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The CompGIT package: a computational tool for Geometric Invariant Theory quotients

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read CompGIT computes the stability boundary of GIT quotients by simple groups.

desk verdict A useful, honestly-scoped software manual for GIT stability computations, but the strictly polystable boundary is claimed more broadly than the T-polystable output actually proves. read the letter →

arxiv 2506.19431 v2 pith:MJGE7L57 submitted 2025-06-24 math.AG cs.CGcs.MSmath.RT

Robert Hanson , Jesus Martinez-Garcia This is my paper · ORCID
classification math.AGcs.CGcs.MSmath.RT MSC 14L2414-04
keywords geometricinvarianttheoryGITquotientunstablelocusnon-stablestrictlypolystableHilbert-Mumfordcriterionsimplealgebraicgroupscomputationalgeometry
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

CompGIT is a computer algebra package that computes, for a simple algebraic group acting linearly on projective space, the three subsets that determine a geometric invariant theory (GIT) quotient: the unstable locus, the non-stable locus, and the strictly polystable locus with respect to a maximal torus. The computation is driven by the Hilbert–Mumford numerical criterion: instead of searching over all points, the package enumerates, up to conjugation, the finitely many one-parameter subgroups whose nonnegative weight subspaces generate the stability boundary. The paper argues that this suffices in principle to solve any GIT quotient of a projective variety by a simple group, provided the variety has an equivariant embedding into projective space. A worked example recovers the classical description of plane cubics: stable forms are exactly smooth cubics, and the unique strictly polystable form is the union of three lines not passing through one point. The package extends the earlier algorithms to the exceptional groups.

What carries the argument

The load-bearing object is the Hilbert–Mumford numerical criterion, implemented through root-system combinatorics. A point of $\mathbb{P}(V)$ is non-stable exactly when some one-parameter subgroup $\lambda$ sends it to a point whose Mumford weight $\mu(p,\lambda)$ is nonnegative; therefore CompGIT never scans points but enumerates the primitive one-parameter subgroups inside the fundamental chamber and, for each, records the set of weights $\Xi_{V,\lambda}^{\geq 0}$ that can appear in a destabilised form. The Weyl group acts on these data, so the final output is a small list of conjugacy classes of maximal states; convex-geometry tests on the weight sets decide polystability. The paper's central manoeuvre is to turn a quotient problem into this finite combinatorial enumeration.

What would settle it

Take a representation with a known answer, e.g. cubic forms in three variables with $G=\mathrm{SL}_3$, and check CompGIT's output against the classical facts: every non-stable family should contain only singular cubics and the strictly polystable family should be $x_0x_1x_2$ up to conjugation. A single listed family containing a smooth cubic, or a missing polystable family, would show the inherited algorithms fail in this implementation.

Watch

Extended reading notes

Core claim

On its own terms, the paper's claim is that the whole stability boundary of a GIT quotient $\mathbb{P}(V)/\!/G$ with $G$ simple can be described by a finite list of 'maximal non-stable states': each state consists of a one-parameter subgroup $\lambda$ of a maximal torus and the set of weights of $V$ on which the Hilbert–Mumford weight is nonnegative, and every non-stable or unstable point is conjugate to a point whose weight coordinates lie in one of these states. The package outputs these states for the unstable locus, the non-stable locus, and the strictly polystable locus, and the paper demonstrates on the moduli of plane cubics that the weight data translate directly into geometric conditions such as singularity type and reducibility. Included for the first time are the exceptional types $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$, whose GIT computations were not previously available in this framework.

Load-bearing premise

The package does not prove the algorithms it implements; it assumes the earlier algorithms and code on which CompGIT is built are correct, and every new output inherits that assumption.

Editorial extensions

If this is right

  • In principle, any GIT quotient $X/\!/G$ with $G$ simple can be solved by embedding $X$ into projective space, running the computation there, and intersecting the resulting stability data with $X$.
  • Up to conjugation, the package gives a complete description of the unstable locus, the non-stable locus, and the strictly polystable locus with respect to a maximal torus.
  • For moduli problems, the weight families translate into concrete geometric conditions, as in the plane-cubic example where stability is exactly smoothness and the only strictly polystable form is the union of three lines not passing through one point.
  • Users can estimate the tractability of a proposed GIT problem from the number of families listed in the complexity tables before attempting the geometric interpretation.
  • The inclusion of the exceptional groups extends computational GIT to problems whose structure group is $G_2$, $F_4$, $E_6$, $E_7$, or $E_8$.

Reading between the lines

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

  • Because the output is a finite list of weight families attached to one-parameter subgroups, it could be fed directly into equivariant cohomology computations for the quotient, a step the paper only mentions in passing.
  • Varying the linearisation changes the Hilbert–Mumford weights, so the same enumeration should produce explicit chamber decompositions and wall-crossing descriptions for variation of GIT without new algorithms.
  • The complexity tables suggest the bottleneck for large representations is not computation but the human analysis of enormous family lists; automated orbit identification or symmetry reduction would be needed before outputs like the half-spin $D_5$ case become usable.
  • Upgrading the strictly $T$-polystable test from a maximal torus to the full group by checking Weyl-group orbits of weight sets would make the package output the genuine closed orbits of the GIT quotient directly.
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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

4 major / 5 minor

Summary. The paper describes CompGIT, a SageMath package for computing the unstable, non-stable, and strictly polystable loci of GIT quotients of projective space by simple algebraic groups, based on algorithms from the authors' earlier preprint [19]. The package is illustrated on the moduli of plane cubics, where its output is translated into the classical stability classification, and complexity tables are given for representations of types B2, B3, D2, and D3. The authors also discuss applications to hypersurface and complete-intersection moduli, outline the inclusion of exceptional groups, and propose future generalizations to semisimple and non-reductive groups.

Significance. If the advertised outputs are correct, CompGIT is a genuinely useful tool: it automates the Hilbert-Mumford criterion for simple groups, and the plane-cubics example correctly reproduces the known classification of stable, semistable, and strictly polystable plane cubics. The paper is strongest where it gives concrete, interpretable output and honest complexity estimates, and the open-source release makes the package available to the community. Its significance is currently limited by three gaps: the strictly polystable output is only proved to be T-polystable, the claimed scope extends beyond the implemented irreducible-representation case, and the new exceptional-group code paths have no demonstrated output. These gaps are fixable, but they must be addressed before the advertised capabilities can be taken at face value.

major comments (4)
  1. [Abstract; Section 1.1; Section 4.2 (Lemma 4.2)] The paper advertises description of the strictly polystable locus X^ps \ X^s, but the computation actually produces strict polystability with respect to a maximal torus T, as the bullet in Section 1.1 and the proof of Lemma 4.2 concede. The passage from T-polystable to G-polystable is handled only for plane cubics, by an argument that is not a general theorem and that relies on the unproved assertion that any G-polystable element must be T-polystable; the compactness argument in that proof does not establish the existence of a G-strictly-polystable element, since the image of a non-compact open subset under a proper map to a compact space can be the whole compact space. Consequently, the quantities |PF_ps| in Tables 2-5 cannot currently be interpreted as counting G-strictly-polystable families unless a general conversion statement or a modified algorithm is supplied.
  2. [Section 5.2 versus Abstract and Sections 1.1/3.1] The abstract and Section 1.1 claim that the package is in principle sufficient for any GIT quotient of a projective variety by a simple group, but Section 5.2 states that the implementation works for irreducible representations only and that reducible representations would require user-supplied weight lists. This is a real scope restriction: for example, the exterior-power problems sketched in Section 4.3.1 are not generally irreducible, and Section 3.1's phrase 'G-reducible representation' (which should be 'irreducible') does not resolve the ambiguity. The advertised scope should be narrowed or the implementation extended.
  3. [Section 4.3.5] The addition of exceptional groups is described as one of the most important contributions of CompGIT, but no computational output, runtime, or validation is provided for any group of type G2, F4, E6, E7, or E8. Because the paper's only external check is the A2 plane-cubics example and the correctness of the underlying algorithms is delegated to the authors' own preprint [19], the exceptional-group claims are unsupported as written; at least one known example with a literature comparison should be included.
  4. [Section 4.2 (paragraph 3)] The statement 'It follows from [19] that our code precisely characterises all the nonstable elements up to G-action' delegates the central correctness claim to an external preprint without stating the precise theorem being instantiated. Since the package modifies and extends the code of [19], the paper should either reproduce the relevant correctness statement or provide a minimal proof for the combinatorial output it relies on, in addition to the plane-cubics sanity check.
minor comments (5)
  1. [Section 3.1] The phrase 'we may assume that V is a G-reducible representation' should read 'irreducible representation'; a reducible representation is not determined by a single highest weight.
  2. [Abstract] 'a well as suggesting' is a typo for 'as well as suggesting'.
  3. [Section 4.1] The phrase 'a maximal one-dimensional torus' is inaccurate for SL3; the relevant object is a maximal torus of rank two.
  4. [Section 4.3.2] 'as it can be seen from Table 4.3.2' should refer to Table 2.
  5. [Tables 3-5] In several rows the reported |PF_ss| exceeds |PF_s| (for example Table 3, d=4; Table 4, d=3; Table 5, d=3), which is surprising given that the unstable locus is contained in the non-stable locus; please clarify the counting convention or correct the column labels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the package's outputs are not fitted parameters disguised as predictions, and the worked plane-cubic example anchors the implementation against an independent known classification.

full rationale

The paper is a software/user-manual rather than a derivation of new first-principles results. CompGIT takes a highest-weight representation of a simple group and produces weight-polyhedron data describing non-stable, unstable, and T-strictly-polystable families; no parameter is fitted to a subset of data and then reported as a prediction. The central worked example, the GIT quotient for plane cubics, is an external benchmark: the code's output is translated into the classical statement that a cubic is stable iff it is smooth, which is known independently of the package. The appeal to the authors' prior work [19] in Section 4.2 ('It follows from [19] that our code precisely characterises all the nonstable elements up to G-action') is a self-citation, but it is not a circular reduction: [19] supplies algorithms with stated assumptions that do not include the target classification, and the plane-cubic example provides external falsifiability for the code. The acknowledged limitation in Section 4.2 that the strictly-polystable output is only T-polystable ('our output only gives us that f is strictly GIT-T-polystable') is a substantive correctness caveat for how the abstract phrases the polystable outputs, but it is not self-definitional; the paper supplies an independent compactness argument to promote the one identified family to G-polystability in the plane-cubic case. No uniqueness theorem is imported from the authors, no ansatz is smuggled in by citation, and no known result is merely renamed. Therefore the derivation chain is not circular, and the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted and no new mathematical entities are introduced. The ledger above records the background facts and external software assumptions on which the package's correctness rests.

assumptions (5)
  • standard math The Hilbert-Mumford numerical criterion correctly characterizes semistable, stable, and polystable points.
    Invoked in Section 1.4 as the basis of the algorithms; it is a standard theorem of GIT.
  • standard math Simple algebraic groups over an algebraically closed field of characteristic 0 are classified by irreducible root systems of Dynkin types.
    Used throughout Section 3 to define inputs for the package.
  • domain assumption SageMath's WeylCharacterRing and WeylGroup classes provide correct representation and root-system data.
    The package relies on these classes as black boxes, as described in Sections 3.2 and 5.2.
  • domain assumption The algorithms in Gallardo, Martinez-Garcia, Moon and Swinarski [19] are correct and are faithfully implemented in CompGIT.
    The paper delegates the correctness of its computations to [19], an arXiv preprint by overlapping authors; no independent proof is given in this paper.
  • domain assumption SageMath's Polyhedron class correctly decides point-in-convex-hull questions used for strictly polystable loci.
    Section 5.1 states that the implementation uses the Polyhedron class as a black box for this purpose.

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Cite this review

Pith. "Pith review of The CompGIT package: a computational tool for Geometric Invariant Theory quotients." pith.science (2026). https://pith.science/paper/MJGE7L57

@misc{pith2026250619431,
  author       = {Pith},
  title        = {Pith review of: The CompGIT package: a computational tool for Geometric Invariant Theory quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJGE7L57}},
  note         = {Machine review of arXiv:2506.19431}
}
read the original abstract

We describe CompGIT, a SageMath package to describe Geometric Invariant Theory (GIT) quotients of projective space by simple groups. The implementation is based on algorithms described by Gallardo--Martinez-Garcia--Moon--Swinarski. In principle the package is sufficient to describe any GIT quotient of a projective variety by a simple group -- in practice it requires that the user can construct an equivariant embedding of the polarised variety into projective space. The package describes the non-stable and unstable loci up to conjugation by the group, as well as describing the strictly polystable loci. We discuss potential applications of the outputs of CompGIT to algebraic geometry problems, a well as suggesting directions for future developments.

Figures

Figures reproduced from arXiv: 2506.19431 by the authors.

Figure 3.1
Figure 3.1. The root system of the exceptional group [PITH_FULL_IMAGE:figures/full_fig_p004_3_1.png] view at source ↗

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