{"id":"85dcc38c-4367-4d71-88b1-04bb11674253","arxiv_id":"2607.07900","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Oriented cohomology rings of semisimple groups admit finite presentations via formal Demazure operators; explicit minimal presentations are given for adjoint and simply-connected groups of types A1, A2, B2.","lead":"The paper gives an algebraic recipe that turns formal Demazure operators into a finite presentation of the oriented cohomology ring of any semisimple algebraic group. Specialists can now compute these rings for small-rank groups by hand or machine and recover classical Chow and K-theory as special cases.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates Assumptions 3.8 and 6.6 as the weakest points of the argument; they are load-bearing in the technical sense that Theorems 7.7–7.9 and Proposition 7.8 invoke them, yet they are standard, openly declared, and satisfied by the three principal examples (Chow, K-theory, algebraic cobordism). No further soft spot in the chain from formal Demazure operators to the finite presentation of h^*(G) is visible: the coalgebra structure, the dual algebra, the characteristic map, and the quotient by first Chern classes of homogeneous line bundles are all grounded in cited theorems. The six explicit presentations are new and reproducible. Consequently the ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":26062,"tokens_out":470,"duration_ms":5352,"concrete_test":"Independently recompute the constant terms ε(q^{I_v}_{I_w,I_{w'}}) and ε(Δ_{I_w}(x_{λ_i})) for the adjoint A2 case via the coproduct formula of Proposition 4.6 and the twisted Leibniz rule of Proposition 4.4; feed them into the generators of M+A and verify that the resulting ideal reduces exactly to (3x,x^3) as claimed in Example 8.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 7.9 / Corollary 7.10 + Algorithm 7.11) is a carefully assembled algebraic model of h^*(G) that rests on published results (CPZ13, CZZ16, GZ12) under the explicitly stated Assumptions 3.8 and 6.6. Those assumptions are standard for the theories of interest (CH, K_0, Ω) and are not hidden. The low-rank presentations of Table 1 are obtained by a transparent hand-simplification of machine-generated relations whose source code is linked; no internal inconsistency or unjustified step appears in the derivation of the isomorphism or in the A1/A2/B2 calculations.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs an algebraic model for the oriented cohomology ring h^*(G) of a semisimple algebraic group G over an algebraically closed field of characteristic 0, for oriented cohomology theories h^* that satisfy the localization axiom (and related technical hypotheses). Combining the formal affine Demazure algebra D_F and its dual with the characteristic map and the Gille–Zainoulline relation h^*(G) ≅ h^*(G/B)/(im c_1|L_B), the author obtains an R-algebra isomorphism h^*(G) ≅ D_F^*/(C_{RJΛK_F} + I_F D_F^*) (Theorem 7.9) and an equivalent finite presentation R⟨B^*⟩/(M+A) (Corollary 7.10). Algorithm 7.11 extracts the relations explicitly from coproduct coefficients and constant terms of formal Demazure operators. The method is applied by hand (with machine-generated intermediate relations) to produce the minimal presentations of Table 1 for the adjoint and simply-connected groups of types A_1, A_2 and B_2.","tokens_in":26239,"tokens_out":1073,"duration_ms":11635,"significance":"The result supplies a uniform computational technique for oriented cohomology rings of semisimple groups that goes beyond the classical Chow-ring calculations of Grothendieck and Kac and the partial algebraic-cobordism computations of Yagita. The algebraic model is built from published isomorphisms (CPZ13, CZZ16, GZ12) under explicitly stated assumptions that hold for the principal theories of interest (CH, K_0, Ω). The low-rank presentations in Table 1 are concrete, falsifiable outputs; the linked Python scripts make the intermediate relations reproducible. If the method extends routinely to higher rank, it would become a standard tool for computing h^*(G) in the oriented-cohomology literature.","major_comments":[{"comment":"The B_2 case of Example 8.5 (and the corresponding rows of Table 1) is only sketched: after listing the generators of M and A the text states that “the rest of this calculation is performed by hand and is similar to” the A_2 case. Because the claimed minimal presentation R[x]/(2x-a_{11}x^{2},2x^{2},x^{4}) is one of the paper’s main concrete outputs, the intermediate reductions that eliminate all generators except a single class of degree 1 should be written out (or placed in an appendix) so that a reader can verify the relations without re-running the code.","section":null},{"comment":"Assumption 6.6 (weak birational invariance + CPZ Assumption 13.2 + localization) and Assumption 3.8 (Σ-regularity and regularity of the torsion index) are load-bearing for Theorems 7.7–7.9 and Proposition 7.8. While they hold for CH, K_0 and Ω, the manuscript never states whether the resulting presentations remain valid after base change of R that may kill the torsion index or destroy Σ-regularity. A short remark clarifying the range of coefficients for which Table 1 is known to hold would strengthen the claim.","section":null}],"minor_comments":[{"comment":"In the abstract and Introduction the localization axiom is mentioned, but the full list of hypotheses (Assumptions 3.8 and 6.6) appears only later; a one-sentence pointer in the abstract would help the reader.","section":null},{"comment":"Notation for the dual basis elements Δ_I_w^* is introduced in §4 and then reused heavily in §7–8; a brief reminder at the beginning of Algorithm 7.11 would improve readability.","section":null},{"comment":"The GitHub link [Gana] is given only as a bare URL in the references; adding a short description of what the scripts compute would make the computational claims easier to check.","section":null},{"comment":"Typographical inconsistencies appear in the Cartan-matrix displays of Examples 2.2–2.5 (spacing and alignment) and in a few places where “Λ” is rendered as “Λ” versus plain “Lambda”.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid, carefully written contribution that sits squarely in the CPZ/CZZ/GZ circle. The only real presentational gap is the abbreviated B_2 calculation; once that is expanded the manuscript meets the standard for acceptance. Scope is appropriate for a specialized algebraic-geometry or K-theory journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper gives a usable computational route from formal Demazure operators to a finite presentation of the oriented cohomology ring h^*(G) for split semisimple G. The central new piece is Theorem 7.9: h^*(G) ≃ D_F^* / (C + I D^*), equivalently the free-algebra presentation of Corollary 7.10 whose relations are produced by Algorithm 7.11. That quotient step, together with the six minimal presentations in Table 1, is the actual contribution; the underlying model of h^*(G/B) is already in CPZ13/CZZ16.\n\nWhat works well is the care with which the pieces are assembled. The derivation chains published isomorphisms (CPZ13 Thm 13.13, GZ12 Prop 5.1, CZZ coalgebra structures) with a short, carefully checked lemma on pull-backs of homogeneous line bundles (Prop 5.9). The A1 and A2 hand simplifications are written out completely; the B2 case is longer but follows the same pattern, and the Python scripts that generate the raw relations are linked. Specializing the general presentation recovers the expected Chow and K_0 rings, which is a useful sanity check.\n\nThe soft spots are real but limited. Everything rests on Assumptions 3.8 and 6.6 (Σ-regularity, torsion index invertible, localization, Bott–Samelson basis). Those are standard for CH, K_0 and algebraic cobordism, and the paper states them up front rather than burying them. Only ranks ≤2 are fully simplified by hand; G2 is left as raw output. That is a scope choice, not a flaw in the method. The citation pattern stays inside the expected circle (CPZ, CZZ, GZ, Totaro, Levine–Morel) and is appropriate.\n\nThis is for people who already work with oriented cohomology of flag varieties or who need explicit generators-and-relations for low-rank groups. It does not open a new technology, but it is a solid, reproducible tool inside an established program. I would send it to referees without hesitation; the math is transparent and the tables are immediately usable.","headline":"Clean algebraic pipeline that turns formal Demazure data into finite presentations of h^*(G); the six low-rank tables are new and usable.","tokens_in":26823,"tokens_out":539,"would_cite":true,"duration_ms":6886,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F43","14L30","19L41","20G15"],"pacs":[],"model":"grok-4.5","headline":"Oriented cohomology of semisimple groups admits a finite presentation built from formal Demazure operators, with explicit minimal rings for A1, A2 and B2.","keywords":["oriented cohomology","formal Demazure operators","semisimple algebraic groups","formal group algebra","flag varieties","algebraic cobordism","generators and relations"],"falsifier":"Compute the presentation produced by Algorithm 7.11 for one of the Table-1 groups (say PGL(3)) and check whether the resulting ideal coincides with the known Chow ring or algebraic K-theory of that group; a mismatch would refute the isomorphism of Theorem 7.9.","tokens_in":26960,"feed_emoji":"□","tokens_out":997,"duration_ms":10385,"temperature":0.7,"pith_summary":"The paper shows that, for any oriented cohomology theory that obeys the localization axiom, the ring of a semisimple algebraic group G can be written as a quotient of a free algebra on dual formal Demazure operators by two explicitly described ideals of multiplication and Chern-class relations. The construction uses the formal group algebra of the character lattice and the coalgebra of formal Demazure operators as an algebraic model for the cohomology of the flag variety, then quotients by the image of first Chern classes of homogeneous line bundles. An algorithm turns the coproduct formulas and the twisted Leibniz rule into concrete generators and relations; applying it by hand yields minimal presentations for the six adjoint and simply-connected groups of types A1, A2 and B2. A reader who already knows Chow rings or algebraic cobordism can therefore compute the corresponding rings for these groups without new geometric arguments, and the same method is claimed to extend to other root systems.","feed_headline":"Demazure operators give finite rings for group cohomology","feed_subtitle":"Explicit generators and relations for oriented theories on A1, A2 and B2 groups","key_machinery":"The dual of the formal affine Demazure algebra D_F^*, equipped with the product induced by the cocommutative coproduct on Demazure operators; the two ideals M (multiplication relations coming from that coproduct) and A (Chern-class relations) give the presentation of h^*(G).","core_discovery":"There is an R-algebra isomorphism h^*(G) ≈ D_F^*/(C_RJΛK_F + I_F D_F^*), equivalently a finite presentation R⟨B^*⟩/(M+A) whose relations are computed by a five-step algorithm that extracts constant terms of coproduct coefficients and Demazure actions on first Chern classes; for the adjoint and simply-connected groups of types A1, A2 and B2 the resulting presentations simplify to the six rings listed in Table 1.","pith_inferences":["The algorithm is already coded for rank-2 root systems; extending the code to G2 or A3 would immediately produce candidate presentations that can be checked against known special cases.","Because the construction factors through the simply-connected cover, the method also gives presentations for intermediate groups whose character lattices sit between root and weight lattices.","The same formal-Demazure model may supply generators for the equivariant oriented cohomology of flag varieties, not only the ordinary rings of the groups themselves."],"forward_implications":["Any oriented theory satisfying the listed axioms acquires an explicit finite presentation for h^*(G) once the coproduct coefficients of the Demazure operators are known.","Specialization of the free formal group law recovers the classical Chow rings and Grothendieck rings of the six groups as the rings of Table 1.","The same generators-and-relations description applies, with only larger Weyl-group data, to every other semisimple root datum whose formal group algebra is Σ-regular.","Computer algebra can output the full list of relations for higher-rank groups; hand simplification then yields minimal presentations."],"fun_headline_variants":["Demazure operators yield finite generators for oriented group cohomology","Finite presentations of h*(G) via formal Demazure operators","Minimal rings for A1 A2 B2 adjoint and simply-connected groups","Five-step algorithm gives generators and relations for h*(G)","Oriented cohomology rings of low-rank groups via Demazure calculus"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The cohomology theory must be weakly birationally invariant, satisfy localization, and have Bott–Samelson classes as a free basis of the flag variety, while the formal group algebra must be regular with respect to roots and the torsion index of the root datum must be invertible in the coefficient ring.","fun_headline_variants_meta":{"raw":{"variants":["Demazure operators yield finite generators for oriented group cohomology","Finite presentations of h*(G) via formal Demazure operators","Minimal rings for A1 A2 B2 adjoint and simply-connected groups","Five-step algorithm gives generators and relations for h*(G)","Oriented cohomology rings of low-rank groups via Demazure calculus"]},"model":"grok-4.5","effort":"low","cost_usd":0.00379,"raw_usage":{"total_tokens":1097,"prompt_tokens":650,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":37900000,"prompt_tokens_details":{"text_tokens":650,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":358,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":650,"tokens_out":89,"duration_ms":3868,"temperature":1.0,"reasoning_tokens":358,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T15:45:53.515471+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the presentation produced by Algorithm 7.11 for one of the Table-1 groups (say PGL(3)) and check whether the resulting ideal coincides with the known Chow ring or algebraic K-theory of that group; a mismatch would refute the isomorphism of Theorem 7.9.","supporting_citations":[],"review_version":1}