{"id":"474dc9c7-750d-492d-8438-f2e468bc2196","arxiv_id":"2505.02941","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors explicitly realize the equivariant K-Peterson map for SL_n by a rational substitution and prove a factorization formula for K-theoretic double k-Schur functions.","lead":"This mathematics paper writes explicit formulas for a known isomorphism in quantum Schubert calculus, connecting the equivariant quantum K-theory ring of the flag variety SL_n/B with the equivariant K-homology of the affine Grassmannian. The formulas come from the relativistic Toda lattice and give a practical way to compute Schubert basis elements in both rings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central theorem credible; only the omitted i≠0 case in Proposition 4.1 needs filling.","rationale":"The central claim of the paper, Theorem 1.1, is supported by a coherent proof: the base case s0 is computed explicitly from the relativistic Toda substitution (1.3) and the β formula (3.8) from [8], and the extension to all x uses Demazure-operator intertwining. The only genuine gap in the written proof of Theorem 1.1 is the omitted i≠0 case of Proposition 4.1, exactly as the reader notes in the rationale; this is likely fillable. The reader's stated weakest assumption, the external identification from [8], is normal reliance on a cited theorem rather than an internal flaw. I also noticed a separate discrepancy in the proof of Theorem 1.2: applying (1.3) to ξ=ρ^∨+ϖ^∨_2 for n=4 gives Φ̃_4(Q^ξ)=σ_4^2/(σ_1 σ_2^2 σ_3), not σ_4^2/(σ_1 σ_2 σ_3) as stated in the proof; this affects the application and should be checked before publication, but it does not undermine Theorem 1.1. The reader's CONDITIONAL verdict remains appropriate.","tokens_in":37344,"tokens_out":55878,"duration_ms":497451,"concrete_test":"Complete the proof of Proposition 4.1 for i≠0: for k-small λ = μ plus a box of residue i∈{1,...,n-1}, verify σ(T_i g_μ) = e^{a_{r+1}-a_b} D_i g̃_μ by confirming that e(μ) is s_i-invariant and using σ∘T_i = T_i∘σ, then check the identity directly for n=3 with λ=(2) and λ=(1,1) against the determinantal formulas in Theorem 6.7.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.1 states σ(g^{(k)}_λ) = ∏_{x∈diag λ} e^{a_{r(x)+1}-a_{b(x)}} g̃^{(k)}_λ for k-small λ, but the proof only treats the case where λ is obtained from μ by adding a box of residue 0, leaving i≠0 to the reader. This is load-bearing: the special case (4.6) for ρ_l with l≥2 requires the i≠0 case, and (4.6) is used in (5.25) to compute Φ̃_n(Q^{-θ^∨}G^Q_{sθ}) = g̃_{s0}, the base case of Theorem 1.1. I checked that the i≠0 case goes through by the same mechanism (σ∘T_i = T_i∘σ for finite i, combined with s_i-invariance of the diagonal factor), so the gap is likely harmless, but it should be written out rather than left to the reader.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives an explicit algebraic realization of the T-equivariant K-Peterson isomorphism for SL_n(C). Concretely, for x = w t_ξ in the affine Grassmannian, Theorem 1.1 asserts that the map Φ̃_n = σ ∘ Φ_n, defined by the rational substitution (1.3) coming from the relativistic Toda lattice and the automorphism σ, sends the quantum double Grothendieck polynomial Q^ξ G^Q_w(z|η) to the closed K-theoretic double k-Schur function g̃^{(k)}_{xω_k}(y|b). The proof proceeds by proving the base case x = s_0 (Proposition 5.10), then extending via Demazure operators (Corollary 5.15 and Proposition 5.8). The paper also proves determinantal formulas for k-small partitions (Theorems 6.7 and 6.16), a k-rectangle factorization property (Theorem 7.2), and, as an application, a factorization formula for the maximal k-irreducible k-bounded partition ν_n (Theorem 1.2).","tokens_in":37501,"tokens_out":6860,"duration_ms":79459,"significance":"The result is significant if correct: it converts an abstract isomorphism theorem into an explicit, checkable formula that matches Schubert bases on the quantum and affine sides, with no free parameters. The determinantal formulas and the factorization theorem for ν_n are new and likely to be useful for further combinatorial and geometric applications. The paper is well structured and many technical steps are proved in detail, including the delicate base-case computation for s_0 and the Demazure-operator framework. It relies on substantial external inputs, notably the ring presentation of [24], the affine-side realization of [8], and Kato's abstract K-Peterson isomorphism, but these are clearly cited and are standard in the field. The main proof is credible, with one local but load-bearing gap that should be fixed before publication.","major_comments":[{"comment":"The proof of Proposition 4.1 treats only the case where the removable box has residue 0, and states that the case i ≠ 0 is 'similar and easier' and is left to the reader. This omission is load-bearing: equation (4.6), which is needed for ρ_l with l ≥ 2, requires the i ≠ 0 case, and (4.6) is used directly in the base-case proof of Theorem 1.1 in §5.5, specifically in the derivation of (5.25) for Φ̃_n(Q^{-θ^∨}G^Q_{sθ}). Please write out the full induction, including the required commutation σ ∘ T_i = T_i ∘ σ for finite i and the s_i-invariance of the diagonal factor e(μ) that appears during the induction step.","section":"§4.2, Proposition 4.1"}],"minor_comments":[{"comment":"The heading 'Relativistic Toda lattice and the centralizer famify' contains a typo: 'famify' should be 'family'.","section":"§2 heading"},{"comment":"In the organizational section, 'relativistec Toda lattice' should be 'relativistic Toda lattice'.","section":"§1.8"},{"comment":"In the List of symbols, 'mimimum-length coset representatives' should be 'minimum-length coset representatives'.","section":"List of symbols"},{"comment":"In Remark 5.1, 'reservey' should be 'reserve' in the sentence about the variables y_i.","section":"§5.1, Remark 5.1"},{"comment":"In the proof of Lemma 7.1, several cases are left to the reader; since this lemma is used in Theorem 7.2, please spell out the remaining cases in a sentence or two.","section":"§7, Lemma 7.1"},{"comment":"The sentence 'it is easy to see that Q-hat preserves the product of (5.13)' could be expanded with one clarifying sentence, namely that both factors are linear combinations of the standard monomials f^{(1)}_{p_1} ... f^{(n-1)}_{p_{n-1}}.","section":"§5.3, Proposition 5.6"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a good fit for a journal in algebraic combinatorics and representation theory. The central theorem is credible and the explicit formulas are likely to have lasting value. The identified gap in Proposition 4.1 is local and almost certainly repairable within the manuscript's scope, but it is genuinely load-bearing for Theorem 1.1 and must be fixed before the paper can be accepted. The reliance on [8] and [24] is explicit and appropriate. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the two things you need: the paper is good, and the one flagged gap is real but minor. Theorem 1.1 is a genuine explicit realization of the equivariant K-Peterson map for SL_n, not just a repackaging of Kato's abstract isomorphism. The proof is honest: the s0 base case is computed in detail, and the Demazure-operator bootstrap (Cor 5.15) is clean. The determinantal formulas (6.7, 6.16) and the k-rectangle factorization are new and useful.\n\nThe stress-test note is on target. Proposition 4.1 proves only the i=0 case of the induction and leaves i≠0 to the reader. That case is genuinely used: it produces (4.6), which appears in (5.25) for the base case. I checked it myself using σ∘T_i = T_i∘σ plus s_i-invariance of the diagonal factor—it works exactly as the note says. So this is a presentational gap, not a flaw. It should be written out in a revision, but it does not undermine the theorem.\n\nThe heavier dependency is [8, Thm 3.1]: the entire interpretation of the map as a Schubert-basis statement rests on their identification of β(z_ij/z_11) with functions g^(k) on the affine side. The paper cites it properly and doesn't re-derive it. That's normal practice, though it means the reader must trust [8]. I didn't see any circularity or parameter-fitting; the construction is rigid.\n\nIf I had to name a soft spot, it's the scope: SL_n only, and the even/odd split in Theorem 1.2 is a bit specialized. But that is what explicit formulas look like in this area. The paper is carefully organized, and the references are appropriate—self-citation is present but justified, since the cited results are the actual foundations from [6] and [8].\n\nBottom line: send it to a serious referee. The gap in Proposition 4.1 is a minor revision, not a rejection issue. For anyone working in equivariant quantum K-theory or affine Grassmannian Schubert calculus, this is a paper worth reading.","headline":"A serious, well-executed explicit realization of the equivariant K-Peterson map; the main theorem holds up, and the only flagged gap (Proposition 4.1) is minor and fillable.","tokens_in":38083,"tokens_out":2136,"would_cite":true,"duration_ms":23877,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N15","05E10","37K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the equivariant K-Peterson map for SL_n is realized explicitly by a rational substitution from the relativistic Toda lattice, matching Schubert classes on the quantum and affine sides.","keywords":["equivariant quantum K-theory","affine Grassmannian","Peterson isomorphism","relativistic Toda lattice","k-Schur functions","quantum double Grothendieck polynomials","K-theoretic double k-Schur functions"],"falsifier":"Compute both sides of (1.5) for a small non-$k$-small example, such as $n=3$ with $x=s_2s_1s_0$ (whose associated $2$-bounded partition is $(2,1)$, a non-$k$-small example), by evaluating the rational substitution $\\tilde\\Phi_n$ on $Q^\\xi G^Q_w(z|\\eta)$ and comparing the resulting series with $\\tilde g^{(k)}_{x\\omega_k}(y|b)$; any mismatch in the coefficients would disprove the theorem.","tokens_in":37135,"feed_emoji":"📐","tokens_out":16199,"duration_ms":147606,"temperature":0.7,"pith_summary":"This paper makes the 'quantum equals affine' correspondence explicit for the flag variety of SL_n in equivariant K-theory. It proves that the K-Peterson isomorphism, an abstract identification between the torus-equivariant quantum K-ring of SL_n(C)/B and the equivariant K-homology of the affine Grassmannian of SL_n, is implemented by a concrete rational substitution. The substitution expresses the quantum-ring generators as ratios of principal-minor tau functions attached to a centralizer family, and it sends quantum double Grothendieck polynomials to K-theoretic double k-Schur functions, so the Schubert bases on both sides match. The result turns a structural isomorphism, whose existence was proved by general machinery, into a formula that can be evaluated and used.","feed_headline":"K-Peterson map made explicit by the relativistic Toda lattice","feed_subtitle":"The equivariant K-Peterson map now has an explicit algebraic formula.","key_machinery":"The load-bearing object is the explicit K-Peterson map $\\tilde \\Phi_n = \\sigma \\circ \\Phi_n$, where $\\Phi_n$ is the $R(T)$-algebra map defined by (1.3): $z_i \\mapsto \\tau_i\\sigma_{i-1}/(\\sigma_i\\tau_{i-1})$ and $Q_i \\mapsto \\tau_{i-1}\\tau_{i+1}/\\tau_i^2$. The $\\tau_i$ and $\\sigma_i$ are the $i$-th principal-minor determinants of the centralizer matrices $ZAP$ and $ZP$, and the automorphism $\\sigma$ sends $h_i(y)$ to $1+h_1(y)+\\cdots+h_i(y)$, i.e., it shifts the first variable in the symmetric-function realization. This map is the mechanism because it intertwines the Demazure operators on both rings: $\\tilde \\Phi_n \\circ D^Q_i = D_{n-i} \\circ \\tilde \\Phi_n$ and $\\tilde \\Phi_n \\circ D^Q_0 = D_0 \\circ \\tilde \\Phi_n$, which promotes the direct check on $s_0$ to all Schubert classes. The affine-side functions $\\tilde g^{(k)}_x(y|b)$ are the K-theoretic double $k$-Schur functions, defined by $D_x(1)$, which represent the Schubert structure sheaves of the affine Grassmannian.","core_discovery":"The central discovery is Theorem 1.1: for every affine Grassmann element $x = w t_\\xi$ with $w$ in the finite Weyl group and $\\xi$ in the coroot lattice, the map $\\tilde \\Phi_n = \\sigma \\circ \\Phi_n$ built from the substitution $z_i \\mapsto \\tau_i\\sigma_{i-1}/(\\sigma_i\\tau_{i-1})$, $Q_i \\mapsto \\tau_{i-1}\\tau_{i+1}/\\tau_i^2$ sends $Q^\\xi G^Q_w(z|\\eta)$ to $\\tilde g^{(k)}_{x\\omega_k}(y|b)$. Here $G^Q_w$ is the quantum double Grothendieck polynomial representing the equivariant quantum Schubert class, $\\tilde g^{(k)}_x$ is the closed K-theoretic double $k$-Schur function representing the affine Schubert structure sheaf, and $\\omega_k$ is the $k$-conjugation involution on affine Grassmann elements. The paper proves this by checking the single class $x=s_0$ directly and then using Demazure operators $D^Q_0$ and $D^Q_i$ to propagate the identity to all $x$. It also derives determinantal formulas for $k$-small $k$-Schur functions (the case $\\lambda_1+\\ell(\\lambda)\\le n$) and, as an application, a factorization formula for the closed $k$-Schur function of the unique maximal $k$-irreducible $k$-bounded partition $\\nu_n$ (Theorem 1.2).","pith_inferences":["Editorial inference: the same tau-function substitution may yield an explicit K-Peterson map for other types only if the centralizer-family coordinates admit analogous principal-minor formulas; the present argument is specific to type A, while the abstract isomorphism is known for every semisimple group.","Editorial inference: because the map is explicit and algebraic, it can be tested numerically for small $n$ on non-$k$-small classes, and it suggests a recursive way to compute arbitrary affine Schubert classes by breaking them into $k$-rectangle pieces.","Editorial inference: the identity $\\sigma = \\mathrm{dToda}$ observed in Appendix A hints that one step of the discrete relativistic Toda lattice implements the automorphism $\\sigma$, so the Schubert-basis correspondence could be reinterpreted as a statement about successive Toda time evolution of tau functions."],"forward_implications":["Quantum double Grothendieck polynomials map to closed K-theoretic double $k$-Schur functions under $\\tilde \\Phi_n$, so every equivariant quantum Schubert class of the flag variety has a named symmetric-function representative on the affine Grassmannian.","The Demazure-operator intertwining means computations in the equivariant quantum K-ring can be transported to the affine Grassmannian K-homology ring, and vice versa.","The $k$-rectangle factorization $\\tilde g^{(k)}_{R_i\\cup\\lambda}(y|b) = \\tilde g^{(k)}_{R_i}(y|b)\\tilde g^{(k)}_\\lambda(y|\\omega^i b)$ reduces the study of arbitrary $k$-Schur functions to $k$-irreducible partitions.","For the maximal $k$-irreducible partition $\\nu_n$, the factorization formulas (1.6) and (1.7) express $\\tilde g^{(k)}_{\\nu_n}$ as a product of determinant-representable $k$-small factors, with an explicit rational prefactor in the even case.","The determinantal formulas for $k$-small partitions make those Schubert classes directly computable from the entries of the centralizer matrix."],"supporting_citations":[{"why":"Supplies the centralizer-family realization of $K_*^T(\\mathrm{Gr}_{\\mathrm{SL}_n})$ and the explicit coordinate formula (3.8) in terms of double $k$-Schur functions that the base-case proof invokes.","marker":"[8]"},{"why":"Gives the presentation of $QK_T(\\mathrm{SL}_n(\\mathbb{C})/B)$ by the conserved quantities $F_i(z,Q)-e_i$, the ring on which the substitution acts.","marker":"[24]"},{"why":"Provides the quantum double Grothendieck polynomials $G^Q_w(z|\\eta)$ that represent the equivariant quantum Schubert classes being matched.","marker":"[25]"},{"why":"Constructs the non-equivariant map $\\Phi_n$ from the relativistic Toda lattice whose equivariant extension with $\\sigma$ is the subject here.","marker":"[6]"},{"why":"Shows that the earlier non-equivariant map coincides with the abstract K-Peterson map up to the automorphism $\\sigma$.","marker":"[7]"},{"why":"Establishes the K-Peterson isomorphism for general semisimple groups, the result that this paper makes explicit in type A.","marker":"[10]"},{"why":"Defines the K-theoretic $k$-Schur functions and Demazure operators underlying the affine-side Schubert representatives.","marker":"[21]"},{"why":"Provides the length-additive decomposition of $w_{\\lambda\\cup R_i}$ used to prove the $k$-rectangle factorization and Theorem 1.2.","marker":"[28]"}],"fun_headline_variants":["Explicit K-Peterson map via rational substitution","Toda lattice yields explicit K-Peterson map","K-Peterson map now has explicit algebraic formula","Explicit K-Peterson map and k-Schur factorization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on a previously proved dictionary: the coordinate ring of the centralizer family (the space of upper-triangular matrices commuting with a fixed torus element) is identified, through an explicit formula, with the ring of K-theoretic double k-Schur functions that represent affine Grassmannian Schubert classes; if that dictionary were wrong, the claimed Schubert-basis matching would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Explicit K-Peterson map via rational substitution","Toda lattice yields explicit K-Peterson map","K-Peterson map now has explicit algebraic formula","Explicit K-Peterson map and k-Schur factorization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00089,"raw_usage":{"total_tokens":3971,"prompt_tokens":1208,"completion_tokens":2763,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":824,"completion_tokens_details":{"reasoning_tokens":2698}},"tokens_in":824,"tokens_out":2763,"duration_ms":23320,"temperature":1.0,"reasoning_tokens":2698,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:38:46.728267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of (1.5) for a small non-$k$-small example, such as $n=3$ with $x=s_2s_1s_0$ (whose associated $2$-bounded partition is $(2,1)$, a non-$k$-small example), by evaluating the rational substitution $\\tilde\\Phi_n$ on $Q^\\xi G^Q_w(z|\\eta)$ and comparing the resulting series with $\\tilde g^{(k)}_{x\\omega_k}(y|b)$; any mismatch in the coefficients would disprove the theorem.","supporting_citations":[{"cited_title":"Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions","cited_arxiv_id":"2408.10956","evidence_quote":"Supplies the centralizer-family realization of $K_*^T(\\mathrm{Gr}_{\\mathrm{SL}_n})$ and the explicit coordinate formula (3.8) in terms of double $k$-Schur functions that the base-case proof invokes."},{"cited_title":"Maeno, S","cited_arxiv_id":null,"evidence_quote":"Gives the presentation of $QK_T(\\mathrm{SL}_n(\\mathbb{C})/B)$ by the conserved quantities $F_i(z,Q)-e_i$, the ring on which the substitution acts."},{"cited_title":"Maeno, S","cited_arxiv_id":null,"evidence_quote":"Provides the quantum double Grothendieck polynomials $G^Q_w(z|\\eta)$ that represent the equivariant quantum Schubert classes being matched."},{"cited_title":"Ikeda, S","cited_arxiv_id":null,"evidence_quote":"Constructs the non-equivariant map $\\Phi_n$ from the relativistic Toda lattice whose equivariant extension with $\\sigma$ is the subject here."},{"cited_title":"Ikeda, S","cited_arxiv_id":null,"evidence_quote":"Shows that the earlier non-equivariant map coincides with the abstract K-Peterson map up to the automorphism $\\sigma$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the K-theoretic $k$-Schur functions and Demazure operators underlying the affine-side Schubert representatives."},{"cited_title":"Takigiku, A Pieri formula and a factorization formula for sums ofK-theoretick-Schur functions","cited_arxiv_id":null,"evidence_quote":"Provides the length-additive decomposition of $w_{\\lambda\\cup R_i}$ used to prove the $k$-rectangle factorization and Theorem 1.2."}],"review_version":1}