{"id":"036b3831-830f-43f8-9372-c99756221968","arxiv_id":"1908.06251","paper_version":3,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors define a minimal parabolic BGG category Omin for Cartan type Lie superalgebras and classify its blocks and the characters of its projective and tilting modules.","lead":"This paper defines a minimal parabolic BGG category for Lie superalgebras of Cartan type and classifies its blocks, then gives character formulas for indecomposable projective and tilting modules. It imports a general categorical framework from Brundan and Soergel and applies it to the W, S, and H families, reobtaining a prior finite-dimensional block result as a special case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The \\bar H block theorem rests on Corollary 4.7; Lemma 4.6's idempotent argument needs an independent check.","rationale":"The reader identified Corollary 4.7 as the weakest assumption, and my reading agrees: it is the essential bridge from CH(n) to \\bar H(n) in the block classification, and its proof via Lemma 4.6 contains a delicate cancellation/linear-independence step that is not fully justified. I did not find a more load-bearing concern: the W(n) and \\bar S(n) block theorems follow different routes, and the character formulas in Theorems 7.2 and 7.3 are derived from the block results plus Soergel reciprocity, so they inherit any weakness in the \\bar H case. The proof of Lemma 4.6 is internal rather than cited, and the specific step where the induction is asserted to work 'by similar arguments' is not written out. An explicit finite-dimensional computation for the smallest Hamiltonian cases would settle whether the lemma is valid. Since this concern is exactly the reader's weakest assumption and does not require changing the already-conditional verdict, I recommend UNCHANGED: the paper should remain CONDITIONAL pending verification of Corollary 4.7.","tokens_in":46615,"tokens_out":7743,"duration_ms":85812,"concrete_test":"Independently verify Lemma 4.6 for the minimal cases n=5 and n=6 by constructing \\Delta(\\lambda)_{CH(n)} explicitly as U(CH(n))\\otimes_{U(P)}L_0(\\lambda) with \\lambda=0 and a generic dominant weight, and solving directly for \\bar H(n)-endomorphisms \\varphi that are idempotent and vanish on \\Delta(\\lambda)_{\\bar H(n)}. In particular, test the k=1 identity used in the proof: check whether D_j\\varphi(D_H(\\xi_1\\cdots\\xi_n)\\otimes v^0_\\lambda) = D_H(\\xi_1\\cdots\\hat\\xi_j\\cdots\\xi_n)\\otimes v^0_\\lambda + \\sum_i[D_j,u_i]\\otimes v_i can hold with nonzero right-hand side, using an explicit basis and weight-space filtration. If any nonzero idempotent with zero restriction to \\Delta(\\lambda)_{\\bar H(n)} exists, Corollary 4.7 and Theorems 4.14 and 4.16 fail; otherwise the transfer argument is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central block classification for \\bar H(n) is transferred from CH(n) through Corollary 4.7, which asserts that every standard CH(n)-module remains indecomposable on restriction to \\bar H(n). This corollary is used in Proposition 4.8(iv) to put all composition factors of I(\\lambda)_{\\bar H(n)} in one block, and it directly underpins Theorems 4.14 and 4.16. The proof of Corollary 4.7 rests entirely on Lemma 4.6, whose key step is not reduced to a standard reference. Specifically, Lemma 4.6 reduces to showing that a \\bar H(n)-endomorphism \\varphi of \\Delta(\\lambda)_{CH(n)} vanishing on \\Delta(\\lambda)_{\\bar H(n)} is zero. The induction on k writes \\varphi((D_H(\\xi_1\\cdots\\xi_n))^k\\otimes v^0_\\lambda) = a(D_H(\\xi_1\\cdots\\xi_n))^k\\otimes v^0_\\lambda + \\sum_i w_i\\otimes\\nu_i with w_i in U(\\bar H(n)_{\\ge1})B and B spanned by lower powers of D_H(\\xi_1\\cdots\\xi_n), and then claims that the case k=1 forces a=0 by a contradiction involving D_j(D_H(\\xi_1\\cdots\\xi_n)\\otimes v^0_\\lambda) = D_H(\\xi_1\\cdots\\hat\\xi_j\\cdots\\xi_n)\\otimes v^0_\\lambda. The contradiction assumes that the resulting weight-space expression cannot cancel, but the linear independence needed here is not explicitly proved. Since \\Delta(\\lambda)_{CH(n)} is free over a suitable \\bar H(n)-submodule with basis the powers of D_H(\\xi_1\\cdots\\xi_n), a nonzero \\varphi could in principle mix the top power into lower powers while vanishing on the \\bar H(n)-submodule; excluding this is exactly the delicate point. Thus the \\bar H block classification is conditional on a lemma whose proof deserves independent verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a parabolic BGG category Omin for the graded Cartan-type Lie superalgebras W(n), \\bar S(n), and \\bar H(n), with P = g_{-1} ⊕ g_0 as a minimal parabolic. The main results are: existence of projective covers of all simple objects with standard flags (Theorem 3.2); a complete block classification for Omin in terms of parameters (c, parity, depth) for the three families (Theorems 4.12, 4.14, 4.16); degenerate BGG reciprocity (Theorem 5.3) and Soergel reciprocity (Propositions 6.4 and 6.6); and explicit character formulas for indecomposable projective and tilting modules (Theorems 7.2 and 7.3). The arguments use Brundan's general framework for graded Lie superalgebras, Soergel's tilting theory for semi-infinite characters, and Serganova's Kac-module character theory.","tokens_in":47013,"tokens_out":10382,"duration_ms":103307,"significance":"If the main theorems are correct, this provides the first systematic block and tilting theory for a non-classical, non-basic Lie superalgebra setting, and it gives concrete, checkable character formulas. The paper makes its external inputs explicit: the semi-infinite characters are verified in Appendix A, the dependence on Serganova's Kac-module multiplicities is clearly identified, and the block description recovers Shomron's finite-dimensional block result for W(n) as a special case. These are genuine strengths and make the statements falsifiable.","major_comments":[{"comment":"The proof of the delicate assertion that every standard CH(n)-module remains indecomposable as an \\bar H(n)-module is incomplete. In the k=1 step, after writing φ(D_H(ξ_1...ξ_n)⊗v^0_λ) = c D_H(ξ_1...ξ_n)⊗v^0_λ + ∑ u_i⊗v_i, the case c=1 is ruled out by applying D_j and comparing with φ(D_H(ξ_1...ξ̂_j...ξ_n)⊗v^0_λ)=0. This comparison only yields a contradiction if the term D_H(ξ_1...ξ̂_j...ξ_n)⊗v^0_λ cannot be cancelled by the remaining sum ∑ [D_j,u_i]⊗v_i; no linear-independence or weight argument is provided, and the same gap is carried into the induction step. Since Corollary 4.7 is the key bridge from CH(n) to \\bar H(n) in Proposition 4.8(iv), Theorems 4.14 and 4.16 are currently conditional on this missing verification.","section":"§4.2, Lemma 4.6"},{"comment":"This proposition is essential for the \\bar S(n) case, but the proof is only sketched. In the two subcases of Case 2, the relations λ∼ν1 and λ∼ν2 are asserted to follow from [14, Theorem 2.10] and 'similar proof as in the Claim'; however, the specific g0-composition factor that appears in the relevant tensor product is not identified, and the existence of the n+-maximal vector of the required weight is not demonstrated. Since Proposition 4.8(ii) depends on Proposition 4.5, this is a load-bearing gap that should be expanded.","section":"§4.3, Proposition 4.5"},{"comment":"The Depth Lemma is used in the converse parts of all three block-classification theorems, but Claim I assumes without proof that a composition factor L(µ) of ∆(λ) has a maximal vector in U(g≥1)_i⊗L0(λ) with dpt(L(µ)) = dpt(L(λ)) + ℓ(µ−λ). For the \\bar H cases, where roots carry both ϵ- and δ-components, the identification of the degree shift with ℓ(µ−λ) needs an explicit argument. Please spell out this bookkeeping.","section":"§4.5, Lemma 4.10"}],"minor_comments":[{"comment":"The abstract states that the authors show there are only two proper parabolic subalgebras containing the Levi g0, but this result is not stated or proved in the body of the paper; please either add the argument or revise the abstract.","section":"Abstract"},{"comment":"There are several typographical and grammatical issues, e.g., 'Sectoin' in Section 0.5, 'Sogerel' in Section 0.4, and 'argumnets' in the proof of Proposition 4.5; the paper would benefit from a careful copyedit.","section":"Throughout"},{"comment":"In Theorems 4.12, 4.14, and 4.16, the quotient notation C/Z and C/2Z is used without explanation; the parametrization by c mod Z (resp. c mod 2Z) should be made explicit in the notation section.","section":"§4.6–4.7"},{"comment":"The phrase 'a block B of g-modf is a subcategory ... satisfying that all its composition factors lie in the same block' is slightly circular because blocks of g-modf are being defined through blocks of F(Omin); please reformulate to avoid ambiguity.","section":"§4.8, Definition 4.20"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are likely correct, but the proof of Lemma 4.6 must be completed before the \\bar H block classification can be fully relied upon. I would also encourage the authors to make Proposition 4.5 self-contained. The paper's reliance on Serganova's character formulas is legitimate and well identified; no novelty concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper gives the first block classification and character formulas for a parabolic BGG category attached to the Cartan type Lie superalgebras W(n), S-bar(n), and H-bar(n), and that is a real result. Read it for the W and S-bar cases; the H-bar case is promising but currently rides on a lemma that lacks a complete proof.\n\nWhat is genuinely new: the category Omin, defined via the minimal parabolic g_{-1}+g_0, is new, and the block classification in terms of depth and parity, plus the projective and tilting character formulas, are new. The paper correctly reobtains Shomron's finite-dimensional W(n) block result as a special case. The machinery from Brundan and Soergel is applied properly, and the semi-infinite character computations are explicit and checkable. The Kac-module realization of co-standard modules is a nice step and makes the BGG and Soergel reciprocities concrete.\n\nNow the soft spot. For H-bar(n), the transfer from CH(n) blocks rests on Corollary 4.7, which says every standard CH(n)-module stays indecomposable on restriction to H-bar(n). The stress-test note is right: Lemma 4.6's induction needs a linear independence statement that is not proved. In the k=1 case, the argument asserts that D_j applied to the expression forces a contradiction because the resulting term cannot cancel with the other terms, but that cancellation is exactly what needs a weight-space or PBW argument. Without it, the idempotent endomorphism could mix the top power into lower powers. The lemma may well be true; it is plausible. But as written, the H-bar block theorems are conditional on that lemma. I would not reject the paper for this, but a referee should demand a complete proof.\n\nMinor issues: Proposition 4.5 partly says 'similar arguments', and Remark 4.22(2) asserts all blocks are wild without proof. Both are minor relative to the main theorems.\n\nWho is this for: anyone working on categories O for Lie superalgebras or on Cartan type representations. It deserves a serious referee. Send it to review, and ask the authors to fix Lemma 4.6 or supply a standard reference. If that lemma gets a clean proof, the paper is a solid contribution.\n\nBest,","headline":"New block classification and character formulas for a minimal parabolic BGG category in three Cartan-type Lie superalgebra families; the W and S-bar cases look solid, while the H-bar case is conditional on an indecomposability lemma that needs a cleaner proof.","tokens_in":47529,"tokens_out":2579,"would_cite":false,"duration_ms":27919,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B10","17B66","17B70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper classifies all blocks of the minimal parabolic BGG category for the Cartan-type Lie superalgebras W(n), \\bar S(n), and \\bar H(n), and computes projective and tilting characters.","keywords":["Lie superalgebras of Cartan type","parabolic BGG category","blocks","projective covers","tilting modules","character formulas","semi-infinite character","minimal parabolic subalgebra"],"falsifier":"Compute $\\operatorname{Hom}_{\\bar H(5)}(\\Delta(\\lambda)_{CH(5)},\\Delta(\\lambda)_{CH(5)})$ for a dominant integral weight $\\lambda$ and check whether it contains any idempotent besides $0$ and the identity; a nontrivial idempotent, or a direct-sum decomposition of $\\Delta(\\lambda)_{CH(5)}$ over $\\bar H(5)$, would disprove Corollary 4.7 and with it the $\\bar H(2r+1)$ block classification. For the even case, finding $L(\\lambda)$ and $L(\\lambda+\\delta)$ sharing a projective cover would contradict Theorem 4.16.","tokens_in":46430,"feed_emoji":"📐","tokens_out":13192,"duration_ms":114644,"temperature":0.7,"pith_summary":"This paper develops a parabolic BGG category $\\mathcal{O}^{\\min}$ for the Cartan-type Lie superalgebras $W(n)$, $\\bar S(n)$, and $\\bar H(n)$, built from the minimal parabolic subalgebra $\\mathfrak{p} = \\mathfrak{g}_{-1} \\oplus \\mathfrak{g}_0$. Its central claim is a complete block classification: simple objects lie in the same block exactly when their weights differ by a root-lattice element and their depth and parity shift by the length of that element, yielding explicit block parameter sets $\\mathbb{C}/\\mathbb{Z} \\times \\mathbb{Z}_2 \\times \\mathbb{Z}$ (with a second $\\mathbb{C}/\\mathbb{Z}$ or $\\mathbb{C}/2\\mathbb{Z}$ factor for the two Hamiltonian families). The paper also proves that every simple object has a projective cover with a standard-module flag, establishes degenerate BGG and Soergel reciprocity, and extracts explicit character formulas for indecomposable projective and tilting modules. A sympathetic reader will care because these results give a uniform block and character theory for the parabolic category of the Cartan-type series, and they include the finite-dimensional $W(n)$ block decomposition as a special case.","feed_headline":"All blocks classified in Cartan-type Lie superalgebra category Omin","feed_subtitle":"Explicit character formulas for projective and tilting modules follow, with finite-dimensional W(n) blocks as a special case.","key_machinery":"The load-bearing construction is the minimal parabolic subalgebra $\\mathfrak{p}=\\mathfrak{g}_{-1}\\oplus\\mathfrak{g}_0$ and the induced standard modules $\\Delta(\\lambda)=U(\\mathfrak{g})\\otimes_{U(\\mathfrak{p})}L_0(\\lambda)$, with $\\lambda$ a dominant integral weight of $\\mathfrak{g}_0$. Around these sits the enveloping projective module $I(\\lambda)=U(\\mathfrak{g})\\otimes_{U(\\mathfrak{g}_0)}L_0(\\lambda)$, which admits a finite $\\Delta$-flag and from which projective covers are extracted as indecomposable summands. Two auxiliary mechanisms carry the block argument: depth and parity lemmas that force the shift conditions in the block parameters, and, for the Hamiltonian case, an idempotent-endomorphism lemma proving that standard modules for $CH(n)$ remain indecomposable over $\\bar H(n)$. On the character side, co-standard modules are realized as the induced modules $K(\\lambda)$ through a Frobenius-extension argument, so the known character formulas for those induced modules feed directly into the final formulas.","core_discovery":"The paper claims that the category $\\mathcal{O}^{\\min}$ associated with the minimal parabolic $\\mathfrak{p}=\\mathfrak{g}_{-1}\\oplus\\mathfrak{g}_0$ has a complete block theory for $\\mathfrak{g}=W(n)$, $\\bar S(n)$, and $\\bar H(n)$. For $W(n)$ and $\\bar S(n)$ the blocks are exactly the sets $\\mathcal{O}^{\\min}(c,\\iota,i)$ indexed by $(c,\\iota,i)\\in\\mathbb{C}/\\mathbb{Z}\\times\\mathbb{Z}_2\\times\\mathbb{Z}$; for $\\bar H(2r+1)$ they are $\\mathcal{O}^{\\min}(c,d,\\iota,i)$ indexed by $(\\mathbb{C}/\\mathbb{Z})^2\\times\\mathbb{Z}_2\\times\\mathbb{Z}$; and for $\\bar H(2r)$ the first parameter is $\\mathbb{C}/2\\mathbb{Z}$ instead of $\\mathbb{C}/\\mathbb{Z}$, so $L(\\lambda)$ and $L(\\lambda+\\delta)$ never share a block. Within a block, depth changes by the integer length $\\ell(\\lambda-\\mu)$ and the parity of a maximal vector changes by the parity of that length. The same framework yields $[P(\\lambda):\\Delta(\\mu)] = (\\nabla(\\mu):L(\\lambda))$ and Soergel reciprocity for tilting modules, and the character formulas express every indecomposable projective and tilting character as a short sum of standard-module characters indexed by the atypical weights $\\Omega$.","pith_inferences":["The maximal parabolic category $\\mathcal{O}^{\\max}$, which the paper sets aside, sits inside $\\mathcal{O}^{\\min}$, so its blocks should be coarser than the ones classified here; comparing the two parameter sets would show exactly which linkage relations the larger parabolic forgets.","The continuous parameter $c\\in\\mathbb{C}/\\mathbb{Z}$ creates families of blocks with no finite-dimensional counterpart; a testable prediction is that translation or shuffling functors act transitively on the depth parameter $i$ inside a fixed $(c,\\iota)$ block.","The same semi-infinite-character and induced-module machinery should extend to infinite-dimensional Lie algebras of vector fields, with the depth parameter recording the degree of polynomial vectors; the authors indicate that such a theory is in preparation.","Because the Hamiltonian classification turns on one delicate indecomposability lemma, a direct computer check for small $n$ (e.g., $n=5,6$) of the endomorphism ring of $\\Delta(\\lambda)_{CH(n)}$ over $\\bar H(n)$ would independently stress-test the transfer before building further theory on it."],"forward_implications":["Every simple object in $\\mathcal{O}^{\\min}$ has a projective cover with a finite standard-module flag, so the subcategory of finitely generated modules has enough projectives and the block relation is well behaved.","For $W(n)$ and $\\bar S(n)$, two simple objects are in the same block exactly when their weights differ by a root-lattice element and their depths and parities shift by the length of that element; the block set is $\\mathbb{C}/\\mathbb{Z}\\times\\mathbb{Z}_2\\times\\mathbb{Z}$.","For the Hamiltonian families an extra complex parameter appears: blocks are indexed by $(\\mathbb{C}/\\mathbb{Z})^2\\times\\mathbb{Z}_2\\times\\mathbb{Z}$ for $\\bar H(2r+1)$ and by $\\mathbb{C}/2\\mathbb{Z}\\times\\mathbb{C}/\\mathbb{Z}\\times\\mathbb{Z}_2\\times\\mathbb{Z}$ for $\\bar H(2r)$, and in the even case $L(\\lambda)$ and $L(\\lambda+\\delta)$ lie in different blocks.","Degenerate BGG reciprocity $[P(\\lambda):\\Delta(\\mu)]=(\\nabla(\\mu):L(\\lambda))$ and Soergel reciprocity expressed through multiplicities in the induced modules $K(\\cdot)$ hold throughout $\\mathcal{O}^{\\min}$.","Indecomposable projective and tilting characters are explicit: outside a short list of atypical weights $P(\\lambda)=T(\\lambda)=\\Delta(\\lambda)$, and the exceptional characters are two- or four-term combinations of standard-module characters."],"supporting_citations":[{"why":"Supplies the character formulas for the induced modules $K(\\lambda)$, the atypical-weight set $\\Omega$, and the irreducibility criterion used in the final character theorems.","marker":"[18]"},{"why":"Provides the general tilting-module and projective-cover framework for $\\mathbb{Z}$-graded Lie superalgebras with a semi-infinite character, from which the paper imports BGG and Soergel reciprocity.","marker":"[7]"},{"why":"Establishes the existence of indecomposable tilting modules and Soergel reciprocity that the paper restates for $\\mathcal{O}^{\\min}$.","marker":"[21]"},{"why":"Gives the classification and root data of the Cartan-type Lie superalgebras $W(n)$, $S(n)$, $H(n)$, including the toral extension setup.","marker":"[13]"},{"why":"Classifies blocks of finite-dimensional $W(n)$-modules, which Theorem 4.21 recovers as a special case.","marker":"[20]"},{"why":"Supplies the idempotent criterion used in Corollary 4.7 to prove indecomposability of $CH(n)$ standard modules over $\\bar H(n)$.","marker":"[1]"},{"why":"Frobenius-extension theory used to realize co-standard modules as the induced modules $K(\\lambda)$ in Proposition 6.5.","marker":"[4]"},{"why":"Provides the Weyl-group action result used in Proposition 4.5 to establish block linkage for $\\bar S(n)$.","marker":"[14]"}],"fun_headline_variants":["Cartan superalgebra blocks fully classified in minimal category","Block decomposition for Cartan-type Lie superalgebras solved","Character formulas for all tilting and projective modules","Minimal parabolic BGG blocks fully classified","Cartan-type superalgebra blocks: exact and explicit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the Hamiltonian series, the block classification rests on the claim that each standard module induced from the larger algebra $CH(n)$ remains indecomposable after restriction to the smaller algebra $\\bar H(n)$; if that indecomposability failed, the listed $\\bar H$ blocks would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Cartan superalgebra blocks fully classified in minimal category","Block decomposition for Cartan-type Lie superalgebras solved","Character formulas for all tilting and projective modules","Minimal parabolic BGG blocks fully classified","Cartan-type superalgebra blocks: exact and explicit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000689,"raw_usage":{"total_tokens":3195,"prompt_tokens":1090,"completion_tokens":2105,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":706,"completion_tokens_details":{"reasoning_tokens":2031}},"tokens_in":706,"tokens_out":2105,"duration_ms":15746,"temperature":1.0,"reasoning_tokens":2031,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:50:59.888247+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\operatorname{Hom}_{\\bar H(5)}(\\Delta(\\lambda)_{CH(5)},\\Delta(\\lambda)_{CH(5)})$ for a dominant integral weight $\\lambda$ and check whether it contains any idempotent besides $0$ and the identity; a nontrivial idempotent, or a direct-sum decomposition of $\\Delta(\\lambda)_{CH(5)}$ over $\\bar H(5)$, would disprove Corollary 4.7 and with it the $\\bar H(2r+1)$ block classification. For the even case, finding $L(\\lambda)$ and $L(\\lambda+\\delta)$ sharing a projective cover would contradict Theorem 4.16.","supporting_citations":[{"cited_title":"Serganova, On representations of Cartan type Lie superalgebras , Amer","cited_arxiv_id":null,"evidence_quote":"Supplies the character formulas for the induced modules $K(\\lambda)$, the atypical-weight set $\\Omega$, and the irreducibility criterion used in the final character theorems."},{"cited_title":"Brundan, Tilting modules for Lie superalgebras , Comm","cited_arxiv_id":null,"evidence_quote":"Provides the general tilting-module and projective-cover framework for $\\mathbb{Z}$-graded Lie superalgebras with a semi-infinite character, from which the paper imports BGG and Soergel reciprocity."},{"cited_title":"Soergel, Character formulas for tilting modules over Kac-Moody alge bras, Representation Theory (1998), 432-228","cited_arxiv_id":null,"evidence_quote":"Establishes the existence of indecomposable tilting modules and Soergel reciprocity that the paper restates for $\\mathcal{O}^{\\min}$."},{"cited_title":"Kac, Lie superalgebra, Adv","cited_arxiv_id":null,"evidence_quote":"Gives the classification and root data of the Cartan-type Lie superalgebras $W(n)$, $S(n)$, $H(n)$, including the toral extension setup."},{"cited_title":"Shomron, Blocks of Lie superalgebras of type W (n), Journal of Algebra 251 (2002), 739-750","cited_arxiv_id":null,"evidence_quote":"Classifies blocks of finite-dimensional $W(n)$-modules, which Theorem 4.21 recovers as a special case."},{"cited_title":"Anderson and K","cited_arxiv_id":null,"evidence_quote":"Supplies the idempotent criterion used in Corollary 4.7 to prove indecomposability of $CH(n)$ standard modules over $\\bar H(n)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Frobenius-extension theory used to realize co-standard modules as the induced modules $K(\\lambda)$ in Proposition 6.5."},{"cited_title":"Kumar, Proof of the Parthasarathy-Ranga Rao-Varadarajan conject ure, Invent","cited_arxiv_id":null,"evidence_quote":"Provides the Weyl-group action result used in Proposition 4.5 to establish block linkage for $\\bar S(n)$."}],"review_version":1}