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Growth in affine Hecke categories

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.09197 v1 pith:BJIUWGYR submitted 2026-08-10 math.RT math.CT

Kevin Coulembier , Jensen O'Sullivan , Daniel Tubbenhauer This is my paper · ORCID
classification math.RTmath.CT MSC 18M0541A6005A1620C08
keywords affineHeckecategoriesKazhdan-LusztigbasisSoergelbimodulestensorpowersasymptoticgrowthSatakeisomorphismWeylgroupsscalar-poly-exp
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $|\Phi^+|$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

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.

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 (1)
  1. [§7, Lemma 7A.4, Case 3] 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.
minor comments (4)
  1. [§2B, Definition 2B.1] 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.
  2. [§5B] 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.
  3. [Lemma 3B.9] 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.
  4. [References] The reference [Tub22] appears in the text as 2022 but the bibliography lists 2024; please make the year consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new asymptotics are proved from external multiplication rules and a published tensor-power theorem, not from the target formula.

full rationale

The derivation chain for the headline result, Theorem 5D.14(b), consists of three independent inputs. The coarse lower bound comes from Theorem 3C.1, proved in Section 3C from the projective-cell growth bound Lemma 3B.9, Shi's cell description, and Weyl's dimension formula. The upper bound comes from the Satake comparison in Theorem 3D.1 and Theorem 3D.4, invoking the published tensor-power theorem [CEO24, Theorem 2.5]; that theorem concerns symmetric tensor categories and representation categories, not affine Hecke categories, and its proof is not derived from the present claims, so it is external evidence rather than a circular input. The sharpened lower bound for one-sided longest elements in affine type A2 uses Lemma 5D.13, the uniform beyond-the-wall estimate nu(b_x b_w0) <= 12. Lemma 5D.13 is proved in Section 7 by direct bookkeeping from the multiplication rules of Libedinsky-Patimo [LP23], with no asymptotic scaling assumed. None of these inputs is defined in terms of nu(b_w^n) or its n^{-3/2} beta^n asymptotics. The scalar-poly-exp Ansatz is explicitly stated as a question in Question 1B.3, tested numerically in Section 5B, and the numerical data are labeled as evidence, not as proof. The skeptical objection concerning Lemma 7A.4 Case 3 is a purported false identity in the manual case analysis; if valid, that is a mathematical correctness issue, not circularity, because the claimed bound does not reduce to the theorem it is used to prove. Accordingly, no load-bearing step in the paper reduces to its own output by construction, and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central claims rest on standard Kazhdan-Lusztig/cell theory, the Satake isomorphism, and the external tensor-power theorem [CEO24]; no new entities or fitted parameters are introduced. The only fitted number in the text is an illustrative estimate of a constant in an example.

free parameters (1)
  • C (Coxeter element scalar estimate) = approximately 0.14 to 0.15
    In Example 1B.4 the authors normalize the computed sequence and estimate C in C n^{-3/2} 8^n; this is illustrative numerical evidence, not part of any theorem.
assumptions (8)
  • standard math Kazhdan-Lusztig basis at v=1 has nonnegative integral structure constants
    Used throughout; positivity is classical [KL79].
  • domain assumption Shi's description of the projective cell as the region outside all Shi strips
    Invoked in Lemma 3B.4 and Theorem 3B.3; taken from [Shi87, Shi88].
  • domain assumption Every element in the projective cell is a uniformly bounded left/right correction of a Satake (spherical) element
    Lemma 3B.5, cited to [LX88, Xi94]; load-bearing for comparing KL growth to Weyl dimension growth.
  • domain assumption Combinatorial Satake isomorphism: H_sph maps b_m_lambda to |W_f|[L(lambda)] and preserves multiplicative structure
    Lemma 2C.3, cited to [Kno05]; connects Hecke powers to representation tensor powers.
  • domain assumption Tensor-power asymptotics for faithful representations of complex reductive groups
    Used in Theorems 3D.1, 3D.4, 4B.1 and 5D.14; the coefficient C_w and exponent -|Phi^+|/2 are imported from [CEO24, Theorem 2.5].
  • domain assumption Libedinsky-Patimo multiplication rules for affine A2
    The A2 analysis and Section 7 case analysis rely on these rules from [LP23].
  • standard math Every nontrivial finite-dimensional representation of a complex adjoint simple group is faithful
    Used in Theorem 3D.1 to apply [CEO24]; correct because the center is trivial and normal subgroups are central.
  • domain assumption The span of the projective cell is a two-sided ideal in the KL cell decomposition
    Used in Lemma 3C.4 proof and Lemma 5D.6; standard cell theory.

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Pith. "Pith review of Growth in affine Hecke categories." pith.science (2026). https://pith.science/paper/BJIUWGYR

@misc{pith2026260809197,
  author       = {Pith},
  title        = {Pith review of: Growth in affine Hecke categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJIUWGYR}},
  note         = {Machine review of arXiv:2608.09197}
}
read the original abstract

This paper studies the asymptotic growth of tensor powers in affine Hecke categories, or equivalently of powers of Kazhdan--Lusztig basis elements in affine Hecke algebras. We prove general bounds in arbitrary affine type, determine precise asymptotics in affine type A1, and establish corresponding results for several natural families in affine type A2.

Figures

Figures reproduced from arXiv: 2608.09197 by the authors.

Figure 1
Figure 1. The affine Ae2 alcove picture, with hyperplane colors 1 = green, 2 = red, and 3 = blue. In addition to the alcoves, the picture shows the two-sided KL cells: the shaded central triangle is the trivial cell, i.e. the fundamental alcove; the purple belt is the cyclic wall; and the green cones form the projective cell. Each alcove is labeled by information about the corresponding element w ∈ W: northwest gives a reduce… view at source ↗
Figure 2
Figure 2. The cyclic words (northwest) and the projective words (northeast). At the southeast, for one cone, we mark the translations T of the fundamental alcove; these are one-sided longest. We also mark the two-sided longest elements S, which lie in Hsph and hence can be mapped to representations under the Satake isomorphism. The inner hexagon consists of the elements corresponding to Wf ; in particular, the longest element… view at source ↗
Figure 3
Figure 3. The (normalized) sequence νn for w = 123. We note two observations: (a) The powers of b123 contain only one summand on the wall. (We will prove this in Section 5D.1 in general.) (b) The data suggest that νn ∼ C · n −3/2 · 8 n (thus, as expected, β = 8). In the computed range, the normalized values νn/(n −3/2 · 8 n ) lie roughly between 0.14 and 0.15, so we estimate C to lie in this range. It is also worth recording … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The affine Ce2 alcove picture, with hyperplane colors 1 = green, 2 = red, and 3 = blue. In addition to the alcoves, the picture shows the two-sided KL cells: the shaded central triangle is the trivial cell, i.e. the fundamental alcove; the purple belt is the reduced ex…
Figure 5
Figure 5. Figure 5: The (normalized) sequence νn for w = 312. Remark 6.3. The exponent is the expected one, but we do not attempt to prove this here. ✸ [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

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