{"id":"73708bc1-09ef-476e-9eb8-a979e29ed103","arxiv_id":"2501.15778","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that changing the Borel subgroup for GL(X) in Ver_p is governed by lowest weights of GL(L_m|L_n), computed via circular weight and cap diagrams.","lead":"This paper works out how to translate between different highest weight labelings for representations of the group GL(X) inside the Verlinde category, a positive-characteristic algebraic structure. It reduces the problem to explicit circle diagram algorithms, opening the door to computational representation theory in these categories.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The transfer of the Shapovalov vanishing criterion in §9.3.3 is asserted without proof; since it underlies Theorem 9.35 and hence the induction in Theorem 9.46, the central multiplicity and lowest-weight claims remain conditional on this step.","rationale":"The reader and I identify the same gap. The central claim of the paper is an explicit, algorithmic translation between Borel labelings (Question 8.6). Its engine is Theorem 9.46, and the induction in Theorem 9.46 starts from projective typical Kac modules. That starting point is established only through the Shapovalov transfer in §9.3.3, which is asserted in one sentence. The rest of the proof follows a coherent strategy: highest weight category structure, translation functors, weight diagrams, and cap diagrams. I see no internal contradiction in those sections, and the categorical sl_p action is a genuinely strong tool. However, because the base case is not independently verified and a semisimplification functor can in principle change scalar morphisms, the main theorem should not be regarded as fully proved until the transfer is either proved or checked by an independent computation. This does not require changing the reader's verdict: CONDITIONAL remains the right assessment.","tokens_in":53777,"tokens_out":9383,"duration_ms":92754,"concrete_test":"For p=5, m=n=2, take λ=(2,0|0,0), for which ⟨λ+ρ,ε_1−δ_2⟩=0, so the classical criterion predicts Sh(λ)=0. Compute Sh(µ|ν) directly in Rep_{Ver_p}(GL(L_2|L_2)): express the map from the top degree 4 component to the degree 0 component in an explicit basis of the weight spaces and evaluate the resulting scalar. If it is nonzero, Corollary 9.17 is false; if zero, repeat for a typical weight such as λ=(1,0|0,0), where the classical constant is nonzero modulo 5, to check that S does not kill a nonzero scalar. This isolates the transfer assertion independently of the cap-diagram induction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing unproved step is the equality Sh(µ|ν)=Sh(λ) in §9.3.3. This is the entire bridge from Kac's classical criterion (Theorem 9.13) to Corollary 9.17, and Corollary 9.17 is what makes Theorem 9.35 ('typical Kac modules are projective') true. Theorem 9.35 is the base case of the induction in Theorem 9.46, so every standard-filtration multiplicity in Theorem 9.46(2) and the lowest-weight formula in Theorem 9.47 ultimately rests on this equality. The paper does not prove that the semisimplification functor S preserves the scalar attached to the Shapovalov pairing on the top component S^{mn}(n_+)⊗S^{mn}(n_-)⊗L_ev(λ)→L_ev(λ). S is a quotient by negligible morphisms; a nonzero scalar in characteristic p can become zero, or a map with the same categorical description could have a different scalar after S because the relevant dual pairing involves evaluation and coevaluation morphisms whose images under S are not tracked. If the constants differed, the irreducibility criterion of Corollary 9.17, the projectivity of typical Kac modules, the cap-diagram multiplicity formula, and the final algorithmic answer to Question 8.6 would all be unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies representations of GL(X) in the Verlinde category Ver_p, where X is an object of Ver_p, and answers the question of how highest weight labels change when the Borel subgroup is changed by permuting non-isomorphic simple summands. The reduction via odd reflections (Theorem 8.10) isolates the category Rep_{Ver_p}(GL(L_m⊕L_r)), reinterpreted as GL(L_m|L_n) with n=p−r. For this category the paper establishes a highest weight structure with Kac modules as standard objects (Theorems 9.3 and 9.4), BGG reciprocity (Corollary 9.5), projectivity and injectivity of projectives (Theorem 9.8), a categorical sl_p-action (Theorem 9.21), an irreducibility criterion for Kac modules (Corollary 9.17 and Theorem 9.35), a circular weight diagram calculus (Section 9.6), cap-diagram formulas for standard filtration multiplicities of projectives (Theorem 9.46), and a lowest weight formula for simple modules (Theorem 9.47), yielding an algorithmic answer to Question 8.6.","tokens_in":54025,"tokens_out":11406,"duration_ms":109620,"significance":"If the proof gaps identified below are closed, the paper is a significant contribution: it gives the first systematic highest weight theory for GL(X) in Ver_p and makes the change-of-Borel labeling explicit and computable. The categorical sl_p-action and the circular weight diagrams are natural and well chosen, and the use of external inputs (Venkatesh's classification, Deligne reconstruction, Kac's theorem, Brundan–Kujawa, Losev, CEOK23) is clearly acknowledged. The central statements are coherent and the overall architecture is convincing, but several load-bearing steps are asserted rather than proved.","major_comments":[{"comment":"The equality Sh(µ|ν)=Sh(λ) is the bridge from Kac's classical irreducibility criterion (Theorem 9.13) to Corollary 9.17, and hence to Theorem 9.35, the base case of the induction in Theorem 9.46. The manuscript asserts this equality with the sentence 'as morphisms map to morphisms under the semisimplification', but the semisimplification functor is a quotient by negligible morphisms and does not by itself preserve the scalar attached to a dual pairing or the evaluation and coevaluation maps used to define Sh. A nonzero scalar in characteristic p can become zero, or the categorical description of the pairing can change after applying S. Since Corollary 9.17, Theorem 9.35, the cap-diagram multiplicity formula in Theorem 9.46, and the lowest weight theorem 9.47 all rest on this transfer, a direct proof or a detailed categorical comparison of the Shapovalov constants is required.","section":"9.3.3"},{"comment":"Definition 9.41 and the surrounding text assert several combinatorial properties of cap diagrams: uniqueness, non-intersection of caps, existence of at least one free ◦, and the absence of free ◦ inside any cap. The paper says these are 'easy to see', but they are load-bearing: the induction in Theorem 9.46 requires, at each step, the existence of an innermost cap with only arrows in its interior, and the tracking of the set P(λ) in part (2) requires that the transformations τ_i act independently on the relevant vertices. Without a proof of the uniqueness and nesting properties of the cap diagram algorithm, the induction and the formula [P(λ):K(α)] ∈ {0,1} in Theorem 9.46(2) are not fully justified. These combinatorial facts should be stated as a lemma and proved.","section":"9.7.3 / Theorem 9.46"},{"comment":"The proof that projective objects are injective asserts that the projection S(n_+⊕n_-)→S^{2mn}(n_+⊕n_-)=1 induces a nondegenerate pairing making Dist(G) a Frobenius extension of Dist(T). This nondegeneracy is not automatic in a symmetric tensor category in positive characteristic, and the statement is used to conclude that P(λ)≃I(λ) and hence that P(λ) has irreducible socle in Corollary 9.11. That socle statement is in turn used in the proof of Theorem 9.47. A proof of the nondegeneracy of the pairing, or a precise reference to a categorical Frobenius-extension lemma, should be supplied.","section":"9.2 / Theorem 9.8"}],"minor_comments":[{"comment":"The displayed formulas 'a_i := µ_i + i − 1' and 'b_j := m − (ν_j − 1)' are inconsistent with Section 9.6 and with Example 9.26; they should read a_i = µ_i − i + 1 and b_j = −m − ν_j + j (mod p). Since these formulas define the weight diagrams used in Theorems 9.46 and 9.47, the correction is important even though the intended convention is recoverable.","section":"Definition 9.24"},{"comment":"The last coordinate of the displayed vector λ should be −18, not 18, to agree with Example 9.49.","section":"Example 9.51"},{"comment":"In the definition of the poset Λ, the condition 'ν_1 ≥ ... ≥ ν_m' should be 'ν_1 ≥ ... ≥ ν_n', since ν has n coordinates.","section":"Section 9.2"},{"comment":"The local rules for the action of F_i and E_i on weight diagrams are stated without proof; since all subsequent diagram computations rely on them, a short derivation from Theorem 9.21 would improve the exposition.","section":"Theorem 9.30"},{"comment":"The reference [CPS88] lacks full publication data; please provide the complete bibliographic information.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is well within the journal's scope and the overall architecture is convincing. The main risk is the Shapovalov-constant transfer in Section 9.3.3; if the author can supply a proof of that equality and of the cap-diagram combinatorial lemma, I would support publication. The typo in Definition 9.24 should be corrected given how central the weight diagrams are."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing you should know: this paper likely gives the right answer, but one load-bearing step is asserted, not proved. The paper develops highest weight theory for GL(X) in Ver_p: translation between Borel labelings via odd reflections, a categorical sl_p-hat action, circular weight diagrams, and explicit cap-diagram formulas for standard filtration multiplicities and lowest weights. Sections 8 and 9 are a genuine advance; the algorithmic answer to the Borel-translation question is new and the analogy to GL(m|n) is used with care. It builds properly on Venkatesh, CEOK23, Brundan–Stroppel and cites them; I don't see a circularity problem.\n\nThe soft spot is exactly where the reader's stress test points. In §9.3.3 the paper says Sh(µ|ν) = Sh(λ) because under semisimplification morphisms map to morphisms. That equality is not automatic: the semisimplification functor is a quotient by negligible morphisms, and the scalar attached to the dual pairing on the top component could change or vanish. This equality is what turns Kac's classical criterion into Corollary 9.17, which gives projectivity of typical Kac modules (9.35), which is the base case of the induction in Theorem 9.46 and hence underlies the lowest-weight theorem 9.47. If the constants don't match, the whole computation is unsupported. This needs a real proof, not an assertion.\n\nTwo smaller soft spots: cap diagram uniqueness and nesting in §9.7.3, and the submodule argument in the induction in 9.46, are also compressed. They look plausible and probably repairable, but they're stated rather than derived. I would not call them fatal on their own; the Shapovalov transfer is the one I'd want checked first.\n\nWho is this for? Specialists in positive-characteristic tensor categories and in supergroup representation theory. The paper is long and notation-heavy, but the payoff is concrete: an explicit, algorithmically usable translation rule. It deserves a serious referee; it should not be desk rejected. My recommendation: send it to review, with the referee explicitly asked to verify §9.3.3 and the cap-diagram induction, and require the author to fill those gaps before acceptance.","headline":"A substantial, mostly convincing advance, but the Shapovalov transfer in §9.3.3 is asserted and everything downstream rests on it.","tokens_in":54615,"tokens_out":2857,"would_cite":true,"duration_ms":28329,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B10","20G05","18M05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives an explicit, computable rule for translating highest weight labels between different Borel subgroup labelings in representations of general linear groups in the Verlinde category Ver_p, reducing the general case to the…","keywords":["Verlinde category","positive characteristic","highest weight category","Kac modules","weight diagrams","categorical sl_p action","odd reflections","lowest weights"],"falsifier":"For a small characteristic p and a weight with at least one cross in its circular diagram (for instance p = 5, m = 1, n = 1, so X = L_1 ⊕ L_4), directly compute the top-component scalar Sh(µ|ν) inside Ver_p by dualizing the action $S^{{1}}$(n_+) ⊗ $S^{{1}}$(n_-) → /BD and compare it with the classical product ∏⟨λ+ρ,α⟩ over positive odd roots reduced modulo p; a mismatch would invalidate Corollary 9.17 and hence Theorems 9.46 and 9.47.","tokens_in":53533,"feed_emoji":"🔁","tokens_out":4571,"duration_ms":42743,"temperature":0.7,"pith_summary":"The paper aims to answer how the highest weight labels of simple modules change when one switches between different Borel subgroups of a general linear group in the Verlinde category Ver_p. It reduces the problem to representations of the group GL(L_m ⊕ L_n), which it treats as an analogue of the supergroup GL(m|n). For this category it builds a full highest weight theory: projective objects, BGG reciprocity, duality, a categorical action of the affine Lie algebra \\hat{\\mathfrak{sl}}_p, and an irreducibility criterion for Kac modules. The main payoff is an algorithmic description, using circular weight diagrams and cap diagrams, of the multiplicities in standard filtrations of indecomposable projective modules and of the lowest weight of every simple module. If correct, the translation between different Borel labelings becomes a concrete, computable procedure rather than an abstract existence statement.","feed_headline":"Odd reflections unlock every Borel labeling in Ver_p","feed_subtitle":"An explicit cap-diagram algorithm computes lowest weights and projective multiplicities for GL(L_m|L_n).","key_machinery":"The central object is the circular weight diagram of an admissible weight (µ|ν) in $Z^{{m|n}}$: the symbols <, >, ×, and ◦ placed on the vertices of a regular p-gon, together with the cap diagram that connects each × to a suitable ◦. These diagrams encode the combinatorial action of translation functors F_i, E_i, which form a categorical \\hat{\\mathfrak{sl}}_p-action on C as shown by the isomorphism [C]_Δ ≅ (Λ^m U)_{loop} ⊗ (Λ^n U^*)_{loop}. The cap diagram provides the transitions between indecomposable projectives, while the Shapovalov-type constant Sh(µ|ν), transferred from the classical supergroup GL(m|n) via semisimplification, supplies the irreducibility criterion for Kac modules.","core_discovery":"For the category C = Rep_{Ver_p}(GL(L_m|L_n)), the paper proves that the standard filtration multiplicities of an indecomposable projective module P(λ) are given by [P(λ):K(α)] = 1 if α belongs to the set P(λ) obtained by swapping crosses with circles along the caps of the weight diagram of λ, and 0 otherwise (Theorem 9.46). Consequently, the lowest weight of the simple module L(λ) is \\hat{λ} − β, where \\hat{λ} is the weight obtained by performing all these cap swaps and β is the sum of all positive odd roots (Theorem 9.47). Since any permutation of non-isomorphic simple summands decomposes into odd reflections, Theorem 8.10 turns these lowest-weight computations into an algorithmic answer to Question 8.6 for arbitrary GL(X) in Ver_p.","pith_inferences":["If the cap-diagram formalism is correct, it suggests that the representation theory of general linear groups in Ver_p admits character formulas of Kazhdan–Lusztig type, but with the affine Weyl group and the p-dependence built into the circular geometry rather than into a straight-line diagram.","The circular diagrams differ genuinely from the classical GL(m|n) diagrams even in characteristic p; the paper itself notes that for some weights the lowest weight computed here differs from the one obtained by Serganova's algorithm, indicating new phenomena specific to the Verlinde setting.","A natural testable extension would be to compare the multiplicities [P(λ):K(α)] with the characters obtained by Brundan's Kazhdan–Lusztig theory for gl(m|n) after reduction modulo p, to see whether the cap-diagram rules interpolate the known classical formulas.","The reduction to odd reflections suggests that the action of the full permutation group on Borel labelings is controlled by a finite set of local lowest-weight moves; one could attempt to package these moves into a braid-group action or a cellular structure on the category."],"forward_implications":["If the main theorem holds, the complete translation between highest weight labelings for different Borel subgroups of GL(X) in Ver_p is explicit: one iterates the lowest-weight computation on consecutive pairs of non-isomorphic simple summands.","The category Rep_{Ver_p}(GL(L_m|L_n)) is a highest weight category in the sense of Cline–Parshall–Scott, with projective objects also injective and with BGG reciprocity relating standard filtration multiplicities to composition multiplicities.","Kac modules are irreducible and projective exactly when the weight is typical, i.e. when its degree of atypicality is zero, matching the classical GL(m|n) criterion after the transfer of the Shapovalov condition.","The categorical \\hat{\\mathfrak{sl}}_p-action identifies the Grothendieck group generated by Kac modules with a tensor product of loop wedge modules, which predicts how translation functors act on the whole category and on its simple objects.","The lowest-weight formula from Theorem 9.47 also gives the dual of L(λ): one has L(λ)^* ≅ L(β − \\hat{λ}), as stated in Corollary 9.52."],"supporting_citations":[{"why":"Supplies the classification of simple GL(X)-modules in Ver_p and the framework of generalized Verma modules that the paper extends to different Borel subgroups.","marker":"[Ven24]"},{"why":"Provides the fiber functor to Ver_p for Frobenius exact symmetric tensor categories, motivating the study of group schemes inside Ver_p.","marker":"[CEOK23]"},{"why":"Introduces the translation functor and weight diagram method for gl(m|n) that is adapted here to the circular Ver_p setting.","marker":"[Bru03]"},{"why":"Gives the highest weight category and diagram framework for the general linear supergroup, the direct source of the diagram conventions used in Section 9.6.","marker":"[BS12]"},{"why":"Provides the model for computing standard filtration multiplicities of indecomposable projective modules for supergroups via cap diagrams, which Theorem 9.46 transfers to Ver_p.","marker":"[GS11]"},{"why":"States the classical irreducibility criterion for Kac modules in terms of the Shapovalov constant, whose transfer to Ver_p is the load-bearing step of Section 9.3.","marker":"[Kac06]"},{"why":"Supplies the method of odd reflections for quasi-reductive supergroups, which the paper adapts to reduce the general Borel-translation question to GL(L_m ⊕ L_n).","marker":"[Ser11]"},{"why":"Provides the template for treating type I weight modules with Kac modules as standard objects, used in the proof that C is a highest weight category.","marker":"[Zou96]"},{"why":"Gives the axiomatic definition of highest weight categories that is verified for Rep_{Ver_p}(GL(L_m|L_n)).","marker":"[CPS88]"},{"why":"Supplies the semisimplification machinery that constructs Ver_p and is used to transfer structures from the classical supergroup setting.","marker":"[EO21]"}],"fun_headline_variants":["Odd reflections decode Borel labelings in Ver_p","Cap swaps give lowest weights for GL in Ver_p","Odd reflections compute projective multiplicities in Ver_p","Borel labelings unified by odd reflections in Ver_p","New formula for projective multiplicities via cap swaps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that when the classical supergroup construction is passed to the Verlinde category, the scalar measuring the Shapovalov form stays the same, so that Kac's irreducibility criterion carries over unchanged.","fun_headline_variants_meta":{"raw":{"variants":["Odd reflections decode Borel labelings in Ver_p","Cap swaps give lowest weights for GL in Ver_p","Odd reflections compute projective multiplicities in Ver_p","Borel labelings unified by odd reflections in Ver_p","New formula for projective multiplicities via cap swaps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000632,"raw_usage":{"total_tokens":2897,"prompt_tokens":900,"completion_tokens":1997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":1922}},"tokens_in":516,"tokens_out":1997,"duration_ms":13827,"temperature":1.0,"reasoning_tokens":1922,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T13:58:07.499152+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small characteristic p and a weight with at least one cross in its circular diagram (for instance p = 5, m = 1, n = 1, so X = L_1 ⊕ L_4), directly compute the top-component scalar Sh(µ|ν) inside Ver_p by dualizing the action $S^{{1}}$(n_+) ⊗ $S^{{1}}$(n_-) → /BD and compare it with the classical product ∏⟨λ+ρ,α⟩ over positive odd roots reduced modulo p; a mismatch would invalidate Corollary 9.17 and hence Theorems 9.46 and 9.47.","supporting_citations":[],"review_version":1}