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Idempotents, traces, and dimensions in Hecke categories

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Pith's one-line read Categorical dimensions in asymptotic Hecke categories reduce to computable scalar ratios.

desk verdict A serious toolbox paper for diagrammatic Hecke categories whose advertised dimension formula is honestly conditional on a deferred theorem in the sequel. read the letter →

arxiv 2507.10061 v1 pith:Y3Z7CXOP submitted 2025-07-14 math.RT math.CTmath.QA

Ben Elias , Liam Rogel , Daniel Tubbenhauer This is my paper
classification math.RTmath.CTmath.QA MSC 18M2020C0818M3018N25
keywords HeckecategoryidempotentscategoricaldimensionasymptoticpartialtraceslocalintersectionformsSoergelbimodulesfusioncategories
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 develops a working calculus for idempotents in the Hecke category — the diagrammatic categorification of the Hecke algebra of a Coxeter group — and uses it to compute categorical dimensions in asymptotic Hecke categories, the semisimple monoidal categories attached to Kazhdan–Lusztig cells. Under the assumption (5F.3) that a reduced expression for $w$ extends one for the Duflo involution $d$, the paper proves that the categorical dimension of $A_w$ equals $\lambda_d(\mathrm{tr}_{d,t}(\rho^a))/\lambda_d(\rho^a)$, where $\lambda_d(f)$ is a scalar extracted from a diagrammatic composition and $\mathrm{tr}_{d,t}$ is an iterated partial trace. The paper also supplies closed formulas for the local intersection forms that control the recursive construction of the idempotents, and a recursive algorithm for the partial traces that appear in the dimension formula. If correct, this reduces computations in the fusion categories promised by cell theory to finite, often computer-checkable diagram calculations.

What carries the argument

The machinery is a trio of diagrammatic notions: top and clasp idempotents (primitive idempotents picking out the indecomposable $B_w$ as a summand of a $w$-object), local intersection forms (the scalar coefficient of the identity in a composition of a projection and an inclusion), and partial traces (closing off an endomorphism of $B_w$ to an endomorphism of $B_x$). Theorem 4E.22 computes the most common local intersection forms as ratios of successive two-colored quantum numbers, determined by the position of the element inside a dihedral coset; Theorem 4I.28 gives a recursive formula for iterated partial traces along linear branching graphs, the bookkeeping graphs of which summands appear as one multiplies by simple reflections; and Theorems 5F.11 and 5F.14 show how the whole dimension calculation for $w$ collapses to the scalar $\lambda_d$ applied to a partial trace of $\rho^a$.

What would settle it

Compute the ratio from Corollary 5F.19 for a diagonal cell in a classical Weyl group where the category $\mathcal{A}_H$ is already known (for example, a cell where it is the category of vector spaces graded by an elementary abelian 2-group), and compare it with the known categorical dimension; a mismatch for any regular dominant $\rho$ would refute Theorem 5E.13.

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Extended reading notes

Core claim

The paper's central claim is that the categorical dimension of an object $A_x$ in an asymptotic Hecke category $\mathcal{A}_H$ can be computed inside the ordinary diagrammatic Hecke category: one constructs the top idempotent for $B_x$ recursively from local intersection forms, forms inclusion maps into $B_x \otimes B_{x^{-1}}$ carrying the polynomial $\rho^a$, and reads off two scalars $\lambda_x$ and $\mu_x$ from Lefschetz pairings. The paper proves (Corollary 5F.19) that when a reduced expression for $x$ extends one for the Duflo involution $d$, the desired dimension is $\lambda_d(\mathrm{tr}_{d,t}(\rho^a))/\lambda_d(\rho^a)$, with the dependence on the whole cell concentrated in $d$. The equality of this ratio with the categorical dimension is delegated to a theorem in the sequel, so the paper's own contribution is the reduction: dimensional data in the fusion category $A_H$ reduce to finite diagrammatic computations in the Hecke category.

Load-bearing premise

The load-bearing premise is a theorem deferred to the sequel: the scalars $\lambda_x$ and $\mu_x$ read off from Lefschetz pairings in the Hecke category are nonzero exactly when the inclusion maps survive in the cell quotient, and their ratio $\mu_x/\lambda_x$ is the categorical dimension of $A_x$; everything in the dimension program hangs on this.

Editorial extensions

If this is right

  • Whenever $w = dt$ with $d$ the Duflo involution and the concatenation reduced, computing the categorical dimension of $A_w$ is reduced to evaluating $\lambda_d$ on one iterated partial trace, a finite diagram computation.
  • For reduced expressions whose branching graph is recursible, all local intersection forms needed to build the idempotent are given by closed formulas, so the construction of idempotents can be automated.
  • For parabolic cells, where $d$ is the longest element of a parabolic subgroup, $\lambda_d(f)$ is the Demazure operator $\partial_d(f)$, giving an explicit base case for the recursion.
  • The techniques are designed to be computer-implementable, and the paper announces that the sequel uses them, with additional machine calculations, to describe the asymptotic Hecke category for finite Coxeter groups except for a small number of cells, notably the middle cell $J_6$ in type $H_4$.

Reading between the lines

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

  • If the deferred Theorem 5E.13 holds as stated, the ratio formula gives a general algorithm for dimensions in every diagonal cell satisfying (5F.3), independent of the classification results that previously identified $A_H$ in Weyl types.
  • The independence of $\rho$ proved in the paper suggests that the ratio is a genuine invariant of the cell; a direct proof of that independence without invoking categorical dimensions would be a strong test of the bridge theorem.
  • Because the same local intersection forms control degenerations in finite characteristic, the closed-form Theorem 4E.22 could be used to locate the first elements where modular Kazhdan–Lusztig theory diverges from characteristic zero; this application is not pursued in the paper.
  • The conjectured isomorphism between $w_0$-dual cells mentioned in the paper would make the three computationally inaccessible cells in type $H_4$ computable by symmetry; proving that conjecture is an immediate next step.
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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

3 major / 4 minor

Summary. The paper develops diagrammatic tools for the Hecke category: a theory of top and clasp idempotents, a recursive construction of idempotents for indecomposable Soergel bimodules, a closed formula for a common family of local intersection forms, and recursive algorithms for iterated partial traces. These tools are then applied to the asymptotic Hecke category: under the assumption that an element w in a diagonal cell is greater than the Duflo involution d in the weak right Bruhat order, the paper reduces the computation of the categorical dimension of A_w to the scalar ratio λ_d(tr_{d,t}(ρ^a))/λ_d(ρ^a). A separate section treats parabolic cells, where the Duflo involution is the longest element of a parabolic subgroup and the relevant scalar is controlled by a Demazure operator.

Significance. If the stated bridge theorems are supplied, the paper would give a substantial and genuinely computational reduction: categorical dimensions in asymptotic Hecke categories would be obtained from finite diagrammatic computations, with no fitted parameters. The recursive treatment of idempotents and partial traces, the closed formula for local intersection forms in Section 4E, and the reduction theorems 5F.11 and 5F.14 are concrete and plausible contributions. The paper is also honest about its two principal deferrals, both of which are load-bearing: Theorem 5E.13, which identifies Lefschetz-pairing scalars with categorical dimensions, and the compatibility data for associators and unitors in the Karoubi-envelope construction. Because these deferrals concern the central claim rather than peripheral examples, the manuscript as submitted is a strong technical foundation but not yet a completed proof of the advertised dimensional computations.

major comments (3)
  1. [Section 5E, Theorem 5E.13] Theorem 5E.13 is explicitly deferred to the sequel [ERT25b], and it is load-bearing for the paper's central claim. The theorem asserts both the nonvanishing criterion for λ_x and μ_x in terms of the lower ideal and the equality μ_x/λ_x = dim(A_x). Without this theorem, Corollary 5F.19 (Eq. (5F.20)) does not compute a categorical dimension; it computes a ratio of Lefschetz-pairing scalars. Theorems 5F.11 and 5F.14 are valid reductions of that scalar ratio, but they do not by themselves connect the ratio to the categorical dimension. The manuscript should either prove Theorem 5E.13 (and Lemma 5E.12, also deferred) or explicitly state that the dimensional conclusion is conditional on the sequel.
  2. [Section 5D, after Eq. (5D.8)] The Karoubi-envelope construction of the asymptotic Hecke category is not fully specified in this paper: the associators, unitors, and the compatibility of the chosen dual bases for the local intersection pairings are acknowledged to be deferred to [ERT25b]. This is not a cosmetic gap. Corollary 5F.19 concerns a categorical invariant, and categorical dimension is only defined once a monoidal structure with consistent associators and unitors is fixed. Without the deferred compatibility statement, the quantity λ_d(tr_{d,t}(ρ^a))/λ_d(ρ^a) is not shown to be independent of the noncanonical choices made in Section 5D, so the identification with the categorical dimension remains incomplete in this manuscript.
  3. [Section 5F, Corollary 5F.19] The scope of the dimensional reduction is narrower than the introduction may suggest: it applies only under assumption (5F.3), i.e. when a reduced expression for w extends one for d in the weak right Bruhat order. The paper notes the w0-dual phenomenon in Example 5F.6 and treats parabolic cells separately, but the general case is not covered here. This is not an error, but the abstract and introduction should be read carefully so that the reader understands that the advertised reduction of categorical dimensions to partial traces is conditional both on assumption (5F.3) and on the unproved bridge Theorem 5E.13.
minor comments (4)
  1. [Section 2C, Remark 2C.13] There is a typo: 'moprhism' should be 'morphism'; similarly, Section 3B contains 'biomdule' for 'bimodule'.
  2. [Section 3C, Lemma 3C.3] The notation I<len(w) is used before Definition 2E.6 is introduced; a forward reference or a brief reminder would improve readability.
  3. [Section 4I, Theorems 4I.9 and 4I.28] Theorem 4I.28 is described as superseding Theorem 4I.9, but the relationship between the two recursion systems is stated only informally; a short explanation of why the later theorem subsumes the earlier one would help the reader.
  4. [Section 5C, Definition 5C.7] The phrase 'L_d A_d is the (local) monoidal identity' is slightly confusing in the non-unital setting; since d ranges over Duflo involutions, it would be clearer to say that the summands A_d serve as a local identity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the dimension formula is conditional on the explicitly deferred Theorem 5E.13, not on a self-referential definition.

full rationale

The paper's advertised derivation is not circular. It contains no fitted parameters, and its two main recursion theorems are genuine inductions: Theorem 4E.22 solves the local-intersection-form recursion using two-colored quantum numbers, and Theorem 4I.28 recursively reduces iterated partial traces using diagrammatic relations, not using the target dimension formula. Theorems 5F.11 and 5F.14 are proven identities in the quotient Hecke category: they reduce the scalars attached to w to scalars attached to d and to iterated partial traces. Corollary 5F.19 then follows formally from Theorem 5E.13. The load-bearing reliance on the authors' own sequel appears exactly where the paper says: 'This Theorem is proven in the sequel [ERT25b]' (Theorem 5E.13), and 'We discuss the associators and unitors in detail in the sequel [ERT25b]' (Section 5D). These are explicit deferrals of missing support, not a reduction of the target to its own input: no equation in this paper defines the categorical dimension of A_x to be μ_x/λ_x, and no fitted scalar is renamed as a prediction. The dimension claim is therefore conditional on an unproven bridge, which warrants a mild self-referential caution, but no circularity finding.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard results in Soergel theory and on the positivity and Lefschetz framework of [EW21]. No free parameters are fitted. The only notable choices are the dominant regular weight rho, whose independence is asserted in Theorem 5E.13, and the normalizations of inclusion maps, which cancel in the ratios.

assumptions (4)
  • domain assumption Soergel conjecture holds for the realization
    Invoked in Sections 2A, 2C, and 4 to control multiplicities and morphism spaces; proven in [EW14] for standard realizations but assumed more generally.
  • domain assumption Lefschetz form on multiplicity spaces is positive definite for dominant regular rho
    This is [EW21, Theorem 3.3] and is the basis for constructing projection maps via M_rho^a and for lambda_x, mu_x being positive scalars in Section 5D.
  • standard math Boundedness of Lusztig's a-function
    Needed for Lusztig's cell theory; referenced to [CH25, Theorem 1.2] in Section 5B.
  • ad hoc to paper Karoubi envelope data for the asymptotic category can be fixed compatibly with associators and unitors
    Section 5D postpones associators and unitors to the sequel [ERT25b]; the dimension computation presumes the monoidal structure exists and is compatible with the chosen dual bases.

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Pith. "Pith review of Idempotents, traces, and dimensions in Hecke categories." pith.science (2026). https://pith.science/paper/Y3Z7CXOP

@misc{pith2026250710061,
  author       = {Pith},
  title        = {Pith review of: Idempotents, traces, and dimensions in Hecke categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3Z7CXOP}},
  note         = {Machine review of arXiv:2507.10061}
}
read the original abstract

We explain how to compute idempotents that correspond to the indecomposable objects in the Hecke category. Closed formulas are provided for some common coefficients that appear in these idempotents. We also explain how to compute categorical dimensions in the asymptotic Hecke category. In many cases, we reduce this to a computation of a partial trace and give recursive formulas for some common partial traces. In the sequel, we apply this technology and perform additional (computer) calculations to complete the description of the asymptotic Hecke category for finite Coxeter groups in all but three cells.

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Pith tools

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