Motivic (Representation) Stability of Representation Varieties and Character Stacks

Jesse Vogel; M\'arton Hablicsek

REVIEW 3 major objections 4 minor 27 references

Motivic (Representation) Stability of Representation Varieties and Character Stacks

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper establishes motivic representation stability: normalized virtual classes of representation varieties and character stacks converge in completed Grothendieck rings, with verified cases including surface groups, commuting tuples…

desk verdict A promising set of conjectures with a nice GL_2 computation, but the main stability theorems rest on a Proposition 4.2 whose proof has an undefined denominator and an unjustified free-locus replacement; the gaps look fixable but are not minor. read the letter →

arxiv 2505.06879 v2 pith:XK5M65YM submitted 2025-05-11 math.AG

Márton Hablicsek , Jesse Vogel This is my paper · ORCID

classification math.AG MSC 14F4514M3520C05

keywords representationstabilityGrothendieckringofvarietiesvarietycharacterstackmotivicdecompositionstackscommutingtuples

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

The paper introduces motivic representation stability, an algebraic counterpart of representation stability: instead of asking whether cohomology groups stabilize, one asks whether the normalized virtual classes $[X_n]/q^{\dim X_n}$ converge in the completed Grothendieck ring of stacks $\widehat K_0(\mathrm{Stck}_k)$. It formulates four conjectures — for surface-group representation varieties, free-group and free-abelian representation varieties with their character stacks, and rank-growing families $C_n(\mathrm{GL}_r)$ — and verifies them in all cases where explicit virtual classes were previously known. The verified results include the surface-group limit for $\mathrm{SL}_2$ and low-rank upper-triangular groups, motivic representation stability of commuting $n$-tuples for every reductive group whose maximal abelian subgroups are connected (and of the $\mathrm{GL}_r$ character stacks), and the limit $\lim_{r\to\infty}[C_2(\mathrm{GL}_r(k))]/(q^r\,[\mathrm{GL}_r(k)])=1$. The payoff of a motivic statement is that one limit packages point counts, E-polynomials, and cohomological stabilization into a single algebraic identity.

What carries the argument

The central object is the completed Grothendieck ring of stacks $\widehat K_0(\mathrm{Stck}_k)$, together with the filtered completion $\widehat M^G_q$ of the localized Grothendieck ring of $G$-varieties; convergence of normalized virtual classes is measured in these rings. The main new tool is motivic decomposition of a $G$-variety with respect to a good set of conjugacy classes of subgroups: for a finite group $G$ one writes $[X]_G=\sum_V [U_V]\otimes V$ in $K_0(\mathrm{Var}_k)\otimes R_{\mathbb{Q}}(G)$, characterized by $\langle T_H,\mathrm{Res}^G_H[X]_G\rangle=[X/H]$, so the trivial-representation piece recovers the quotient class. Around this sit three computational devices: Lemma 2.7 comparing symmetric powers of a variety and of an open complement, Lemma 4.1 identifying motivic stability of $\mathrm{Sym}^n(X)$ with the condition that the top-degree coefficient of $[X]$ is 1, and branching matrices that track how centralizers of commuting tuples change when a new element is added. Specialness of $\mathrm{GL}_r$ and its centralizers is used to convert torsor classes into products in the rank-stability proof.

What would settle it

Compute the virtual class of the free locus $X_n$ in Proposition 4.2 for $r=2$ and increasing $n$: if $[X_n]/q^{2n}$ does not tend to 1, or if $[X_n/\mathrm{GL}_2]$ differs from $[\mathrm{Sym}^n(\mathbb{A}^2)]$, then the replacement at the heart of Proposition 4.2 fails and the stability theorems built on it lack their stated proof. For Theorem 5.2, a finite-rank check is available: the proof gives $[C_2(\mathrm{GL}_r)]=[\mathrm{GL}_r][C_r]$, so computing $[C_r]$ for $r=3,4$ from the Jordan-block parametrization and comparing with the claimed $q^r$ asymptotics would test the rank limit directly.

Watch

Extended reading notes

Core claim

The central claim, stated as Conjectures A–D and proved in the listed cases, is that representation stability is visible at the level of virtual classes: the limits of normalized classes in $\widehat K_0(\mathrm{Stck}_k)$ exist and have the expected closed forms. Conjecture A is verified for $\mathrm{SL}_2(k)$ and for upper-triangular groups of ranks up to 5; Conjecture B is verified for $\mathrm{GL}_r$ and $U_r$ character stacks; Conjecture C is verified through Theorem 4.12: if $G$ is a linear reductive group over an algebraically closed field of characteristic 0 and every maximal abelian subgroup of $G$ is connected, then $C_n(G)$ is motivically representation stable, and Corollary 4.13 extends this to the $\mathrm{GL}_r$ character stacks. Conjecture D is proved for $n=2$ in Theorem 5.2. The proof machinery is a motivic decomposition with respect to irreducible rational representations of finite groups, which turns the condition that the $S_{\lambda[n]}$-quotients stabilize into the condition that the motivic decomposition is stable.

Load-bearing premise

The load-bearing premise is the free-locus replacement in Proposition 4.2 — that the free part of $\mathrm{Sym}^n(\mathbb{A}^r)$ has the same asymptotic class as the whole symmetric power and that $[X_n/\mathrm{GL}_r]$ can be replaced by $[\mathrm{Sym}^n(\mathbb{A}^r)]$ — which the proof asserts without justification and which is expressed in a formula containing the zero factor $q^r-q^r$ in the denominator.

Editorial extensions

If this is right

For the verified groups, the surface-group representation varieties satisfy $\lim_{g\to\infty}[\mathrm{Rep}_G(M_g)]/[G]^{2g}=[G/[G,G]]/[G]$, so the motivic measure of the commutator equation stabilizes to the abelianization quotient class.
For $\mathrm{GL}_r$ and the upper-triangular groups, the free-group character stacks are motivically representation stable, meaning every $S_{\lambda[n]}$-quotient has a well-defined limit.
For any reductive $G$ with connected maximal abelian subgroups, the commuting-tuple varieties $C_n(G)$ satisfy motivic representation stability; in particular the symmetric group acts motivically stably on the space of commuting $n$-tuples.
The rank-stability theorem for $n=2$ gives $[C_2(\mathrm{GL}_r(k))]/(q^r[\mathrm{GL}_r(k)])\to 1$, so the class of commuting pairs is asymptotically the class of the group times $q^r$.
The $\mathrm{SL}_r$ example shows the connectedness hypothesis is necessary: with a finite center, non-monic factors in the relevant virtual classes prevent motivic representation stability.

Reading between the lines

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

The branching-matrix recursion shown for $\mathrm{GL}_2$ should give explicit virtual classes for $C_n(\mathrm{GL}_3)$ as well; carrying that computation out would provide a finite-rank check of Theorem 4.12 before the rank limit of Conjecture D is addressed.
Because the E-polynomial is a motivic measure on the completed Grothendieck ring of stacks, the verified motivic limits should imply the corresponding stabilization of mixed Hodge numbers for the same families; the paper records this link for surface groups, and the same implication would hold for the commuting-tuple results.
The unproven free-locus replacement inside Proposition 4.2 is a genuine gap: if the asymptotic class of the free locus differs from that of the full symmetric power, then the character-stack stability for free groups and the general commuting-tuple theorem would need a different proof, even if the conjectures themselves are true.
A plausible route to Conjecture D for all $n$ is to show that $[C_n(\mathrm{GL}_r)]/[\mathrm{GL}_r]$ converges by a simultaneous-centralizer version of the Jordan-block parametrization used for $n=2$; the paper leaves this open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

You should know this paper mainly for its conjectures and for a solid GL_2 computation; the general stability theorems are not supported as written because a key proposition has a zero denominator and an unjustified step.

What is actually new: Definition 2.10 of motivic representation stability, the motivic decomposition for finite group actions (Sections 2.2–2.3), and Conjectures A–D. Those are reasonable algebraic analogues of Church–Farb and Ramras–Stafa, and the paper makes the case that they would imply representation stability in many cohomology theories. The GL_2 branching matrix in Section 4.2.1 is self-contained; Theorem 4.10's closed form for commuting n-tuples checks out. The SL_2 surface-group example in Section 3.3 also gives the claimed limit. The non-connected counterexample in Section 3.4 is a good sanity check.

Where it falls short: Proposition 4.2 is load-bearing. The displayed denominator contains q^r - q^r for every r, so the formula is undefined even for r=1. That is likely a typo, but the proof has a deeper problem: it replaces [X_n/GL_r] by [Sym^n(A^r)] without justification. The stress-test note is right that X_n/GL_r is not Sym^n(A^r); it is the unordered configuration quotient of the Grassmannian, and no argument is supplied for that substitution. Since Corollary 4.3, Theorem 4.4, Theorem 4.12, and Corollary 4.13 all invoke Proposition 4.2, the central theorems for free-group character stacks and commuting tuples inherit the gap. Theorem 4.12 is also only a sketch—it hand-waves the negligibility and the structure of maximal abelian subgroups for general reductive groups. Proposition 4.14 depends on that framework. Separately, Corollary 4.3 has a typo "q^q - q^i" in the denominator.

On the positive side, Theorem 5.2 (rank stability for commuting pairs in GL_r) does not rely on Proposition 4.2; its proof via conjugacy class counts looks credible. So the paper has real content, but the main general theorems need repair.

Who is this for: people working on motivic analogues of representation stability, and to a lesser extent character variety enumerators. It deserves a serious referee, but the referee should demand a rewritten Proposition 4.2 and a fuller proof of Theorem 4.12. As it stands, I'd read it as a set of interesting conjectures plus partial evidence, not as a finished proof of the stability theorems.

Recommendation: send to peer review, but expect heavy revision.

Referee Report

3 major / 4 minor

Summary. The paper introduces a motivic analogue of representation stability in Grothendieck rings of varieties, together with a motivic decomposition for finite group actions. It formulates Conjectures A–D on the motivic stability of representation varieties and character stacks of surface groups, free groups, and free abelian groups, and proves them in special cases. The main positive results claimed are Theorem 4.12 (motivic representation stability for commuting tuples in reductive groups with connected maximal abelian subgroups), Corollary 4.13 (the same for GL_r character stacks), and Theorem 5.2 (a rank-stability limit for C_2(GL_r)). The paper also contains explicit computations for SL_2, upper triangular groups, and a branching-matrix computation for GL_2.

Significance. If fully correct, the paper would provide a useful algebraic counterpart to representation stability and give the first motivic-level verification of several stability conjectures for character stacks. Strengths include the self-contained GL_2 branching matrix computation, the explicit SL_2 surface-group limit, and the formulation of precise conjectures with concrete testable predictions. However, the main stability theorems currently depend on an unjustified step in Proposition 4.2 and on sketched proofs in Theorems 4.12 and 5.2, so the significance is conditional on repairing those arguments.

major comments (3)

[Section 4.1, Proposition 4.2] The statement of Proposition 4.2 contains an undefined expression: the denominator \prod_{i=1}^r(q^r - q^i) includes the factor q^r - q^r = 0, so the asserted limit in \hat{M}^G_q is not defined for any r. In the proof, after defining X_n as the free locus in Sym^n(A^r), the text claims that [X_n/GL_r] can be replaced by [Sym^n(A^r)]; however, X_n/GL_r is the quotient of the full-rank unordered configuration space of n vectors in A^r by GL_r, not the symmetric power, and no argument is given for this replacement. Since Corollary 4.3, Theorem 4.4, and Theorems 4.12/Corollary 4.13 all rely on Proposition 4.2, the central stability results for free-group character stacks and commuting tuples are not supported as written.
[Section 4.2.2, Theorem 4.12] The proof of Theorem 4.12 asserts an equality of limits with a finite sum over maximal abelian subgroups, but it does not prove that C_n(G) admits a stratification compatible with the S_{\lambda[n]}-quotient into pieces of the form Ind^G_{N_G(A)}(A^n), nor that the error terms are negligible in \hat{M}_q. The worked GL_2 example does not give the general proof, and the displayed formula even contains a typo (Ind^{GL_2}_{N_G(A)} should be Ind^G_{N_G(A)}). As written, Theorem 4.12 is not established, and Corollary 4.13 inherits this gap.
[Section 5, Theorem 5.2] The proof of the identity [C^J_2(GL_r)] = [E_J/H_J][GL_r] uses the formula [E_J \times GL_r \times Z_J/H_J] = [E_J/H_J][GL_r][Z_J]. This product formula requires the H_J-quotient to be a Zariski-locally trivial fiber bundle over E_J/H_J with fiber GL_r \times Z_J; when H_J acts on E_J with nontrivial stabilizers, the fiber over a point is (GL_r \times Z_J)/Stab(A), which has smaller dimension, and the specialness of GL_r and Z_J does not by itself justify the formula. Consequently the key identity [C_2(GL_r)] = [GL_r][C_r] is not proven, so Theorem 5.2 (and Theorem 5.6) are not fully supported.

minor comments (4)

[Corollary 4.3] The denominator in the statement contains the typo q^q instead of q^r.
[Section 4.1, proof of Proposition 4.2] The sentence 'we will show that X_n can be covered by varieties of negligible dimension' appears to refer to the complement of X_n; the described cover uses elements of GL_r \setminus \{id\} and therefore covers the non-free locus, not X_n. Please rephrase.
[Section 5, proof of Theorem 5.3] In the paragraph parametrizing Jordan blocks, the index in q^{ni} should be q^{n_l} to match the product over l.
[Example 3.4] The expression for [Rep_{SL_2(k)}(M_g)] should be checked for typographical errors; the factor (q^{2g-1}+q) looks unusual and is not discussed in the surrounding text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the main stability results are not forced by their inputs; Proposition 4.2 issues are correctness gaps, not circularity.

full rationale

I walked the claimed derivation chain. The core positive results (Theorems 4.12, 4.13, and 5.2) are proved from structural lemmas (Lemma 4.1, Corollary 2.6, the branching-matrix computation, and the conjugacy-class count), not by re-inserting the conjectures as inputs. No parameter is fitted and then renamed a prediction. The self-citations to [26], [13], [24], and [25] supply framework theorems and previously computed virtual classes; these are explicit, parameter-free, externally checkable computations, so under the stated rules they are real evidence rather than circularity. The one seriously under-supported step in the text is in Proposition 4.2, where the proof asserts that [X_n/GL_r] "can be replaced" by [Sym^n(A^r)]^{GL_r} and displays a denominator containing the zero factor q^r - q^r; this is a correctness and editing gap, not a circular reduction, because the asserted limit is not definitionally equal to the input and no fitted data re-enter the argument. Accordingly no circularity step meets the evidentiary bar.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper's results rest on the completed Grothendieck-ring framework, quoted lemmas from the authors' thesis [26], special-group quotient formulas, and the connectedness assumption on maximal abelian subgroups. There are no data-fitted free parameters; q is the Lefschetz class, not an ad hoc constant.

assumptions (6)

domain assumption K0(Var_S^G) admits the q-adic filtration and completion in which limits are computed.
Section 2.1.1 defines the completed ring \hat{M}^G_q following Vakil-Wood; all stability statements depend on this completion, and no proof of its convergence properties is given.
domain assumption Restriction and induction maps are continuous for closed subgroups, and special groups satisfy [X] = [X/G][G].
Used in Corollary 2.6, Proposition 4.2, and Section 5; relies on closed subgroup quotients existing and on GL_r and Jordan centralizers being special.
domain assumption The framework lemmas quoted from [26] (Theorem 2.19, Lemma 2.20, Theorem 2.21) are valid.
These lemmas underpin the definition of motivic representation stability and the Kunneth formula, but are only cited, not reproved.
domain assumption All maximal Abelian subgroups of the reductive group G are connected.
Explicit assumption in Theorem 4.12; known for GL_r but false for SL_r, as noted in Proposition 4.14.
domain assumption The E-polynomial is a motivic measure on the completed stack ring, and PORC point counts imply E-polynomial identities.
Section 3.2 relies on Bridger-Kamgarpour's PORC result and Katz's theorem; these are invoked without proof.
standard math Frobenius's character formula and Lang's theorem justify the finite-field heuristics for connected groups.
Section 3.1 uses these classical results to suggest the surface-group limits.

invented entities (2)

Motivic decomposition [X]_G in K0(Var_S) tensor RQ(G)
purpose: Decomposes the virtual class of a G-variety into pieces indexed by irreducible rational G-representations.
Defined in Section 2.3; it exists only after choosing a good set of subgroups and has no external falsifiable handle.
Motivic representation stability
purpose: Defines stability of a sequence of G x S_n-varieties via the motivic stability of [X_n/S_lambda[n]].
Core definition from Definition 2.10, inherited from the authors' thesis [26].

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Pith. "Pith review of Motivic (Representation) Stability of Representation Varieties and Character Stacks." pith.science (2026). https://pith.science/paper/XK5M65YM

@misc{pith2026250506879,
  author       = {Pith},
  title        = {Pith review of: Motivic (Representation) Stability of Representation Varieties and Character Stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XK5M65YM}},
  note         = {Machine review of arXiv:2505.06879}
}

abstract

In this paper, we introduce the notions of motivic representation stability that is an algebraic counterpart of the notion of representation stability. In the process, we also introduce the notion of motivic decomposition for varieties equipped with an action of a finite group $G$. This motivic decomposition decomposes the virtual class of the variety with respect to irreducible rational representations of $G$. We also formulate conjectures on motivic representation stability in the context of representation varieties and character stacks, and we verify the conjectures for groups whose virtual classes have been extensively studied.

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Behrend and A

K. Behrend and A. Dhillon. On the motivic class of the stack of bund les. Advances in Mathe- matics, 212(2):617–644, 2007

work page 2007

[2] [2]

Bridger and M

N. Bridger and M. Kamgarpour. Character stacks are porc cou nt. Journal of the Australian Mathematical Society, 115(3):289–310, 2023

work page 2023

[3] [3]

Chevalley, A

C. Chevalley, A. Grothendieck, and J.-P. Serre. Anneaux de cho w et applications, s´ eminaire c. Chevalley. Chevalley, Ecole Normale Superieure, Paris , 1958

work page 1958

[4] [4]

Church and B

T. Church and B. Farb. Representation theory and homological stability. Advances in Mathe- matics, 245:250–314, 2013

work page 2013

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T. Ekedahl. The Grothendieck group of algebraic stacks. arXiv e-prints , pages arXiv–0903, 2009

work page 2009

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G. Frobenius. ¨Uber Gruppencharaktere. Sitzungsberichte der K¨ oniglich Preußischen Akademie der Wissenschaften zu Berlin , 1896

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W. Fulton and J. Harris. Representation theory. Graduate Texts in Math , 129, 1991

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work page 2020