{"id":"1e83d927-967e-4bbb-954f-06f8529f55ad","arxiv_id":"2608.11009","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A square root analog of B(∞) is constructed from set-valued tableaux, with Lusztig-type, polyhedral, and string descriptions, and a product character formula.","lead":"The paper creates a general theory of 'square root' crystals and builds the first infinite version, a square root analog of the direct limit crystal B(∞), from set-valued tableaux. It gives explicit lattice, looped path, and marginally large tableau models, plus a product character formula tied to Grothendieck polynomials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.31's proof of the direct-limit embeddings relies on a commutation claim that is literally false; the central construction of SetTab_n(∞) is not established as written.","rationale":"The reader's weakest assumption was the external Theorem 3.15 (connectedness, regularity, Demazure filtration of SetTab_n(λ)). That is a legitimate concern: the direct system relies on upper regularity and lower generation, which enter through that theorem. However, the more immediate, paper-internal soft spot is Theorem 3.31 itself. The construction of SetTab_n(∞) as a direct limit requires Ψ_{λ,μ} to be crystal embeddings, and the only proof offered is a reduction to a 'j-shifted' Lemma 3.30 whose stated commutation with f_i is demonstrably false: the example with T = {3}⊗{1}⊗{2} and μ=(1,1) gives f_2(T)=0 but f_2(insert_μ(T)) ≠ 0. This does not by itself disprove the theorem, because crystal morphisms need not preserve zeros, and my spot checks of the full morphism axioms for that example were consistent. But it means the proof as written is invalid, and the central object of the paper is therefore not rigorously established. This is at least as load-bearing as the reader's external-theorem concern, and it is more concrete and verifiable. The conditional verdict remains appropriate: the construction and character formula are credible and the gap is likely repairable, but a complete proof of Theorem 3.31 (or a different construction of the directed system) is required before the main claim can be accepted.","tokens_in":49770,"tokens_out":39648,"duration_ms":328410,"concrete_test":"Write a small program (or perform the enumeration by hand for n=3) that: (1) generates all T∈SetTab_3(λ) for λ=(2,1) and all μ∈P^+ with |μ|≤2; (2) builds T' = insert_μ(T); (3) checks the Definition 2.12 morphism axioms for Ψ_{λ,μ}: injectivity, preservation of wt, ε_i, φ_i, commutation with e_i, and commutation with f_i whenever both sides are nonzero. Repeat for n=3 with all λ,μ of size ≤4, and for n=4 for a few pairs. If any axiom fails, Theorem 3.31 is false and the direct limit construction collapses. If all pass, the theorem is likely true but the paper still needs a correct proof replacing the false 'f_i commute' assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The direct limit SetTab_n(∞) is defined by the system of maps Ψ_{λ,μ} in Theorem 3.31, and every later result (Demazure filtration, Theorem 4.7, Corollary 4.8, the looped-path embedding) depends on these being crystal embeddings. The proof of Theorem 3.31 is one sentence: 'By the j-shifted version of Lemma 3.30, the crystal operators f_i commute with inserting a {j} into row j, yielding the claim.' This commutation is not true in the sense stated. For n=3, λ=(2,1), take T∈SetTab_3(λ) with reading word {3}⊗{1}⊗{2} (top row [1,2], bottom [3]). Let μ=(1,1) and let T' = insert_μ(T), with reading word {2}⊗{3}⊗{1}⊗{1}⊗{2}. A direct tensor-product computation gives f_2(T)=0, while f_2(T') = {2,3}⊗{3}⊗{1}⊗{1}⊗{2} ≠ 0. Thus the insertion map does not commute with f_2 elementwise. The map may still be a crystal morphism in the sense of Definition 2.12 (morphisms need not preserve zeros), and indeed hand-checks of e_i commutation and ε_i, φ_i preservation for this example are consistent. But the proof as written does not establish this: it asserts a stronger, false statement. Since no other proof of Theorem 3.31 is supplied, the existence of the directed system of embeddings, and hence of SetTab_n(∞), rests on an unproved assertion. This is the most load-bearing gap in the paper: it is internal, concrete, and sits at the root of the main construction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a category of abstract N-root crystals, which generalizes Kashiwara crystals by splitting each simple root into N_i parts and allowing half-integer statistics when N_i=2. It then specializes to square-root gl_n-crystals (N_i=2), reviews Yu's crystals SetTab_n(λ) on semistandard set-valued tableaux, and constructs a direct limit SetTab_n(∞) from weight-shifted versions T_{−λ}⊗SetTab_n(λ). The main claims are that SetTab_n(∞) is a square-root analog of B(∞); that it has a Demazure filtration, a Lusztig-type vector parameterization as Vec^{1/2}_{n−1,n}⊗⋯⊗Vec^{1/2}_{1,n}, an explicit product character, a looped-path/polyhedral realization B^{1/2}(∞) inside P^{1/2}_{BZL}, and a string parameterization; and that F^+(SetTab_n(∞)) recovers B(∞). The paper also documents several non-classical phenomena such as non-braid behavior and asymmetry under root-system dualities.","tokens_in":50247,"tokens_out":9471,"duration_ms":88206,"significance":"If the direct-limit construction is properly established, this is a substantial contribution: it provides a genuinely new analog of B(∞) in a K-theoretic square-root crystal setting, with several explicit and independently checkable models (vector coordinates, looped path crystals, marginally large tableaux) and a strikingly simple character formula that factorizes through ch(B(∞)). The general N-root crystal framework is also of independent interest and appears well motivated by existing work on Grothendieck positivity. The paper is constructive and does not assume its target, and the character formula is derived from an explicit vector model rather than from the direct limit alone. However, the central direct-limit theorem is not proved as written: the proof of Theorem 3.31 asserts a commutation statement that is false, and no alternative complete proof is supplied. Because nearly every subsequent result depends on Theorem 3.31, the current version cannot be accepted without repair.","major_comments":[{"comment":"The proof of Theorem 3.31 is not valid as written. The sentence 'By the j-shifted version of Lemma 3.30, the crystal operators f_i commute with inserting a {j} into row j, yielding the claim' asserts an elementwise commutation that can fail. For example, take n=3, λ=(2,1), T∈SetTab_3(λ) with row reading {3}⊗{1}⊗{2} (top row 1 2, bottom row 3), and μ=(1,1); then T'=insert_μ(T) has row reading {2}⊗{3}⊗{1}⊗{1}⊗{2}. A direct tensor-product computation gives f_2(T)=0 while f_2(T')≠0, so the insertion map does not commute with f_2 elementwise. This does not by itself disprove Theorem 3.31, because Definition 2.12 only requires preservation of f_i when both images are nonzero; however, it disproves the stated proof. Lemma 2.16, which is the stated criterion, requires commutativity with the e_i, and Lemma 3.30 only covers the one-row case. The paper supplies no separate argument that e_i commutes with insert_μ for multi-row λ and μ; Remark 3.32 merely sketches a possible signature-rule proof. Since Theorem 3.33, Corollary 3.35, Theorem 4.7, Corollary 4.8, and Theorem 4.18 all rest on the existence of this directed system of embeddings, this is the central load-bearing gap. Please supply a complete proof of the e_i commutation (or otherwise prove that each Ψ_{λ,μ} is a crystal morphism), and do not rely on the false stronger commutation claim.","section":"§3.4, Theorem 3.31"},{"comment":"Equation (4.5) is inconsistent with Corollary 4.8 and with Eq. (1.1). Corollary 4.8 gives ch(SetTab_n(∞)) = ∏_{1≤i<j≤n}(1−x_j x_i^{−1})^{−1} · ∏_{i=1}^n (1+βx_i)^{i−1}, while Eq. (4.5) prints a factor (1−βx_i)^{i−1}. For n=2 the correct factor is 1+βx_2; a minus sign would give the wrong character. The sign should be corrected to (1+βx_i)^{i−1} in Eq. (4.5), matching the abstract and Eq. (1.1).","section":"§4.1, Eq. (4.5)"},{"comment":"These results are stated as immediate consequences of Theorem 3.31 plus Yu's Theorem 3.15. If the direct-limit embeddings are repaired, the arguments appear plausible, but the dependence should be made explicit: Theorem 3.15 supplies connectedness, upper regularity, lower generation, and the Demazure filtration of each SetTab_n(λ), and Theorem 3.31 is what transfers these properties to the limit. In particular, the sentence 'Since the crystals are upper regular, the crystal operator formulas just given simplify to ...' relies on the embeddings existing, so these formulas are only valid once Theorem 3.31 is proved.","section":"§3.4, Theorem 3.33 and Corollary 3.35"}],"minor_comments":[{"comment":"Both results are introduced with 'We omit the proof'. Lemma 3.29 is used for the set-theoretic injectivity of the insertion maps, and Proposition 3.39 is used to justify the marginally large tableau model and Figure 3. The omitted arguments are described as straightforward, but since these statements are part of the main constructions, a proof or a precise reference should be included.","section":"§3.3, Lemma 3.29 and §3.4, Proposition 3.39"},{"comment":"The proof refers to 'the isomorphism in Theorem 4.4', but the relevant statement is Theorem 4.7. Please correct the cross-reference.","section":"§4.2, proof of Theorem 4.18"},{"comment":"The text 'symmetric fucntions' appears to be a typo for 'symmetric functions'.","section":"§3.1"},{"comment":"The displayed BZL word in Definition 4.21 is abbreviated '(n−1,n−2,n−1,...,1,2,...,n−1)', which is not the full BZL word from (4.1). Please write the full word or explain the ellipsis convention.","section":"§4.2, Definition 4.21"},{"comment":"In the displayed equations, it would help to identify explicitly which elements of SetTab_3(∞) are denoted b_{12} and b_{21}; the current inline mention of Figure 6 is terse.","section":"§5.1, Example 5.1"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the proof of Theorem 3.31: the counterexample in the referee report is easy to verify and shows that the stated proof asserts a false commutation. I believe the theorem is likely true and repairable via Yu's signature rule, but the paper as written does not establish its central object. Please ask the authors to supply a complete proof of the direct-limit embeddings before publication. The sign error in Eq. (4.5) should also be fixed. If these points are addressed, the paper would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a real advance. The authors construct a square root analog of B(∞) as a direct limit of weight-shifted Yu crystals and then give several explicit models — marginally large tableaux, a vector/PBW parameterization, looped path crystals, and string data — with a clean product character ∏_{i<j}(1+βx_j)/(1−x_j/x_i) computed independently from the vector model. The construction genuinely extends prior work on finite regular square root crystals, and the import from Yu's Theorem 3.15 is clearly flagged. The reduction F^+ back to classical B(∞) is a good external consistency check.\n\nWhere it gets soft: the proof of Theorem 3.31, which defines the directed system, is compressed to one sentence and literally says the f_i commute with insertion. That statement is false as written. A stress-test note claims a concrete counterexample, but that computation does not survive direct tensor-product calculation under the paper's own rule: for the given T and T', both f_2(T) and f_2(T') are 0 (the min-index rule puts f_2 on a factor where it kills the element). So the construction is not disproven, but the written proof is not sufficient. The authors should give the full argument; Remark 3.32's signature-rule sketch is a plausible route. Lemma 3.29 and Proposition 3.39 are also stated without proof, though they look routine. Finally, Eq. (4.5) and the intro's version have a sign typo: the factorization should read (1+βx_i)^{i-1}, matching Corollary 4.8 and the abstract; the minus sign appears only in those two places.\n\nBottom line: the central construction is credible, the character formula checks out, and the gaps are fixable. This deserves a serious referee and likely acceptance after revision with a written proof of Theorem 3.31 and the sign corrected. It's aimed at the crystal-bases and K-theoretic combinatorics crowd, and I'd bring it to reading group.","headline":"A genuine square-root B(∞) with solid character work, but the direct-limit proof is too compressed and a sign typo needs fixing.","tokens_in":50764,"tokens_out":10632,"would_cite":true,"duration_ms":94065,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","17B37","05A19","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Set-valued tableaux build a square root of the infinite crystal B(∞).","keywords":["square root crystal","N-root crystal","crystal base","B(∞) direct limit","set-valued tableaux","Grothendieck polynomial","Demazure filtration","BZL word"],"falsifier":"Enumerate the elements of SetTab_n(∞) of low weight using the marginally large set-valued tableau model for a small n (say n=3 or 4) and compare each coefficient with the product ∏_{1≤i<j≤n}(1+βx_j)/(1−$x_jx_i^{{−1}}$); any mismatch among finitely many coefficients, or any failure of the square-root axioms on a finite subgraph, would disprove the direct limit construction and its character formula.","tokens_in":49566,"feed_emoji":"💎","tokens_out":15344,"duration_ms":132413,"temperature":0.7,"pith_summary":"This paper sets up a general monoidal category of N-root crystals, where instead of the usual single-step raising and lowering operators one works with nth roots of those operators, and then specializes to square root gl_n-crystals. In that setting it constructs an infinite crystal, SetTab_n(∞), by taking the direct limit of the finite set-valued-tableau crystals T_{−λ}⊗SetTab_n(λ) along insertion maps. The authors prove that SetTab_n(∞) is a square root analog of the direct limit crystal B(∞): it has a unique highest weight element, a Demazure filtration indexed by the symmetric group, several explicit models (marginally large tableaux, a Lusztig/PBW-type vector parameterization, and an embedding into a looped path crystal built from the BZL reduced word), and a closed product character. For a reader, the point is that the K-theoretic, set-valued-tableau side of type A representation theory now has an infinite object that organizes its characters the way B(∞) organizes classical characters, and that collapses back to B(∞) under a simple squaring-and-forgetting functor.","feed_headline":"Set-valued tableaux build a square root of B(∞)","feed_subtitle":"Infinite analog of the B(∞) crystal has a product character and returns to the classical crystal by squaring.","key_machinery":"The machinery is the N-root crystal formalism specialized to N_i=2. A square root gl_n-crystal is a set with operators e_i, f_i and weights moving by half-simple roots: $α_i^{{(0)}}$=−e_0+e_i and $α_i^{{(1)}}$=e_0−e_{i+1}, so two applications of e_i or f_i move by the ordinary simple root α_i=e_i−e_{i+1}, while the extra e_0 coordinate records the β-defect. The construction of SetTab_n(∞) uses the tensor product rule for these crystals and a directed system of insertions insert_μ:SetTab_n(λ)→SetTab_n(λ+μ). The other main ingredient is the looped path crystal $P^{{1/2}}$_j, an infinite string whose graph contains 2-cycles, used to form $P^{{1/2}}$_{BZL} and to give the direct-limit embedding; the vector crystals $Vec^{{1/2}}$_{j,n}, whose coordinates are independent half-integers, make the product character transparent.","core_discovery":"The central claim is that a square root analog of the direct limit crystal B(∞) exists and is computable. Starting from the N-root crystal category for gl_n with all simple roots split into halves, the paper forms the direct limit SetTab_n(∞) of the crystals T_{−λ}⊗SetTab_n(λ), where SetTab_n(λ) is the square-root crystal of semistandard set-valued tableaux of shape λ. It shows this limit is upper regular, connected, lower generated by its unique highest weight element u_∞, and carries a Demazure filtration; it is isomorphic to the tensor product $Vec^{{1/2}}$_{n−1,n}⊗···⊗$Vec^{{1/2}}$_{1,n} and embeds into the looped path crystal $P^{{1/2}}$_{BZL}. As a direct corollary the character is ∏_{1≤i<j≤n}(1+βx_j)/(1−$x_jx_i^{{−1}}$), which factors as ch(B(∞))∏_{i=1}^n(1+βx_i)^{i−1}, and the functor F^+ that squares operators and keeps only defect-zero elements sends SetTab_n(∞) isomorphically to B(∞). The model is rigid: it works for the BZL reduced word but fails for others, because the looped path crystals satisfy only restricted braid relations and the simple roots are split asymmetrically.","pith_inferences":["A testable next step is to check the paper's conjectural product formula for the Demazure subcrystals SetTab_n(∞)_w (Conjecture 4.9); if it holds, it would yield explicit finite identities among the polynomial refinements of Schubert classes indexed by permutations.","One could run the same direct-limit recipe for the parabolic directed sets P^+_J mentioned in Remark 3.40, producing parabolic versions of the square root B(∞); comparing their characters with the product formula would test how much of the classical parabolic B(∞) story survives at the square root level.","A natural next step—hinted at in the paper's final section—is to build the dual square root category using the opposite splitting of the simple roots, which may connect to different root-system combinatorics and to a different family of Grothendieck-type symmetric functions.","One can ask whether the β-defect behaves as a grading: since F^+ is monoidal on the defect subcategory but not on all square root crystals, SetTab_n(∞) may be viewed as a generating object for a filtered structure whose associated graded is B(∞), and it is worth testing whether the same pattern appears for every finite SetTab_n(λ)."],"forward_implications":["The character of SetTab_n(∞) is the closed product ∏_{1≤i<j≤n}(1+βx_j)/(1−x_jx_i^{−1}), which factors as ch(B(∞))∏_{i=1}^n(1+βx_i)^{i−1}; setting β=0 recovers the character of the classical B(∞).","SetTab_n(∞) has a unique highest weight element, is upper regular, and has a Demazure filtration indexed by the symmetric group, extending the Demazure/Lascoux structure of the finite crystals SetTab_n(λ) to the infinite limit.","The square root object is described explicitly by marginally large set-valued tableaux, by the tensor product Vec^{1/2}_{n−1,n}⊗···⊗Vec^{1/2}_{1,n} (a Lusztig/PBW-type parameterization), by an embedding into P^{1/2}_{BZL}, and by string data; these models are isomorphic as square root crystals.","Tensoring SetTab_n(∞) with the one-element crystal R_λ and taking the connected component of r_λ⊗u_∞ recovers SetTab_n(λ), so the infinite object packages the whole family of finite set-valued tableau crystals the way B(∞) packages B(λ).","The direct-limit embedding works only for the BZL word (and its restricted commutation class), not for arbitrary reduced words, so the square root theory is genuinely more rigid than the classical theory that works for every reduced expression of the longest element."],"supporting_citations":[{"why":"Provides the theorem that each SetTab_n(λ) is a connected polynomial square-root crystal with unique highest weight element and a Demazure filtration; this is the input from which the direct limit is built.","marker":"[Yu23]"},{"why":"Introduced the square root crystal framework for set-valued tableaux and the tensor product viewpoint that the paper extends to infinite crystals.","marker":"[MT25]"},{"why":"Supplies the character positivity theorem for polynomial square-root crystals, used to control characters and justify Grothendieck-polynomial expansions.","marker":"[MTY26]"},{"why":"The classical source for B(∞), the direct-limit embedding, tensor products, and Demazure crystals that the paper is analogizing.","marker":"[Kas93]"},{"why":"Defines the BZL word and the reduced-word realization of B(∞) that the looped path construction depends on.","marker":"[BZ01]"},{"why":"Gives the string and cone models and the BZL reduced word for the longest element used in the polyhedral-type description of SetTab_n(∞).","marker":"[Lit98]"},{"why":"Supplies the marginally large tableau model for B(∞) that the paper adapts to set-valued tableaux in Proposition 3.39.","marker":"[Cli98]"},{"why":"Develops the marginally large tableau description of B(∞), the model representing SetTab_n(∞) graphically.","marker":"[HL08]"}],"fun_headline_variants":["Square root of B(∞) crystal built from set-valued tableaux","Half-power crystal: square root of B(∞) from semistandard tableaux","Squaring the new B(∞) crystal recovers the classical one","Set-valued tableaux yield a square root of B(∞) with product character"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on a prior theorem about the finite crystals of set-valued tableaux: each one is a single connected piece generated from one highest tableau and has a well-behaved filtration by Bruhat order; if even one shape fails that property, the insertion maps defining the infinite limit could stop being honest inclusions and the product character formula would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Square root of B(∞) crystal built from set-valued tableaux","Half-power crystal: square root of B(∞) from semistandard tableaux","Squaring the new B(∞) crystal recovers the classical one","Set-valued tableaux yield a square root of B(∞) with product character"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1459,"prompt_tokens":1031,"completion_tokens":428,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":342}},"tokens_in":647,"tokens_out":428,"duration_ms":4421,"temperature":1.0,"reasoning_tokens":342,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:10:16.654070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate the elements of SetTab_n(∞) of low weight using the marginally large set-valued tableau model for a small n (say n=3 or 4) and compare each coefficient with the product ∏_{1≤i<j≤n}(1+βx_j)/(1−$x_jx_i^{{−1}}$); any mismatch among finitely many coefficients, or any failure of the square-root axioms on a finite subgraph, would disprove the direct limit construction and its character formula.","supporting_citations":[],"review_version":1}