displayed equations, and \"fomula\" in the title of reference [21]. The paper should be carefully proofread.","section":"Throughout"},{"comment":"The notation [ℓ]_λ is introduced only in (75), but it is used implicitly in the introduction and in Theorem 1.1; the paper would be easier to read if the definition were moved earlier or if the introduction included a pointer to (75).","section":"§1.3 and §2"},{"comment":"The example is helpful but the subscript notation for partitions of length 10 is unwieldy; a Young-diagram or abbreviated notation, as used in the displayed grid, would improve readability.","section":"§3.3, Example 3.20"},{"comment":"The line \"(1 - e_1^⊥ + e_2^⊥ + ⋯ + (-1)^ℓ e_ℓ^⊥)\" is correct but could be written as ∑_{d=0}^ℓ (-1)^d e_d^⊥ to avoid ambiguity about the signs in the displayed sum.","section":"§4.3, equation (102)"},{"comment":"The reference [3] is to the authors' own previous preprint arXiv:2501.04200; since results from [3] (e.g., Proposition 2.10 used in Corollary 3.21) are load-bearing, the paper should state explicitly which facts are quoted from that preprint and whether they are available in a peer-reviewed form.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main gap is the surjectivity missing from Proposition 4.4 when passing from Ω sets to the full Bruhat interval in Claim 4.6. This is not a small presentation issue; it is central to Theorem 4.5. However, the approach is plausible and the gap may be fillable using strong-cover properties and the existing Γ recursion. I recommend major revision rather than rejection, with the expectation that the authors either supply a proof of the reverse inclusion or adjust the statement of Theorem 4.5 to the sub-sum that is actually established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the paper. The main theorem is new and the paper does real work, but there is a load-bearing gap in the proof that needs to be fixed before the theorem is established.\n\nWhat is good: Theorem 4.5 (also Theorem 1.1) is a genuinely new lowering-operator formula for closed K-k-Schur functions, and it gives a clean route to the Ikeda–Iwao–Naito theorem. The paper is carefully organized around the Mirror Lemmas, and the authors put real effort into controlling the recursive straightening process via the sets Gamma. The background material is well integrated, and the writing is methodical.\n\nThe soft spot: the proof of Claim 4.6, specifically equation (88) and its use in the induction for equation (84). In Case 1 (and again in equation (96) and in the induction for (98)), the authors replace the union over n of Omega_{nu(0),{z}^n} with the full Bruhat interval {mu in P^k_ell : w_mu <= w_nu(0)}, citing Proposition 4.4. But Proposition 4.4 proves only the forward inclusion: if nu is in Omega_{lambda,z}, then w_nu < w_lambda. It does not prove that every mu with w_mu <= w_nu appears in the union over all n of the recursively defined Omega sets. That surjectivity is exactly what is needed to make the equality in (88) hold. Without it, the lowering-operator sum on the left side equals a sum over some subset of the Bruhat interval, not necessarily the full closed K-k-Schur function. This gap is load-bearing: Theorem 4.5 and the subsequent proof of Theorem 1.2 both rest on it.\n\nA smaller point: the proof of Corollary 3.21 relies on a vanishing statement from the authors' previous preprint [3]; this is auxiliary, but the dependency should be stated more prominently.\n\nOverall: the paper deserves peer review. The gap is real but may be fillable from strong-cover properties (Lemma 4.2) and known behavior of lowering operators. A referee should ask for a precise proof of the reverse inclusion or, if it fails, a corrected statement. As it stands, I would not cite it as established, but I would send it to review and expect a revised version to be a solid contribution.","headline":"New lowering-operator formula for closed K-k-Schur functions, but the proof of Claim 4.6 assumes without proof that the reachable set equals the full Bruhat interval.","tokens_in":44294,"tokens_out":4154,"would_cite":false,"duration_ms":36870,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","14N15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a sum of lowering operators acting on a K-k-Schur function reproduces the closed K-k-Schur function, and uses this to give a combinatorial proof of the closed k-Schur Katalan conjecture.","keywords":["symmetric functions","K-k-Schur functions","closed K-k-Schur functions","lowering operators","Katalan functions","Bruhat order","affine Schubert calculus","K-theory"],"falsifier":"In a small explicit case, expand both sides of Theorem 4.5 in the Schur basis and compare coefficients; a mismatch falsifies the theorem. To target the weak point directly, enumerate the $\\Gamma$-recursion for a small $\\lambda$ and check that every $\\mu$ with $w_\\mu \\le w_\\lambda$ appears in the union of the recursively generated sets; any missing $\\mu$ refutes the reverse inclusion used at equations (96) and (98).","tokens_in":43266,"feed_emoji":"🧮","tokens_out":9045,"duration_ms":81281,"temperature":0.7,"pith_summary":"The paper establishes a lowering-operator formula for closed K-$k$-Schur functions: for every $k$-bounded partition $\\lambda$, summing all lowering-operator products supported on a certain subset $[\\ell]_\\lambda$ applied to the K-$k$-Schur function $g^{(k)}_\\lambda$ reproduces the closed K-$k$-Schur function $\\sum_{\\mu \\in P^k_\\ell,\\, w_\\mu \\le w_\\lambda} g^{(k)}_\\mu$, where $\\le$ is Bruhat order. The formula turns a Bruhat-interval sum of many symmetric functions into a single operator expression, so closed functions inherit the same lowering-operator calculus that defines Katalan functions. A direct consequence is a combinatorial proof of the closed $k$-Schur Katalan conjecture, previously established by a different method. A reader should care because the paper locates the closed functions inside the operator mechanics of Katalan functions, making the K-theoretic Schubert representatives computable from root-ideal data alone.","feed_headline":"Lowering operators rebuild closed K-k-Schur functions","feed_subtitle":"One identity turns a Bruhat-interval sum of symmetric functions into a single operator product.","key_machinery":"The machinery is the Katalan-function calculus introduced in [2]. A Katalan function $K(\\Psi;M;\\gamma)$ is built from a root ideal $\\Psi$, a multiset $M$ of lowering operators, and an integer vector $\\gamma$ by applying $\\prod_{z\\in M}(1-L_z)\\prod_{(i,j)\\in\\Psi}(1-R_{ij})^{-1}$ to $g_\\gamma$, and the K-$k$-Schur function $g^{(k)}_\\lambda$ is the special case $K(\\Delta_k(\\lambda);\\Delta_{k+1}(\\lambda);\\lambda)$. The paper's workhorse is a structural analysis, Theorem 3.18 and Corollary 3.21, of $L_z^n g^{(k)}_\\lambda$: after straightening via the Mirror Lemma and the Mirror Straightening Lemma, each power reduces to a positive sum of K-$k$-Schur functions indexed by the sets $\\Omega_{\\lambda,z}$, with a remainder term that telescopes once one sums over $n$. The closed-function formula then follows from a recursive bookkeeping device, the sets $\\Gamma^i_{\\nu^i}$, which repeatedly removes the forbidden ``down'' columns and re-expresses sums over multisets supported there; Bruhat order enters through Proposition 4.4, which shows every partition produced by the lowering recursion lies in the interval below $\\lambda$.","core_discovery":"The central claim is Theorem 4.5: for $\\lambda \\in P^k_\\ell$, $$\\sum_{\\operatorname{supp}(S) \\subseteq [\\ell]_\\$\\lambda$} L_S $g^{{(k)}}$_\\$\\lambda$ = \\sum_{\\mu \\in P^k_\\ell,\\, w_\\mu \\le w_\\$\\lambda$} $g^{{(k)}}$_\\mu.$$ Here $L_S$ is a product of lowering operators $L_z$, each of which acts on the determinant representative $g_\\gamma$ by sending $\\gamma$ to $\\gamma-\\varepsilon_z$, and the support restriction $[\\ell]_\\lambda$ excludes the columns $\\operatorname{down}_{\\Delta_k(\\lambda)}(z)$ singled out by the root ideal $\\Delta_k(\\lambda)$. The right-hand side is the closed K-$k$-Schur function. The paper then derives Theorem 1.2, $\\widetilde{g}^{(k)}_\\lambda = (1 - G_1^\\perp)\\left(\\sum_{\\mu: w_\\mu \\le w_\\lambda} g^{(k)}_\\mu\\right)$, from this identity and the elementary identity $1 - G_1^\\perp = \\sum_{d \\ge 0} e_d^\\perp$, giving a purely combinatorial proof of the closed $k$-Schur Katalan conjecture that had recently been proved by another route.","pith_inferences":["The reverse-inclusion gap suggests the $\\Gamma$-recursion may be more generous than necessary; a smaller generating set of multiset supports might already saturate the Bruhat interval, which would simplify the algorithm.","Since the straightening propositions in Section 3 already operate on the generalized set $\\widetilde{P}^k_\\ell$, the same lowering-operator formula may hold for generalized K-k-Schur functions indexed by $\\widetilde{P}^k_\\ell$, not only by $k$-bounded partitions.","Interpreting the multiset sum as a path count would define a statistic on Bruhat intervals of $k$-bounded partitions, with the multiplicity of each $g^{(k)}_\\mu$ counting lowering-path ways to reach $\\mu$ from $\\lambda$.","The telescoping step in Corollary 3.21 relies only on the vanishing of high powers of lowering operators, so the argument may adapt to inhomogeneous variants where $(1-G_1^\\perp)$ is replaced by other alternating sums of $e_d^\\perp$."],"forward_implications":["The closed K-k-Schur function can be computed by iterating lowering operators on a single function, without summing over a Bruhat interval, making the K-homology representatives algorithmically accessible from root-ideal data.","Theorem 1.2 follows as an immediate corollary, so the formerly conjectural closed k-Schur Katalan formula now has a proof inside the same lowering-operator calculus that defines the functions.","The identity is manifestly positive as a sum over multisets, indicating that closed K-k-Schur functions carry a positive combinatorial model in terms of lowering paths.","The support condition $\\operatorname{supp}(S) \\subseteq [\\ell]_\\lambda$ gives a root-ideal characterization of which lowering columns generate the Bruhat interval, connecting affine Bruhat order to data attached to $\\Delta_k(\\lambda)$."],"supporting_citations":[{"why":"Introduces Catalan functions and the subset lowering operator, supplying the straightening lemmas and cover-based strong cover result used in Section 3.","marker":"[1]"},{"why":"Defines Katalan functions, the lowering operator $L_z$, the Mirror Lemmas, the closed k-Schur Katalan function, and the conjecture that the paper proves combinatorially.","marker":"[2]"},{"why":"Establishes the vanishing of high powers of lowering operators on generalized K-k-Schur functions, used in Corollary 3.21 to telescope remainder terms.","marker":"[3]"},{"why":"Proved the target theorem by a different method; this paper re-proves it combinatorially via Theorem 4.5.","marker":"[8]"},{"why":"Provides the bijection between $k$-bounded partitions and $k+1$-cores used to translate Bruhat order into core containment in Lemma 4.1.","marker":"[9]"},{"why":"Introduced K-k-Schur functions $g^{(k)}_\\lambda$ as K-theoretic Schubert representatives of the affine Grassmannian.","marker":"[10]"},{"why":"Defines Bruhat order on the affine symmetric group and the core-containment criterion used throughout Section 4.","marker":"[15]"},{"why":"Motivates closed K-k-Schur functions through $k$-rectangle factorization and provides the Pieri-type context they are expected to satisfy.","marker":"[21]"}],"fun_headline_variants":["Combinatorial proof of Katalan conjecture","Lowering operators prove Katalan conjecture","Closed K-k-Schur functions via lowering operators","New formula for closed K-k-Schur functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that every partition whose associated affine permutation lies weakly below $\\lambda$'s is actually reached by the recursive lowering process; only the reverse direction—that everything reached lies below—is proved, and the unproved direction is the one that identifies the generated set with the full Bruhat interval.","fun_headline_variants_meta":{"raw":{"variants":["Combinatorial proof of Katalan conjecture","Lowering operators prove Katalan conjecture","Closed K-k-Schur functions via lowering operators","New formula for closed K-k-Schur functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001007,"raw_usage":{"total_tokens":4241,"prompt_tokens":912,"completion_tokens":3329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":3271}},"tokens_in":528,"tokens_out":3329,"duration_ms":27639,"temperature":1.0,"reasoning_tokens":3271,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:37:10.435658+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a small explicit case, expand both sides of Theorem 4.5 in the Schur basis and compare coefficients; a mismatch falsifies the theorem. To target the weak point directly, enumerate the $\\Gamma$-recursion for a small $\\lambda$ and check that every $\\mu$ with $w_\\mu \\le w_\\lambda$ appears in the union of the recursively generated sets; any missing $\\mu$ refutes the reverse inclusion used at equations (96) and (98).","supporting_citations":[{"cited_title":"Blasiak, J","cited_arxiv_id":null,"evidence_quote":"Introduces Catalan functions and the subset lowering operator, supplying the straightening lemmas and cover-based strong cover result used in Section 3."},{"cited_title":"Blasiak, J","cited_arxiv_id":null,"evidence_quote":"Defines Katalan functions, the lowering operator $L_z$, the Mirror Lemmas, the closed k-Schur Katalan function, and the conjecture that the paper proves combinatorially."},{"cited_title":"Ikeda, S","cited_arxiv_id":null,"evidence_quote":"Proved the target theorem by a different method; this paper re-proves it combinatorially via Theorem 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bijection between $k$-bounded partitions and $k+1$-cores used to translate Bruhat order into core containment in Lemma 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced K-k-Schur functions $g^{(k)}_\\lambda$ as K-theoretic Schubert representatives of the affine Grassmannian."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines Bruhat order on the affine symmetric group and the core-containment criterion used throughout Section 4."},{"cited_title":"Takigiku, A Pieri formula and a factorization fomula for sums of K-theoretic k-Schur functions, Algebraic Combinatorics 2 (2019), 447-480","cited_arxiv_id":null,"evidence_quote":"Motivates closed K-k-Schur functions through $k$-rectangle factorization and provides the Pieri-type context they are expected to satisfy."}],"review_version":1}