{"id":"c7c7a33e-56a4-40c6-89d4-0c7f43bd5c14","arxiv_id":"2608.09197","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In affine Hecke categories, high tensor powers of a fixed object have a number of indecomposable summands of order n^{-|Phi^+|/2} times an exponential; proved in type A1, and for longest elements in type A2, with coarse bounds in all affine types.","lead":"This paper proves how the number of summands in repeated tensor powers of Kazhdan-Lusztig basis elements grows in affine Hecke algebras. It derives exact asymptotics in affine type A1, sharp growth rates for natural families in affine type A2, and general bounds in all affine types.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7A.4 Case 3 contains a false identity, so the proof of Lemma 5D.13 — the key input for Theorem 5D.14(b) — is invalid as written.","rationale":"The reader identified Lemma 5D.13 as the weakest assumption, and I agree that the central claim Theorem 5D.14(b) depends on it. My stress-test goes further: the proof of Lemma 5D.13 itself contains a concrete false equality in Lemma 7A.4, Case 3. The contradiction is not about missing machine verification but about the validity of the displayed computation: the left side has a standard-basis term of length 13, while the right side is supported in lengths at most 12. This is a mathematical error in the argument as written. I am not claiming Lemma 5D.13 is false; the uniform bound may well be true and fixable with a correct case analysis. But the paper's present proof is incomplete at the exact point that upgrades the coarse n^{-3} bound to the sharp n^{-3/2} bound for one-sided longest elements. Since the rest of the paper, including the general bounds, the affine A1 analysis, and the two-sided longest case, appears sound, a conditional acceptance — pending independent verification or correction of Lemma 5D.13 — is the appropriate verdict. The proposed concrete test settles the matter by direct computation: it first falsifies the printed identity and then checks whether the needed uniform bound survives.","tokens_in":28044,"tokens_out":38340,"duration_ms":360284,"concrete_test":"Use the accompanying code [COT26] or any SageMath implementation of the affine A2 Hecke algebra at v=1 to compute the two sides of the displayed identity in Lemma 7A.4 Case 3 for theta(2,0)=123123212, u=3, w0=121. Compare their standard-basis expansions: the left side contains delta_{1231232123121} with coefficient 1, while the right side has no standard term of length 13, so the identity fails. Then, independently of that identity, verify Lemma 5D.13 by computing nu(b_x b_w0) for all beyond-the-wall elements x up to length, say, 30; if a counterexample to the bound 12 appears, Theorem 5D.14(b) needs revision, and if no counterexample appears, a corrected expansion for Case 3 must be supplied before the proof is complete.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 5D.14(b) obtains the sharp n^{-3/2} lower bound by sandwiching nu(b_w^n) against nu(e b_w^n) via the uniform estimate nu(b_x b_w0) <= 12 (Lemma 5D.13). That lemma is proved in Section 7. In Lemma 7A.4, Case 3, the proof uses the displayed identity b_{theta(m,n)u} b_{w0} = b_{theta(m,n)} b_{s_{2m-2n}} b_{s_{2m-2n+1}} (b_t b_s - 1). This identity is false as written. Take m=2, n=0, so theta(2,0)=123123212, which ends in st=12; here u=3 and w0=121 with s=1, t=2. The word theta(2,0)*3*121 = 1231232123121 is reduced (braid-triplet distances are 1 and 3, both odd by Lemma 5D.9), so the left side contains the standard-basis element delta_{theta*3*121} of length 13 with coefficient 1. The right side is b_theta times an element of the finite Hecke algebra of W_f={1,2}, supported in lengths at most 3; every standard term on the right therefore has length at most ell(theta)+3 = 12. Hence the coefficient of delta_{1231232123121} is 1 on the left and 0 on the right. Since the bound nu(b_x b_w0) <= 12 for exactly this class of beyond-the-wall elements is what Lemma 5D.13 must provide, the proof of the uniform bound collapses at this case. Consequently the headline one-sided-longest result in affine A2, Theta(n^{-3/2} beta^n), is not fully proved in the text as it stands.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops asymptotic growth bounds for tensor powers in affine Hecke categories, formulated as growth of the coefficient sum ν(b_w^n) for Kazhdan–Lusztig basis elements in affine Hecke algebras. In arbitrary affine type the authors prove a general lower bound C n^{-|Φ^+|} β^n ≤ ν(b_w^n) ≤ β^n for elements in the projective cell (Theorem 3C.1), and use the Satake isomorphism to obtain sharp asymptotics for two-sided longest elements (Theorem 3D.4). In affine type A1 they determine ν(b_w^n) explicitly up to the scalar prefactor (Theorem 4B.1). In affine type A2 they establish the expected n^{-3/2} polynomial correction for two-sided longest elements and, via the uniform beyond-the-wall estimate ν(b_x b_{w0}) ≤ 12, the weaker Θ(n^{-3/2} β^n) form for one-sided longest elements (Theorem 5D.14). A computational section on affine type C2 is explicitly labeled as experimental.","tokens_in":28444,"tokens_out":8703,"duration_ms":78367,"significance":"If the proofs are correct, the paper makes an important contribution by establishing the expected n^{-|Φ^+|/2} polynomial correction in a non-spherical affine Hecke setting, going beyond the directly representation-theoretic cases. The general framework of Section 3C, the Satake reduction, and the explicit A1 computations are clean and well motivated. The paper also ships reproducible code and distinguishes clearly between proved results and experimental observations, especially in Section 6. The decisive external input [CEO24] is published and independent of the paper's own claims, so the overall strategy is not circular. The main obstruction is not the architecture of the proof but a specific technical identity in the key uniform-bound lemma, which is load-bearing for Theorem 5D.14(b).","major_comments":[{"comment":"The displayed identity b_{θ(m,n)u} b_{w0} = b_{θ(m,n)} b_{s_{2m-2n}} b_{s_{2m-2n+1}} (b_t b_s − 1) is false. For m=2, n=0 one has θ(2,0)=123123212, which ends in st=12, and u=3. The word θ(2,0)3 121 = 1231232123121 is reduced: the braid-triplet distances are 1 and 3, both odd by Lemma 5D.9. Hence the left side contains the standard-basis element δ_{1231232123121} with coefficient 1 (length 13). On the right, the factor b_s b_t (b_t b_s − 1) lies in the finite Hecke algebra of {1,2}, whose standard-basis support has length at most 3; so every standard term on the right has length at most ℓ(θ)+3 = 12. The coefficient of that length-13 element is therefore 1 on the left and 0 on the right. Since this is precisely the case that must establish the uniform bound for beyond-the-wall elements ending in st, the proof of Lemma 5D.13 is incomplete, and Theorem 5D.14(b) is not proved as written.","section":"§7, Lemma 7A.4, Case 3"}],"minor_comments":[{"comment":"The sentence “in general, ν(b_x)ν(b_y) = 1” is confusing because ν(b_x) is not generally 1; this phrasing should be clarified or removed.","section":"§2B, Definition 2B.1"},{"comment":"The numerical statement “|a_{n+2} − a_{n+1}| < |a_{n+1} − a_n| < 0.0001 for n∈{20,...,30}” should specify precisely how the normalized sequence a_n is indexed and which values were checked; as written it is hard to verify from the displayed data.","section":"§5B"},{"comment":"The quantity d_k is used in the table of values of d, but it is only defined inside the proof; defining d_k in the statement would improve readability.","section":"Lemma 3B.9"},{"comment":"The reference [Tub22] appears in the text as 2022 but the bibliography lists 2024; please make the year consistent.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The false identity in §7 is the single blocking issue for the paper's main type-A2 theorem. If the authors can supply a correct proof of Lemma 5D.13, or replace the manual case analysis with a machine-checkable computation, the paper would be a strong candidate for acceptance. The reliance on [CEO24] is legitimate and not circular, and Section 6 is clearly experimental, which is appropriate. I would encourage the editor to treat the requested revision as focused on the uniform-bound lemma rather than on the overall framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper, but the sharp one-sided-longest theorem in affine A2 is not proved as written. The load-bearing uniform bound Lemma 5D.13 rests on Lemma 7A.4, and Lemma 7A.4 Case 3 uses an identity that is simply false.\n\nThe concrete failure: in Case 3, the proof asserts b_{θ(m,n)u}b_{w0} = b_{θ(m,n)}b_{s_{2m-2n}}b_{s_{2m-2n+1}}(b_t b_s − 1). Take m=2, n=0. Then θ=123123212, u=3, s=1, t=2, w0=121. The left side contains δ_{1231232123121} with coefficient 1; that word is reduced (all braid-triplet distances are odd), so it is a standard basis element of length 13. The right side is b_θ times an element of the finite Hecke algebra of {1,2}, which has support only in lengths at most 3. So every standard term on the right has length at most 12. The identity cannot hold. Since this is exactly the case needed to bound ν(b_x b_{w0}) for the elements x = θ(m,n)u, the proof of Lemma 5D.13 collapses. And Lemma 5D.13 is the only input that upgrades the coarse n^{-1} lower bound to the sharp n^{-3/2} for one-sided longest elements.\n\nThat is disappointing, because the rest of the paper is genuinely good. Theorem 3C.1's general bounds in arbitrary affine type look clean and are new; the affine A1 asymptotics are explicit and correct; the two-sided longest case in A2 follows directly from Satake plus [CEO24], with no hand-checked cases; and the type C2 section is honestly flagged as experimental/conditional. The writing is careful about what is proved and what is not, and the code is referenced.\n\nThe soft spot is not a minor typo: it is load-bearing for Theorem 5D.14(b). The result might well be true, but as it stands the Θ(n^{-3/2}) lower bound is unsupported. The A1 and general-bound parts are unaffected, and the reliance on [CEO24] (with overlapping authorship) is not a problem because that theorem is published and independent of the present claims.\n\nRecommendation: do send this to peer review. The ideas and the correct parts are valuable, and a referee can reasonably ask for a repaired or replaced proof of the uniform bound. The paper deserves a serious referee, but the A2 one-sided theorem should not be accepted without a correct proof.","headline":"A genuinely new paper on growth in affine Hecke categories, but the headline A2 one-sided-longest theorem rests on a provably false identity in Lemma 7A.4.","tokens_in":28987,"tokens_out":17470,"would_cite":true,"duration_ms":143998,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18M05","41A60","05A16","20C08"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that tensor powers in affine Hecke categories grow exponentially with a polynomial correction fixed by the root system: $n^{-1/2}$ in affine type $A_1$ and $n^{-3/2}$ in affine type $A_2$ for longest elements.","keywords":["affine Hecke categories","Kazhdan-Lusztig basis","Soergel bimodules","tensor powers","asymptotic growth","Satake isomorphism","affine Weyl groups","scalar-poly-exp"],"falsifier":"Directly compute $\\nu(b_x b_{w_0})$ for the infinite beyond-the-wall family $x=\\theta(m,n)$ in affine type $A_2$ for, say, $m+n$ up to $50$ with a computer algebra system; the paper's Section 7 proof is a manual case check with no machine verification, so a single value above $12$ would falsify Lemma 5D.13 and hence the sharp $n^{-3/2}$ lower bound. A milder check is to compute the first few dozen terms of $\\nu(b_{123}^n)$ from the KL multiplication rules and test whether the normalized sequence $\\nu(b_{123}^n)/(n^{-3/2}8^n)$ stays within two positive constants; a different polynomial exponent would falsify the claimed growth law.","tokens_in":27852,"feed_emoji":"🧮","tokens_out":7363,"duration_ms":71529,"temperature":0.7,"pith_summary":"This paper asks how many indecomposable summands appear in the $n$-th tensor power of an object in an affine Hecke category, equivalently in the $n$-th power of a Kazhdan--Lusztig basis element of an affine Hecke algebra. The authors establish that in arbitrary affine type, for elements in the projective cell, the number of summands is at least a constant times $n^{-|\\Phi^+|}\\beta^n$ and at most $\\beta^n$. In affine type $A_1$, every non-finite element satisfies the sharp asymptotic $C_w n^{-1/2}\\beta^n$, and in affine type $A_2$, two-sided longest elements satisfy $C_w n^{-3/2}\\beta^n$ while one-sided longest elements lie in $\\Theta(n^{-3/2}\\beta^n)$. The paper thereby proves the expected root-system correction $n^{-|\\Phi^+|/2}$ beyond the directly representation-theoretic spherical case, for the first time in rank two. A reader should care because this pins down a structural pattern: the polynomial correction to exponential tensor growth is dictated by the ambient root system, not by the particular element chosen.","feed_headline":"Root-system growth law proven for affine Hecke powers","feed_subtitle":"In affine A1 powers split like n^{-1/2}; in affine A2 longest elements give Θ(n^{-3/2}·8^n).","key_machinery":"The load-bearing tool is the combinatorial Satake isomorphism, which identifies the spherical subalgebra of the affine Hecke algebra with the Grothendieck ring of finite-dimensional representations of the adjoint group. Under this isomorphism, two-sided longest basis elements $b_w$ become representations $V_w$, and powers $b_w^n$ become ordinary tensor powers $V_w^{\\otimes n}$, which have known asymptotics from tensor-category methods. For one-sided longest elements, the paper sandwiches $b_w^n$ between its spherical projection $(eb_w)^n$ and a bounded multiple of it, using a uniform bound on $\\nu(b_x b_{w_0})$ for beyond-the-wall elements $x$ in type $A_2$. The projective cell (the lowest two-sided cell) is described by inequalities on root pairings and by finite left-right corrections around spherical elements, and explicit local multiplication rules in affine type $A_2$ carry the reduction from the cyclic wall to the projective cell. Weyl's dimension formula converts length growth along a ray into the polynomial degree $|\\Phi^+|$.","core_discovery":"For a fixed element $w$ of an affine Weyl group, write $b_w^n = \\sum_x a_x(n) b_x$ in the Kazhdan--Lusztig basis and let $\\nu(b_w^n)=\\sum_x a_x(n)$ count summands with multiplicity; the exponential scale is $\\beta=\\nu_\\delta(b_w)$, the sum of standard-basis coefficients. The paper's central claim is that this count grows like a constant times $\\beta^n$ times a polynomial correction controlled by the positive root system: in affine type $A_1$, $\\nu(b_w^n)\\sim C_w n^{-1/2}\\beta^n$ for every non-finite $w$; in affine type $A_2$, two-sided longest elements satisfy $\\nu(b_w^n)\\sim C_w n^{-3/2}\\beta^n$, and one-sided longest elements satisfy $c_1(w)n^{-3/2}\\beta^n\\le \\nu(b_w^n)\\le c_2(w)n^{-3/2}\\beta^n$ for large $n$. More coarsely, in arbitrary affine type every element $w$ in the projective cell obeys $C n^{-|\\Phi^+|}\\beta^n\\le \\nu(b_w^n)\\le \\beta^n$. The main discovery is that the exponent $-|\\Phi^+|/2$ — already known for tensor powers of faithful representations of complex reductive groups — survives the passage to affine Hecke categories for the longest classes of elements, despite the non-semisimple and infinite setting.","pith_inferences":["The numerical evidence in affine type $C_2$ — $\\nu(b_{312}^n)\\sim C n^{-2}8^n$ — suggests the expected exponent $|\\Phi^+|/2=2$ holds there too, but the paper's methods would need a uniform beyond-the-wall bound analogous to the twelve-summand bound of affine $A_2$.","Because the one-sided longest argument uses only the uniform bound $\\nu(b_x b_{w_0})\\le C$, the $n^{-|\\Phi^+|/2}$ result should extend to any affine type where such a bound can be verified, plausibly all rank-two affine Weyl groups.","The Perron--Frobenius block-triangular mechanism is general: for wall elements, the projective cell is the unique final class, so the wall contribution is exponentially negligible; confirming this in other affine types would settle the stronger form of the growth question."],"forward_implications":["For two-sided longest elements in affine type $A_2$, $\\nu(b_w^n)\\sim C_w n^{-3/2}\\beta^n$, so the growth problem reduces exactly to tensor powers of $PGL_3$-representations.","For one-sided longest elements in affine type $A_2$, the weaker sharp form $\\nu(b_w^n)\\in\\Theta(n^{-3/2}\\beta^n)$ holds, extending the expected $n^{-|\\Phi^+|/2}$ correction beyond the spherical case.","In any affine type, every element in the projective cell has $C n^{-|\\Phi^+|}\\beta^n\\le\\nu(b_w^n)\\le\\beta^n$, giving an affirmative answer to the exponential part of the growth question and bounding the possible polynomial loss.","For cyclic on-the-wall elements in affine type $A_2$, the wall's own summands are asymptotically negligible; the projective cell governs the growth, so the full asymptotic is recovered from the projective part alone."],"supporting_citations":[{"why":"Introduces the Kazhdan--Lusztig basis and cell preorders on which the growth quantity $\\nu(b_w^n)$ is defined.","marker":"[KL79]"},{"why":"Supplies the combinatorial Satake isomorphism identifying the spherical Hecke algebra with the representation ring of the adjoint group.","marker":"[Kno05]"},{"why":"Provides the asymptotic theorem on tensor powers in symmetric tensor categories used to get $C_w n^{-|\\Phi^+|/2}\\beta^n$ for both-sided and one-sided longest elements.","marker":"[CEO24]"},{"why":"Gives the explicit multiplication rules and cell descriptions in affine type $A_2$ that drive the wall-negligibility and the uniform bound behind the sharp one-sided estimate.","marker":"[LP23]"},{"why":"Describes the lowest two-sided cell via root-pairing inequalities, used to show regular rays eventually lie in the projective cell.","marker":"[Shi87]"},{"why":"Provides the finite left-right correction description of the projective cell around spherical elements, used in the length and dimension comparisons.","marker":"[LX88]"},{"why":"Continues the description of the lowest two-sided cell and its based ring, supporting the projective-cell geometry used in the general bounds.","marker":"[Xi94]"}],"fun_headline_variants":["Affine Hecke growth exponent: half the root count","n^{-1/2} growth in A1, n^{-3/2} in A2 for longest","Root size determines Hecke power growth law","Exact growth for longest affine Hecke powers","Kazhdan-Lusztig powers scale with root system"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The sharp $n^{-3/2}$ lower bound for one-sided longest elements rests entirely on the hand-verified claim that multiplying any beyond-the-wall Kazhdan--Lusztig basis element by the longest finite element produces at most twelve summands; if one of the cases in the Section 7 analysis is wrong, the lower bound would degrade.","fun_headline_variants_meta":{"raw":{"variants":["Affine Hecke growth exponent: half the root count","n^{-1/2} growth in A1, n^{-3/2} in A2 for longest","Root size determines Hecke power growth law","Exact growth for longest affine Hecke powers","Kazhdan-Lusztig powers scale with root system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000968,"raw_usage":{"total_tokens":4097,"prompt_tokens":901,"completion_tokens":3196,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":3108}},"tokens_in":517,"tokens_out":3196,"duration_ms":22945,"temperature":1.0,"reasoning_tokens":3108,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:36:00.016427+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute $\\nu(b_x b_{w_0})$ for the infinite beyond-the-wall family $x=\\theta(m,n)$ in affine type $A_2$ for, say, $m+n$ up to $50$ with a computer algebra system; the paper's Section 7 proof is a manual case check with no machine verification, so a single value above $12$ would falsify Lemma 5D.13 and hence the sharp $n^{-3/2}$ lower bound. A milder check is to compute the first few dozen terms of $\\nu(b_{123}^n)$ from the KL multiplication rules and test whether the normalized sequence $\\nu(b_{123}^n)/(n^{-3/2}8^n)$ stays within two positive constants; a different polynomial exponent would falsify the claimed growth law.","supporting_citations":[],"review_version":1}