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arxiv: 2606.26767 · v1 · pith:ETHAPBILnew · submitted 2026-06-25 · 🧮 math.AG · math.RT

e-polynomials of character varieties

Alfonso Zamora This is my paper

Pith reviewed 2026-06-26 02:50 UTC · model grok-4.3

classification 🧮 math.AG math.RT
keywords character varietiese-polynomialsmixed Hodge structuresparabolic stratificationmotivic expressionsreductive groupsmirror symmetry
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The pith

Extending parabolic stratification to general reductive groups gives explicit motivic expressions for their character varieties.

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

This paper presents lecture notes introducing e-polynomials of character varieties via mixed Hodge structures. It covers arithmetic techniques such as counting points over finite fields and geometric techniques such as stratification into parabolic types, with a full worked example for the GL_3-character variety of the free group. The notes then generalize the stratification method to an arbitrary reductive group G. This produces explicit motivic expressions for the G-character varieties and reduces certain topological mirror symmetry conjectures for the corresponding moduli spaces.

Core claim

The geometric stratification into parabolic types extends from GL_n to a general reductive group G, which yields explicit motivic expressions for the G-character varieties and reduces certain topological mirror symmetry conjectures for these moduli spaces.

What carries the argument

Stratification into parabolic types, extended from the GL_n case to arbitrary reductive groups G.

If this is right

  • Explicit motivic expressions become available for character varieties attached to any reductive group.
  • Topological mirror symmetry conjectures for these moduli spaces reduce to statements about the motivic expressions.
  • Arithmetic point-counting methods apply uniformly across character varieties of general reductive groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same stratification could be tested on other invariants of these moduli spaces beyond e-polynomials.
  • Explicit computations for groups such as SL_n or exceptional groups become feasible with the generalized method.
  • If the reduced mirror symmetry statements hold in examples, they would support the original conjectures.

Load-bearing premise

That the parabolic stratification and point-counting methods developed for GL_n carry over directly to arbitrary reductive groups without new obstructions.

What would settle it

A concrete reductive group G other than GL_n for which the parabolic stratification fails to produce a valid motivic expression for the character variety.

Figures

Figures reproduced from arXiv: 2606.26767 by Alfonso Zamora.

Figure 5.1
Figure 5.1. Figure 5.1: Core (Y, H) of (X, G). Definition 61. A core for (X, G) is a pair (Y, H) given by a subvariety Y ⊆ X and a subgroup H ⊆ G such that (i) Y is orbitwise-closed for the H-action, i.e. H ⋅ y ⊆ Y , for all y ∈ Y . (ii) For every x ∈ X, we have G ⋅ x ∩ Y ≠ ∅. (iii) For every two W1,W2 ⊆ Y disjoint closed (in Y ) H-invariant subsets, we have that G ⋅ W1 ∩ G ⋅ W2 = ∅. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_5… view at source ↗
read the original abstract

These lecture notes contain the material presented at one of the mini-courses of the Workshop on Character Varieties and Higgs Bundles held in Liberia, Guanacaste, Costa Rica, in August 2025. They also contain some exercises for the students attending the conference. This manuscript contains the basic ideas and constructions about $e$-polynomials in character varieties and the state of the art of certain research in the field, plus some new further directions. We introduce mixed Hodge structures and $e$-polynomials, together with a series of arithmetic (counting points over finite fields) and geometric (stratification into parabolic types) techniques to compute them. We include a complete example of the calculation of the $e$-polynomial for the ${\rm GL}_3$-character variety of the free group. Finally, we extend the geometric stratification into parabolic types to a general reductive group $G$ to obtain explicit motivic expressions for the $G$-character varieties, and reduce certain topological mirror symmetry conjectures for these moduli spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. These lecture notes introduce mixed Hodge structures and e-polynomials of character varieties, along with arithmetic techniques (point counting over finite fields) and geometric techniques (stratification into parabolic types). They provide a complete example computing the e-polynomial for the GL_3-character variety of the free group, then extend the parabolic stratification to a general reductive group G to derive explicit motivic expressions for the G-character varieties and reduce certain topological mirror symmetry conjectures.

Significance. If the claimed extension of the stratification and point-counting techniques from GL_n to arbitrary reductive G holds without additional hypotheses, the explicit motivic expressions and reduction of mirror symmetry conjectures would constitute a useful advance in the study of character varieties, building directly on the provided GL_3 example.

major comments (1)
  1. [final paragraph of the abstract / section on general reductive group G] Final paragraph of the abstract (and the corresponding section extending the stratification): the central claim that the geometric stratification into parabolic types extends directly to general reductive G, yielding valid motivic expressions without extra hypotheses on root systems or parabolic subgroups, is load-bearing for the reduction of the mirror symmetry conjectures. The lecture-notes format raises the possibility that this extension is sketched rather than fully derived with all necessary checks for non-GL_n groups; a concrete verification (e.g., for a simple non-GL_n example) is needed to support the claim.
minor comments (1)
  1. The manuscript would benefit from explicit cross-references between the GL_3 example and the general-G extension to clarify which steps carry over unchanged.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive comment on the extension to general reductive groups. We address the concern below.

read point-by-point responses

  1. Referee: [final paragraph of the abstract / section on general reductive group G] Final paragraph of the abstract (and the corresponding section extending the stratification): the central claim that the geometric stratification into parabolic types extends directly to general reductive G, yielding valid motivic expressions without extra hypotheses on root systems or parabolic subgroups, is load-bearing for the reduction of the mirror symmetry conjectures. The lecture-notes format raises the possibility that this extension is sketched rather than fully derived with all necessary checks for non-GL_n groups; a concrete verification (e.g., for a simple non-GL_n example) is needed to support the claim.

    Authors: The section on general reductive G derives the extension explicitly by replacing the GL_n-specific data with the standard parabolic subgroup structure, Bruhat decomposition, and Levi decompositions that hold for any reductive group. The stratification into parabolic types is defined uniformly via the root system and Weyl group action, without additional hypotheses; the motivic expressions and point-counting arguments then carry over verbatim from the GL_3 case. The lecture-notes format presents this as a direct generalization rather than a sketch. Nevertheless, to address the request for explicit verification, we will insert a short worked example for SL_2 (or another simple group) in the revised notes. revision: yes

Circularity Check

0 steps flagged

No circularity: extension to general G presented as independent contribution

full rationale

The manuscript introduces mixed Hodge structures, e-polynomials, and arithmetic/geometric techniques (point counting, parabolic stratification) with a complete GL_3 example, then claims an extension of the stratification to arbitrary reductive G yielding motivic expressions. No equations, fitted parameters, or self-citations appear in the provided text that would make the extension reduce by construction to prior inputs or definitions within the paper. The techniques are described as developed externally and applied to produce new expressions, rendering the derivation self-contained against external benchmarks rather than tautological. This matches the reader's assessment of reliance on external constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work relies on standard mixed Hodge structures and character variety constructions from prior literature.

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discussion (0)

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Reference graph

Works this paper leans on

50 extracted references

  1. [1]

    , journal=

    Deligne, P. , journal=. Th

  2. [2]

    Proceedings of the London Mathematical Society , volume=

    Arithmetic of singular character varieties and their E-polynomials , author=. Proceedings of the London Mathematical Society , volume=. 2017 , publisher=

    2017

  3. [3]

    On the cohomology of Hilbert schemes of points , author=. J. Algebraic Geom. , volume=

  4. [4]

    1997 , publisher=

    Algebraic groups and Lie groups , author=. 1997 , publisher=

    1997

  5. [5]

    Journal of Geometry and Physics , volume=

    Serre polynomials of SLn-and PGLn-character varieties of free groups , author=. Journal of Geometry and Physics , volume=. 2021 , publisher=

    2021

  6. [6]

    Katz , author=

    Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz , author=. Inventiones mathematicae , volume=. 2008 , publisher=

    2008

  7. [7]

    Revista matem

    Hodge polynomials of -character varieties for curves of small genus , author=. Revista matem. 2013 , publisher=

    2013

  8. [8]

    International Journal of Mathematics , volume=

    Arithmetic of character varieties of free groups , author=. International Journal of Mathematics , volume=. 2015 , publisher=

    2015

  9. [9]

    Journal of Pure and Applied Algebra , volume=

    On the number of stable quiver representations over finite fields , author=. Journal of Pure and Applied Algebra , volume=. 2009 , publisher=

    2009

  10. [10]

    2008 , publisher=

    Mixed Hodge structures , author=. 2008 , publisher=

    2008

  11. [11]

    and Nozad, A

    Florentino, C. and Nozad, A. and Zamora, A. , journal=. Generating series for the E -polynomials of (n,. 2023 , publisher=

    2023

  12. [12]

    Inventiones mathematicae , volume=

    The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli , author=. Inventiones mathematicae , volume=. 2003 , publisher=

    2003

  13. [13]

    International Mathematics Research Notices , volume=

    Counting rational points of quiver moduli , author=. International Mathematics Research Notices , volume=. 2006 , publisher=

    2006

  14. [14]

    2002 , publisher=

    Algebraic Topology , author=. 2002 , publisher=

    2002

  15. [15]

    , title =

    Heinloth, J. , title =. 2024 , url =

    2024

  16. [16]

    , title =

    Totaro, B. , title =. Vol. 2, Proceedings of the ICM, Beijing , pages=

  17. [17]

    and Lehrer, G.I

    Dimca, A. and Lehrer, G.I. , title =. Austral. Math. Soc. Lect. Ser. , pages=

  18. [18]

    , title =

    Grothendeck, A. , title =. Séminaire Claude Chevalley 3 , volume=

  19. [19]

    , title =

    Simpson, C. , title =. Publications Mathématiques de l'IHÉS , volume =

  20. [20]

    , title =

    Corlette, K. , title =. Journal of Differential Geometry , volume =

  21. [21]

    Donaldson, S. K. , title =. Proceedings of the London Mathematical Society (3) , volume =

  22. [22]

    2012 , publisher=

    Introduction to Moduli Problems and Orbit Spaces , author=. 2012 , publisher=

    2012

  23. [23]

    1994 , publisher=

    Geometric Invariant Theory , author=. 1994 , publisher=

    1994

  24. [24]

    , title =

    Procesi, C. , title =. Adv. Math. , issn =. 1976 , language =

    1976

  25. [25]

    , title =

    Artin, M. , title =. J. Algebra , issn =. 1969 , language =

    1969

  26. [26]

    and Fricke, R

    Klein, F. and Fricke, R. , title =. 2017 , publisher =

    2017

  27. [27]

    Hitchin, N. J. , title =. Proc. Lond. Math. Soc. (3) , issn =

  28. [28]

    Pseudo-quotients of algebraic actions and their application to character varieties , fjournal =

    Gonz. Pseudo-quotients of algebraic actions and their application to character varieties , fjournal =. Commun. Contemp. Math. , issn =

  29. [29]

    Lawton, S. and Mu. E -polynomial of the (3,. Pac. J. Math. , issn =

  30. [30]

    E -polynomials of the (2,

    Mart. E -polynomials of the (2,. Int. Math. Res. Not. , issn =

  31. [31]

    , title =

    Mereb, M. , title =. Math. Ann. , issn =

  32. [32]

    and Silva, J

    Florentino, C. and Silva, J. , title =. Open Math. , issn =

  33. [33]

    and Nozad, A

    Florentino, C. and Nozad, A. and Silva, J. and Zamora, A. , title =. Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019. Trends in Mathematics, Birkhäuser, Cham. 99-110 , pages =

    2019

  34. [34]

    and Zamora, A

    Gonz\'alez-Prieto, \'A. and Zamora, A. , title =. 2024 , archivePrefix =. 2408.03111 , primaryClass =

    arXiv 2024

  35. [35]

    1998 , publisher=

    Linear Algebraic Groups , author=. 1998 , publisher=

    1998

  36. [36]

    , TITLE =

    Borel, A. , TITLE =. 1991 , PAGES =

    1991

  37. [37]

    and Thaddeus, M

    Hausel, T. and Thaddeus, M. , title =. Inventiones Mathematicae , volume =. 2003 , publisher =

    2003

  38. [38]

    and Wyss, D

    Groechenig, M. and Wyss, D. and Ziegler, P. , year =. Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration , volume =

  39. [39]

    Journal of Algebra , volume=

    A computational criterion for the Kac conjecture , author=. Journal of Algebra , volume=. 2007 , publisher=

    2007

  40. [40]

    and Singh, R

    Li, R. and Singh, R. , title =. arXiv:2410.10008 , year =

    arXiv

  41. [41]

    , title =

    Schiffmann, O. , title =. Ann.of Math. (2) , volume=

  42. [42]

    Mellit, A. , id =. Poincar. Inventiones mathematicae , number =

  43. [43]

    and Witten, E

    Kapustin, A. and Witten, E. , title =. Commun. Num. Theor. Phys. 2007

    2007

  44. [44]

    , title =

    Gothen, P. , title =. International Journal of Mathematics , volume =

  45. [45]

    and Lawton, S

    Florentino, C. and Lawton, S. , title =. Topology and Its Applications , volume =. 2024 , doi =

    2024

  46. [46]

    and Lawton, S

    Florentino, C. and Lawton, S. , title =. Mathematische Annalen , volume =

  47. [47]

    and Lawton, S

    Florentino, C. and Lawton, S. , title =. Topology and Its Applications , volume =

  48. [48]

    and Florentino, C

    Casimiro, A. and Florentino, C. and Lawton, S. and Oliveira, A. , title =. Forum Mathematicum , volume =

  49. [49]

    Pacific Journal of Mathematics , volume=

    E-polynomial of the SL(3,C)-character variety of free groups , author=. Pacific Journal of Mathematics , volume=. 2016 , publisher=

    2016

  50. [50]

    and Hanany, A

    Feng, B. and Hanany, A. and He, Y.-H. , title =. Journal of High Energy Physics , year =