{"id":"2f48b169-dc21-4b33-a54e-d0f37dd301a9","arxiv_id":"2506.19431","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"CompGIT computes GIT stability loci for projective-space quotients by simple groups, based on Gallardo-Martinez-Garcia-Moon-Swinarski algorithms and adding support for exceptional groups.","lead":"This paper presents CompGIT, a SageMath package that automatically describes the unstable, non-stable, and strictly polystable parts of geometric invariant theory quotients of projective space by simple groups. It is a user manual with worked examples, complexity tables, and a list of open moduli problems the tool could address.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Strictly-polystable output is only T-polystable; no general conversion to G-polystable is proved, so the advertised description of the polystable boundary overreaches.","rationale":"The reader's conditional verdict is appropriate. The reader's weakest assumption was inherited correctness of the algorithms from [19]; that is one legitimate concern, but the more specific and textually supported soft spot is the semantic gap between the G-strictly-polystable locus that GIT applications need and the T-strictly-polystable locus that the package actually outputs. Section 4.2 explicitly flags this distinction and supplies only an ad hoc argument for one example, not a general theorem. The SL2/C^2 example shows T-polystability is strictly weaker than G-polystability in general, so the overreach is real unless the package first restricts to the G-semistable locus and a further theorem is supplied. This does not invalidate the useful unstable/non-stable outputs or the worked plane-cubic example, so the paper remains conditionally acceptable, but the polystable claims and the tables in Section 4.3 need either a proof of the conversion or a clear caveat.","tokens_in":16445,"tokens_out":22075,"duration_ms":246212,"concrete_test":"Run P = GITProblem(WeylCharacterRing('A1')(1,0)) and P.solve_strictly_polystable(). The GIT quotient P^1//SL2 has empty semistable locus (the invariant ring is C), so any nonempty strictly-polystable output would prove that the third output is T-polystable rather than G-polystable. If the output is empty, then instead run SL2 on binary quartics (highest weight (4,0)) and compare solve_strictly_polystable() against the known closed orbit of x^2 y^2; any additional family would show the T-output overcounts the G-boundary, and a match would still leave the general conversion unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The third advertised output is not the G-strictly-polystable locus defined in Section 1.1 as X^ps\\X^s, but only the strictly polystable locus with respect to a maximal torus T. Section 4.2 concedes this: 'our output only gives us that f is strictly GIT-T-polystable'. For unstable and non-stable loci this distinction is harmless because those sets are G-invariant and the families are stated up to conjugation; for polystable loci it matters. A T-polystable point need not be G-polystable: for G=SL2 acting on its standard representation V=C^2, the point [1:1] is T-polystable for a diagonal maximal torus (weights 1 and -1, with 0 in the interior of their convex hull), yet the GIT quotient is empty and no point is G-polystable. In Section 4.2 the authors bridge the gap for plane cubics by a dimension/compactness argument; no general theorem is given that would let the outputs in Section 4.3 (Tables 2-5) be read as G-polystable families. The abstract and the central claim that CompGIT 'describes the strictly polystable loci' therefore go beyond what is established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16661,"tokens_out":20425,"duration_ms":204930,"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":[{"comment":"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.","section":"Abstract; Section 1.1; Section 4.2 (Lemma 4.2)"},{"comment":"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.","section":"Section 5.2 versus Abstract and Sections 1.1/3.1"},{"comment":"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.","section":"Section 4.3.5"},{"comment":"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.","section":"Section 4.2 (paragraph 3)"}],"minor_comments":[{"comment":"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.","section":"Section 3.1"},{"comment":"'a well as suggesting' is a typo for 'as well as suggesting'.","section":"Abstract"},{"comment":"The phrase 'a maximal one-dimensional torus' is inaccurate for SL3; the relevant object is a maximal torus of rank two.","section":"Section 4.1"},{"comment":"'as it can be seen from Table 4.3.2' should refer to Table 2.","section":"Section 4.3.2"},{"comment":"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.","section":"Tables 3-5"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially a user manual for a software package, and its mathematical novelty is largely inherited from [19]; the main value is the package itself and the worked plane-cubics example. I am not recommending rejection, but the advertised strictly-polystable capability and the 'any GIT quotient' scope need to be corrected. The editors may also want to verify that the package is actually available and runnable from the GitHub URL before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"If you use computational GIT, this is worth having on the radar. The genuinely new pieces are support for exceptional groups in CompGIT and the complexity tables for types B2, B3, D2, D3. Everything else is a clean user manual built on the algorithms of [19] by an overlapping group. The package itself is the contribution, and it ships as real software with a worked example that reproduces the classical plane-cubics story: smooth cubics stable, unions of three non-concurrent lines strictly polystable, etc. That example is correct and gives external grounding that the implementation is not fantasy.\n\nThe paper's best trait is honesty: it says repeatedly that the computational core comes from [19], that correctness is inherited rather than proved here, and that some large applications are infeasible. The complexity tables, even if preliminary, are a practical guide for people choosing GIT problems to attack.\n\nTwo soft spots. First, there is no commit hash, test suite reference, or independent check of the exceptional-group outputs; the reader cannot tell which version of the code produced Tables 2–5. Minor, but fixable with one GitHub link and a run log. Second, and more substantial, the strictly polystable output is genuinely over-sold. The implementation computes strictly T-polystable points for a maximal torus, not strictly G-polystable points. For unstable and non-stable loci the distinction is harmless because those loci are G-invariant and stated up to conjugation. For polystable loci it is not automatic: a T-polystable point can fail to be G-polystable. The paper only bridges the gap for plane cubics, via a dimension/compactness argument, and no general theorem is given that lets the PF_ps entries in Tables 2–5 be read as G-polystable families. That leaves the abstract's claim that CompGIT describes the strictly polystable loci one step beyond what is established.\n\nNet: this is a legitimate software-plus-description paper, not a research breakthrough, and it should not be held to the standard of a proof paper. It earns a serious referee. I would send it to peer review with the request that the polystable claims be either proved in the needed generality or substantially qualified, and that version/test data be included. The package is usable and the worked example is solid; with those revisions, the paper is a good contribution to computational invariant theory.","headline":"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.","tokens_in":17198,"tokens_out":2627,"would_cite":true,"duration_ms":27589,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14L24","14-04"],"pacs":[],"model":"deepseek-v4-flash","headline":"CompGIT computes the stability boundary of GIT quotients by simple groups.","keywords":["geometric invariant theory","GIT quotient","unstable locus","non-stable locus","strictly polystable locus","Hilbert-Mumford criterion","simple algebraic groups","computational algebraic geometry"],"falsifier":"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.","tokens_in":1558,"feed_emoji":"🧮","tokens_out":10464,"duration_ms":158458,"temperature":0.7,"pith_summary":"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.","feed_headline":"Computer package maps the stability boundary of GIT quotients","feed_subtitle":"The tool reduces a hard quotient problem to explicit families of weight vectors.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $E_8$."],"supporting_citations":[{"why":"Supplies the core algorithms and the original code that CompGIT implements and extends to exceptional groups.","marker":"[19]"},{"why":"States the Hilbert–Mumford numerical criterion that reduces stability to one-parameter subgroup weights.","marker":"[33]"},{"why":"Earlier work that the implemented algorithms systematise and generalise.","marker":"[17]"},{"why":"Earlier computational treatment of the nullcone that the paper compares against CompGIT.","marker":"[12]"}],"fun_headline_variants":["CompGIT computes GIT stability for exceptional types","SageMath tool maps GIT quotient stability boundary","Exceptional GIT quotients now explicitly described","Package details stable, unstable, polystable loci"],"cache_read_input_tokens":19328,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["CompGIT computes GIT stability for exceptional types","SageMath tool maps GIT quotient stability boundary","Exceptional GIT quotients now explicitly described","Package details stable, unstable, polystable loci"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1182,"prompt_tokens":866,"completion_tokens":316,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":482,"completion_tokens_details":{"reasoning_tokens":254}},"tokens_in":482,"tokens_out":316,"duration_ms":3722,"temperature":1.0,"reasoning_tokens":254,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:32:34.122040+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Computation of GIT quotients of semisimple groups","cited_arxiv_id":"2308.08049","evidence_quote":"Supplies the core algorithms and the original code that CompGIT implements and extends to exceptional groups."},{"cited_title":"Mumford, J","cited_arxiv_id":null,"evidence_quote":"States the Hilbert–Mumford numerical criterion that reduces stability to one-parameter subgroup weights."},{"cited_title":"Gallardo and J","cited_arxiv_id":null,"evidence_quote":"Earlier work that the implemented algorithms systematise and generalise."},{"cited_title":"Derksen and G","cited_arxiv_id":null,"evidence_quote":"Earlier computational treatment of the nullcone that the paper compares against CompGIT."}],"review_version":2}