{"id":"f10dda23-5f82-44bd-bf1d-64c4d18a7b99","arxiv_id":"2507.16215","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The asymptotic character of a module in category O for truncated shifted Yangians equals the bar-character of its KLR module, giving a weak change-of-basis formula between the dual canonical and MV bases.","lead":"This paper defines a new invariant, the asymptotic character, for modules in category O of certain quantized algebras, and proves it computes equivariant multiplicities of characteristic cycles. In the case of truncated shifted Yangians, this yields a formula relating the dual canonical basis and the Mirkovic-Vilonen basis of the coordinate ring C[N].","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.38 proof uses an inconsistent partition of GT-weights: F' and F'' overlap, and lemma 5.44's singular statement is proved via the regular parametrization (5.41); as written the bridge between χ∞ and KLR bar-characters is unproven.","rationale":"The reader's weakest assumption was the external black-box equivalence of [Kam+19b]; my concern is internal to the proof of Theorem 5.38, which is the essential bridge between the asymptotic character invariant and the KLR bar-character. As printed, the three-way partition of GT-weights is not disjoint and lemma 5.44's proof contradicts its statement regarding regularity. This is load-bearing because Theorem 5.38 is used verbatim to prove the main Theorem 6.1 and hence the headline change-of-basis formula Theorem 2.13. The gap may be repairable, and the surrounding framework (Theorem 3.55, the geometric computations, the worked examples) is valuable and mostly independent, but the central claim as written is not established. A verdict of UNVERDICTED is appropriate until the partition and limit lemmas are corrected. I do not see evidence of fraud or deliberate circularity; the issue is a concrete proof gap in a long combinatorial argument.","tokens_in":51249,"tokens_out":12323,"duration_ms":123218,"concrete_test":"Recompute the decomposition of χ(M) in the proof of Theorem 5.38 for a concrete SL3 case with λ=ϖ1+ϖ2, µ=0, and a parameter set R whose minimum is smaller than some GT-weight entries (for instance R1={0}, R2={5}). Enumerate the regular and singular S in (Λν)_R, sort them by the paper's definitions of E(i), F', F'', and verify that no S is assigned to two subsets and that lemma 5.25 is only applied when ecyc e(S)=e(S). Then check lemma 5.44 for a singular S with a repeated longitude: equation (5.41) does not parametrize it, so the claimed limit has no basis. If the partition cannot be made disjoint and the lemmas re-proven, Theorem 5.38 is unproven.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central equality (Theorem 2.13, via Theorem 6.1) depends on Theorem 5.38, which identifies the asymptotic character with the KLR bar-character. Its proof partitions the set (Λν)_R of GT-weights into three disjoint subsets. As printed, E(i) is all regular weights in Λν(i)∩(Λν)_R, while F' = (Λ^<R)^∁ ∩ Λsing ∩ (Λν)_R and F'' = Λsing ∩ (Λν)_R. Thus F' ⊂ F'', and every singular S outside the cyclotomic range is counted twice in the displayed decomposition of χ(M), once with coefficient dim W_S and once with coefficient (1/σ(S)) dim W_S. Moreover, lemma 5.44 is stated for this singular F' but its proof invokes the bijection Qi of (5.41), which only parametrizes regular GT-weights by strictly increasing integers; for singular S it does not apply. If F' was intended to be regular, then E(i) and F' overlap on regular weights with max S ≥ min R, again contradicting the disjoint partition. Finally, the proof applies lemma 5.25 to S∈E(i), although the required idempotent identity ecyc e(S)=e(S) holds only for S∈Ecyc(i)=E(i)∩(Λν)^<R. These inconsistencies mean the three limit lemmas 5.43-5.45 do not establish the displayed formula for χ∞_d(M), so Theorem 5.38 and hence the main formula are not proven as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an asymptotic character map $\\chi^\\infty_d$ on category $\\mathcal{O}$ of filtered quantizations satisfying axioms H1--H11 and proves that it computes the equivariant multiplicity of the characteristic cycle (Theorem 3.55). It then verifies these hypotheses in detail for truncated shifted Yangians associated to simply-laced simple Lie algebras (Corollaries 4.23, 4.30, 4.41), and uses the equivalence of categories of [Kam+19b] to relate the asymptotic character to the KLR bar-character (Theorem 5.38). Assembling these results gives the main commutative diagram (Theorem 6.1), which is reformulated as the weak change-of-basis formula $D^-(d_L)=\\sum_Z n_{L,Z}D^-(b_Z)$ (Theorem 2.13), with $n_{L,Z}$ the multiplicities of MV cycles in the characteristic cycle. The paper also discusses injectivity of the characteristic cycle map, gives worked examples including Nakada's hook formula, and sketches consequences for a conjecture of Nakajima in the Kac--Moody setting.","tokens_in":51555,"tokens_out":14294,"duration_ms":158297,"significance":"If the main theorem is established, this is a valuable contribution: it provides a new, geometrically meaningful mechanism for comparing Lusztig's dual canonical basis with the Mirkovi\\'c--Vilonen basis of $\\mathbb{C}[N]$, and it predicts integrality and non-negativity of the change-of-basis coefficients through characteristic cycles. The abstract framework of Section 3 is clearly formulated and reusable, the hypotheses H1--H11 are checked explicitly for truncated shifted Yangians, and the paper is careful to flag sign and convention issues (Remarks 4.20, 4.32, 5.33). The derivation of the asymptotic character from equivariant Hilbert polynomials is largely independent of the KLR side, which speaks against circularity. However, the proof of Theorem 5.38, which is the bridge between $\\chi^\\infty_d$ and the KLR bar-character, contains serious gaps as written; since Theorem 6.1 and hence Theorem 2.13 depend on it, the main claim is not fully proven in the present form. The flaws appear local and repairable, so the appropriate response is a major revision rather than rejection.","major_comments":[{"comment":"The claimed partition of $(\\Lambda_\\nu)_R$ into three disjoint subsets is not disjoint as stated. The displayed definitions give $F'=(\\Lambda^{<R}_\\nu)^\\complement\\cap \\Lambda^{\\mathrm{sing}}_\\nu\\cap (\\Lambda_\\nu)_R$ and $F''=\\Lambda^{\\mathrm{sing}}_\\nu\\cap (\\Lambda_\\nu)_R$, so $F'\\subset F''$. Consequently, in the decomposition of $\\chi(M)$ in the proof of Theorem 5.38, every singular GT-weight outside the cyclotomic range is counted twice: once in the $F'$-term with coefficient $\\dim W_S(M)$ and once in the $F''$-term with coefficient $(1/\\sigma(S))\\dim W_S(M)$. If $F'$ was instead intended to consist of regular weights, then $F'$ overlaps $E(i)$ on the regular weights with $\\max S\\ge \\min R$. Thus the triple $(E(i),F',F'')$ is not a partition in either reading, and the limit computation that follows cannot be valid as written.","section":"§5.5 (definitions before Theorem 5.38)"},{"comment":"Lemma 5.44 is stated for the singular set $F'(k,j)_t$, but its proof invokes the bijection $Q_i$ of (5.41), which parametrizes regular GT-weights by strictly increasing integers. For a singular $S$ with repeated longitudes, the summation over $q_1<\\cdots<q_k<N$ does not enumerate the elements of $F'(k,j)_t$ bijectively, and the multiplicity factor $\\sigma(S)$ is not taken into account. Hence the conclusion of Lemma 5.44 is not established for the singular locus it is stated for. Relatedly, the second case in the proof of Lemma 5.45 applies Lemma 5.44 to the regular set $\\tilde S$ obtained by forgetting multiplicities, so the cited lemma does not actually cover that case. These gaps mean that Lemmas 5.43--5.45 do not, as written, prove the displayed formula for $\\chi^\\infty_d(M)$ used in Theorem 5.38.","section":"§5.5 (Lemma 5.44 and its proof)"},{"comment":"In the proof of Theorem 5.38, Lemma 5.25 is applied to every $S\\in E(i)$ to obtain $\\dim e(S)N=\\dim e(i)(e_{\\mathrm{cyc}}N)$. The required idempotent identity $e_{\\mathrm{cyc}}e(S)=e(S)$ holds only for $S\\in E_{\\mathrm{cyc}}(i)=E(i)\\cap(\\Lambda_\\nu)^{<R}$. For regular $S$ with $\\min R\\le \\max S<\\max R$, the right-hand side is zero while $e(S)$ need not annihilate $N$, so the rewriting of the first sum in $\\chi(M)$ is false. These regular outside-cyclotomic weights are not covered by Lemmas 5.44--5.45 (which are formulated for singular weights), so their contribution to $\\chi^\\infty_d(M)$ is not shown to vanish. A corrected proof must restrict the Lemma 5.25 step to $E_{\\mathrm{cyc}}(i)$ and handle $E(i)\\setminus E_{\\mathrm{cyc}}(i)$ separately, for instance by showing that its asymptotic contribution is of order $O(n^{d-1})$ or by including it in a correctly partitioned singular/regular outside-cyclotomic case.","section":"§5.5 (proof of Theorem 5.38)"}],"minor_comments":[{"comment":"The sentence introducing Definition 4.1 contains a duplicated phrase: \"It first appeard in first appeared in\" should be corrected to \"It first appeared in\".","section":"§4.1 (Definition 4.1)"},{"comment":"In the proof of Lemma 4.51, the set on which $|\\varpi^\\vee_{i_1,\\dots,i_N}(c_1,\\dots,c_N)|=r$ is the complement of a union of finitely many affine hyperplanes, not itself a union of affine hyperplanes; the wording should be adjusted to say that the desired set is the complement of a finite union.","section":"§4.7 (Lemma 4.51)"},{"comment":"The displayed denominator \"$\\rho_{i_1}!\\cdot\\!\\cdot\\!\\cdot\\!\\cdot\\!\\cdot\\rho_{i_q}!$\" appears to be a typesetting artifact and should read $\\rho_{i_1}!\\cdots\\rho_{i_q}!$.","section":"§5.5 (Lemma 5.45)"},{"comment":"The formula \"$\\lambda(S)=\\frac12(s_{i_1}\\alpha_{k+1}+\\cdots+s_{i_k}\\alpha_{i_d})+\\mu_r$\" in the proof of Lemma 5.44 appears to contain an indexing/notation error; it should refer to the longitudes of the $k$ left black strands in a way consistent with (5.42).","section":"§5.5 (Lemma 5.44 proof)"}],"recommendation":"major_revision","confidential_remarks":"The flaws identified in Section 5.5 are local and appear repairable: the partition of $(\\Lambda_\\nu)_R$ needs to be corrected, the singular cases in Lemmas 5.44--5.45 need a parametrization that accounts for multiplicities, and the application of Lemma 5.25 must be restricted to $E_{\\mathrm{cyc}}(i)$ with a separate argument for the remaining regular weights. I do not see evidence that the main framework is unsound; the paper's reliance on [Kam+19b] for the category equivalence is standard and clearly flagged. The manuscript is original and contains a substantial amount of correct and useful material, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the asymptotic character formalism is a real contribution, and the first half of the paper (sections 2–4) is careful and largely convincing. But the proof of Theorem 5.38, which is the bridge from χ∞ to KLR bar-characters and hence the main formula, has a genuine technical gap. As printed, the claimed partition of the GT-weights in the proof is not a partition: F′ and F″ overlap, and the singular non-cyclotomic weights are counted twice with different coefficients. Lemma 5.44 also invokes the regular parametrization (5.41) for singular sets of parameters, where it does not apply. And the application of Lemma 5.25 to all S∈E(i) requires ecyc e(S)=e(S), which only holds in the cyclotomic range. These are not cosmetic slips; they are load-bearing for the equality ς(χ∞d(M)) = ch(Θcyc(M)). So Theorem 5.38 and the main theorem are not proven as written.\n\nThe good news is that the core idea is worth taking seriously. The definition of the asymptotic character (Definition 3.47) and the theorem that it computes equivariant multiplicities of characteristic cycles (Theorem 3.55) are genuinely new and are proved from the hypotheses in a way that mostly checks out. The verification of H1–H11 for truncated shifted Yangians is careful, including the sign conventions which are flagged explicitly. The paper also does the right thing by stating clearly that the full change-of-basis conjecture remains open and positions Theorem 2.13 as evidence. The reliance on the Kamnitzer–Tingley–Webster–Weekes–Yacobi equivalence is a black box, but an honest one; that is deep prior work, not a defect of this paper.\n\nThe fixes for the proof might be local — redefine F″ as the singular cyclotomic weights, restrict the use of (5.41) to regular subsequences, and apply Lemma 5.25 only to Ecyc(i). But as of now, the central computation is not established, and a referee would need to verify those repairs carefully. I would send this to a serious referee, because the stakes and the novelty are high, but the revision should be substantial, not cosmetic.","headline":"The asymptotic character formalism is a real contribution, but the proof of Theorem 5.38 has a genuine, load-bearing gap in the partition of GT-weights.","tokens_in":52136,"tokens_out":6217,"would_cite":false,"duration_ms":57008,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B37","17B10","14M15","17B67"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves the D-image version of a conjectured non-negative change of basis between the dual canonical and Mirkovi\\'c–Vilonen bases of $\\mathbb{C}[N]$, with coefficients counting MV cycles in characteristic cycles of category…","keywords":["category O","truncated shifted Yangians","asymptotic characters","Mirkovic-Vilonen cycles","dual canonical basis","KLR algebras","characteristic cycles","equivariant multiplicities"],"falsifier":"Pick a concrete case not forced by minuscule symmetry, e.g. $\\mathfrak{g}=\\mathfrak{sl}_4$, $\\lambda=\\varpi_1+\\varpi_3$, $\\mu=0$, and an integral parameter set $R$ in general position. For each simple cyclotomic KLR module $L$, compute independently the KLR character side $D^-(d_L)$ and the geometric side: construct the corresponding category $\\mathcal{O}$ module, pass to its associated graded module, read off the multiplicities $n_{L,Z}$ of the top-dimensional components, and evaluate $\\sum_Z n_{L,Z}D^-(b_Z)$. Any difference between the two rational functions at a generic point of the Cartan would refute Theorem 2.13.","tokens_in":51002,"feed_emoji":"🧮","tokens_out":13745,"duration_ms":138725,"temperature":0.7,"pith_summary":"The paper's aim is to compare two well-known bases of the coordinate ring $\\mathbb{C}[N]$ of a unipotent group in the simply-laced case: the dual canonical basis, built from representation theory of KLR algebras, and the Mirkovi\\'c–Vilonen (MV) basis, built from geometry of the affine Grassmannian. The proposed comparison is that each dual canonical basis vector $d_L$ is a non-negative integer combination of MV basis vectors $b_Z$, with the coefficient $n_{L,Z}$ equal to the multiplicity of the cycle $Z$ in the characteristic cycle of a category $\\mathcal{O}$ module for a truncated shifted Yangian. The paper proves the weak form of this statement (Theorem 2.13): after applying the rational-function map $D^-$ to both sides, $D^-(d_L)=\\sum_Z n_{L,Z}D^-(b_Z)$. A new asymptotic character $\\chi^\\infty$ carries the argument, since it turns ordinary weight multiplicities into rational functions and, by Theorem 3.55, equals the equivariant multiplicity of the characteristic cycle, while an existing equivalence of categories identifies it with a KLR bar-character. If the main theorem is right, the conjectured basis change is not a formal coincidence: its coefficients are computable geometric invariants of a concrete module-support construction.","feed_headline":"Character limit derives the canonical-to-MV change of basis","feed_subtitle":"Weight multiplicities through a limit become cycle data, making the canonical-to-MV change explicit.","key_machinery":"The load-bearing construction is the asymptotic character map $\\chi^\\infty_d$, defined for a module $M$ of Gelfand–Kirillov dimension $d$ by $$\\chi^\\infty_d(M)(h)=\\lim_{n\\to\\infty}\\frac{1}{n^d}\\sum_\\mu \\dim M(\\mu)\\, $e^{{\\langle\\mu,h\\rangle/n}}$.$$ The limit is not a formal device: by Theorem 3.55 it computes the equivariant multiplicity of the characteristic cycle of $M$, namely the sum $\\sum_Z n_Z\\,\\varepsilon_T(Z)$ over the $d$-dimensional irreducible components $Z$ of the support of the associated graded module. Once the paper verifies that truncated shifted Yangians satisfy the required finiteness and integrality hypotheses, this same $\\chi^\\infty$ is rewritten, through the equivalence of Theorem 5.31, as the bar-character of a cyclotomic KLR module, yielding the main identity.","core_discovery":"On its own terms, the central discovery is an equality of rational functions that ties two previously separate computational languages. For $\\mathfrak{g}$ simple, simply-laced, with $\\lambda$ dominant and $\\mu$ a weight of $V(\\lambda)$, every simple module $L$ over the cyclotomic KLR algebra $R^\\lambda_{\\lambda-\\mu}$ gives a module in category $\\mathcal{O}$ of the truncated shifted Yangian $Y^\\lambda_\\mu(R)$. The theorem asserts $$D^-(d_L)=\\sum_Z n_{L,Z}D^-(b_Z),$$ where $d_L$ is the dual canonical basis vector attached to $L$, $b_Z$ is the MV basis vector attached to a stable MV cycle $Z$, and $n_{L,Z}\\in\\mathbb{Z}_{\\geq 0}$ is the multiplicity of $Z$ in the top-dimensional characteristic cycle of that module. The equality is established through a commutative diagram whose three arrows are the characteristic cycle map, the category equivalence to parity KLRW modules followed by the cyclotomic idempotent, and the asymptotic character map; the latter is shown to compute equivariant multiplicities of supports. The paper therefore treats Conjecture 2.15, the unshadowed basis change $d_L=\\sum_Z n_{L,Z}b_Z$, as a strictly stronger statement that its methods support but do not reach.","pith_inferences":["The paper only proves the $D^-$-shadow of the conjectured basis change; a natural next step that the paper does not take is to test whether the same coefficients $n_{L,Z}$ satisfy the stronger identity $d_L=\\sum_Z n_{L,Z}b_Z$ inside $\\mathbb{C}[N]$, which would fully resolve Conjecture 2.15.","Because Section 3's asymptotic character is developed for any filtered quantization satisfying hypotheses H1–H9, the same limit construction could be applied to other quantizations with a Hamiltonian torus action; the paper's checks for truncated shifted Yangians suggest a general recipe for turning characters into equivariant multiplicities.","The equivalence used is simply-laced; if the corresponding B-integrality and category equivalence were extended to non-simply-laced parameters, the main identity would presumably extend to the symmetrized Coulomb-branch setting, where the paper can currently only give evidence."],"forward_implications":["If Theorem 2.13 holds, the conjectured change of basis from the dual canonical basis to the MV basis has the explicit, computable coefficients $n_{L,Z}$, and these coefficients are non-negative integers because they are cycle multiplicities.","The asymptotic character map $\\chi^\\infty_d$ becomes a general invariant of filtered quantizations: on category $\\mathcal{O}$ it is a group homomorphism from $K_0(\\mathcal{O}_{\\leq d})$ to rational functions that kills objects of lower GK dimension and equals the equivariant multiplicity of the characteristic cycle (Theorem 3.55).","In the minuscule case, the paper's setup recovers Nakada's colored hook formula and the Peterson–Proctor hook formula as special cases, showing that the new formalism reproduces known combinatorial identities (Example 2.16).","When $\\lambda$ is a sum of minuscule coweights and the parameters avoid finitely many affine hyperplanes, the characteristic cycle map is an isomorphism between the top category $\\mathcal{O}$ and the top Borel–Moore homology of the repelling set (Corollary 4.52).","In the symmetric Kac–Moody setting, the same commutative diagram shows that the top homology of the repelling set of the Coulomb branch is non-zero for every weight of $V(\\lambda)$ (Corollary 6.8), evidence toward the conjectured Kac–Moody geometric Satake."],"supporting_citations":[{"why":"Supplies the equivalence of categories $\\Theta:\\mathcal{O}^\\lambda_\\mu(R)\\to P^R_\\mu\\text{-mod}$ and the parametrization of maximal ideals by the product monomial crystal, used to identify asymptotic characters with KLR bar-characters.","marker":"[Kam+19b]"},{"why":"Provides the D map, the equality $D^-(b_Z)$ equals the equivariant multiplicity at the bottom fixed point of an MV cycle, and the biperfect-basis framework that the main identity compares.","marker":"[BKK21]"},{"why":"Proves the T-equivariant isomorphism between the repelling set $(W^\\lambda_\\mu)^-$ and $\\mathrm{Gr}^\\lambda\\cap S^\\mu_-$, which identifies irreducible components of the repelling set with MV cycles.","marker":"[Kry18]"},{"why":"Establishes Proposition 9.19, that the cyclotomic idempotent functor induces an equivalence on the top GK subcategory, allowing passage to simple KLR modules in the main diagram.","marker":"[Kam+24]"},{"why":"Introduces KLRW algebras, their steadied quotients, and the cyclotomic idempotent isomorphism used to define the functor $\\Theta_{\\mathrm{cyc}}$.","marker":"[Web17]"},{"why":"Gives the KLR algebra categorification of the $U(\\mathfrak{n})$-module $\\mathbb{C}[N]$ and connects simple KLR modules with the dual canonical basis.","marker":"[KL09; Rou08; KL11]"},{"why":"Categorifies the irreducible representation $V(\\lambda)$ by cyclotomic KLR modules and shows that the pullback functor categorifies the map $\\psi_\\lambda$.","marker":"[KK12]"},{"why":"Supplies the geometric Satake isomorphism identifying weight spaces with top Borel–Moore homology of MV cycles, the geometric basis $b_2$ used on the right-hand side of the main formula.","marker":"[MV07]"}],"fun_headline_variants":["Character limit links canonical and MV bases explicitly","Asymptotic character map computes equivariant multiplicities","Basis change via characteristic cycles: non-negative integers","Category O character map explains MV cycle multiplicities","New formula: asymptotic characters give KLR module coefficients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's chain of reasoning rests on a previously proved equivalence between category $\\mathcal{O}$ for truncated shifted Yangians and modules over parity KLRW algebras, assumed as a black box for every integral set of parameters; if that equivalence fails in any needed case, the identification of $\\chi^\\infty$ with the KLR bar-character, and with it the main formula, would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Character limit links canonical and MV bases explicitly","Asymptotic character map computes equivariant multiplicities","Basis change via characteristic cycles: non-negative integers","Category O character map explains MV cycle multiplicities","New formula: asymptotic characters give KLR module coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000478,"raw_usage":{"total_tokens":2396,"prompt_tokens":1004,"completion_tokens":1392,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":1329}},"tokens_in":620,"tokens_out":1392,"duration_ms":12498,"temperature":1.0,"reasoning_tokens":1329,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:15:49.331857+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a concrete case not forced by minuscule symmetry, e.g. $\\mathfrak{g}=\\mathfrak{sl}_4$, $\\lambda=\\varpi_1+\\varpi_3$, $\\mu=0$, and an integral parameter set $R$ in general position. For each simple cyclotomic KLR module $L$, compute independently the KLR character side $D^-(d_L)$ and the geometric side: construct the corresponding category $\\mathcal{O}$ module, pass to its associated graded module, read off the multiplicities $n_{L,Z}$ of the top-dimensional components, and evaluate $\\sum_Z n_{L,Z}D^-(b_Z)$. Any difference between the two rational functions at a generic point of the Cartan would refute Theorem 2.13.","supporting_citations":[],"review_version":1}