{"id":"5fc71fa2-a3f5-47b6-a2c0-c34fadf1b56b","arxiv_id":"2505.07099","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable higher Specht polynomials indexed by infinite Ferrers diagrams give irreducible representations of the infinite symmetric group, and a conjectural f-indexed filtration describes all non-completely-reducible layers.","lead":"This paper constructs stable, infinite versions of higher Specht polynomials inside a newly defined ring of eventually symmetric functions, and shows they form irreducible representations of the infinite symmetric group. It then describes the maximal completely reducible parts of the polynomial ring and, assuming a stated conjecture, builds explicit filtrations whose graded pieces are those parts.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's advertised filtrations rest on Conjecture 4.9; if the f-support bound or injectivity of p^f_n fails, the f-parameter and all of Section 4 after Definition 4.14 collapse.","rationale":"Good-faith reading: Sections 2 and 3 build explicit objects, including stable higher Specht polynomials, the representations V_hatM, and Theorem 3.23 identifying the maximal completely reducible subrepresentations, and those parts are largely self-contained modulo finite-n results imported from [Z2] and [Z3]. The paper is honest that Section 4 is conditional. The reader's weakest_assumption matches mine. I do not see a fatal flaw in Theorem 2.19 or Theorem 3.23: the limit arguments reduce infinite relations to finite ones by substituting finitely many variables to zero, and the construction of F_hatM,hatT is well-defined because the relevant orbit sums have finite-support monomials with coefficient 1. The main risk is that the final advertised filtration is not an established theorem. Conjecture 4.9 is not an isolated technicality; it is what makes the f-parameter intrinsic for arbitrary polynomials in infinitely many variables. The paper's own examples only reach d = 4, and the dimension condition in part (iii) is a nontrivial combinatorial identity that deserves a proof. For these reasons I would keep the reader's CONDITIONAL verdict: accept the unconditional structural results, but do not present the Section 4 filtrations as proved until Conjecture 4.9 is settled or clearly isolated as a conditional appendix. The proposed computational check for d = 5 and d = 6 is a feasible way to seek a counterexample; proving the dimension identity would be even stronger.","tokens_in":64816,"tokens_out":23430,"duration_ms":237184,"concrete_test":"Implement the decomposition of Qrxn+1_5 from Theorem 1.23 for n = 6 (or n = 7) using the explicit bases of Theorem 1.13; for every M in SSYT_5(lambda) with lambda of size n and every T in SYT(lambda), express F_{M,T} in the Qrxn+1_5 basis and compute supp_{n+1}F_{M,T}. Check that (i) every N in the support has f_N in {f_M, f_M - 1}; (ii) the matrix of p^5_n on the direct sum of the V_M with f_M = 5 is injective; and (iii) the composite p^4_n + q^5_n is injective, equivalently the dimension inequality A_n^5 + A_n^4 <= B_{n+1}^4 holds. A counterexample to any of these disproves Conjecture 4.9; if all pass for d = 5 and d = 6, the conditional section gains real support. This directly tests the exact unproved premise rather than merely enlarging the example count.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Conjecture 4.9 is the load-bearing unproved premise of the paper's final goal. It asserts that when a polynomial F lying in a single Specht-module component V_M within Qrxnsd, with f_M = f, is viewed in Qrxn+1sd, every tableau N in supp_{n+1}F has f_N in {f, f-1}, and that the maps p^f_n (projection to the f-part) and p^{f-1}_n + q^f_n are injective. These assertions are used in Lemma 4.11 to prove Qrxns^f_d is contained in Qrxn+1^f_d; Corollary 4.13 then defines f_F as an intrinsic parameter; Definition 4.14 defines Qrx8s^f_d on that basis; and Proposition 4.15, Theorem 4.17, Theorem 4.20, and the abstract's claim of explicit filtrations are all conditional on it. The paper says so explicitly after Definition 4.14. The author reports verification only for d at most 4 and selected additional tableaux. Part (iii) is especially strong: injectivity of p^{f-1}_n + q^f_n requires the dimension inequality A_n^f + A_n^{f-1} <= B_{n+1}^{f-1} for the f-isotypic dimensions, and no general proof is given. If any support contains f_N <= f-2, or if q^f_n + p^{f-1}_n has a kernel, then f_F can depend on n and the filtration and graded pieces do not exist as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the ring Λ∼ of eventually symmetric formal power series of bounded degree, defines infinite Ferrers diagrams and infinite (semi-)standard tableaux, and constructs stable generalized higher Specht polynomials F_{M̂,T̂}. It proves that the spaces V_{M̂} are irreducible S_N- and S_∞-representations indexed by infinite Ferrers diagrams (Theorem 2.19), and that the maximal completely reducible subrepresentations of Q[x_∞]_d and Λ∼_d are the finite direct sums of these irreducible pieces over suitable indices (Theorem 3.23). In Section 4 the paper states Conjecture 4.9 about the behavior of the f-parameter under adding one variable and derives, conditional on it, filtrations of Q[x_∞]_d and Λ∼_d whose graded pieces are maximal completely reducible (Definitions 4.14 and 4.18; Theorems 4.17 and 4.20). The paper explicitly labels all results after Definition 4.14 as dependent on Conjecture 4.9.","tokens_in":65230,"tokens_out":5542,"duration_ms":52786,"significance":"The unconditional portions give an explicit, basis-level construction of irreducible representations of the infinite symmetric group inside a natural completion of the polynomial ring, and Theorem 3.23 answers a well-posed question about the failure of complete reducibility. The paper is honest about the conditional nature of Section 4. However, the advertised final filtrations are not theorems as submitted: they depend on an unproved conjecture, and the foundations rely on two unpublished preprints by the author. If Conjecture 4.9 can be proved, this will be a substantial contribution; in its current form, the central claim of the abstract is not established.","major_comments":[{"comment":"Conjecture 4.9 is the load-bearing unproved premise of the final goal. Lemma 4.11 uses it to prove Q[x_n]^f_d ⊆ Q[x_{n+1}]^f_d; Corollary 4.13 uses it to define f_F independently of n; Definition 4.14 uses f_F to define Q[x_∞]^f_d and Λ∼^f_d; and Proposition 4.15, Theorem 4.17, Theorem 4.20, and Corollaries 4.16, 4.22, and 4.27 all inherit this dependence. The paper explicitly states after Definition 4.14 that the results are conditional, and reports verification only for d ≤ 4 and selected additional tableaux. Since the abstract presents the filtrations and the form of the maximal completely reducible subrepresentations without this caveat, the conjecture must be proved, or the advertised claims must be re-labeled as conjectural, before the paper can be accepted.","section":"§4, Conjecture 4.9 and Definition 4.14"},{"comment":"The proof skeleton relies at essential points on results quoted from the author's unpublished preprints [Z2] and [Z3]: Theorem 1.13, Proposition 1.16, Theorem 1.22, and Theorem 1.23. These are used in the proofs of Proposition 2.15, Theorem 2.19, Lemma 2.23, Theorem 2.28, Proposition 2.31, Theorem 3.11, Theorem 3.15, and Theorem 3.23. A journal referee cannot verify unpublished preprints, so either full proofs must be included in this paper or the dependence on [Z2] and [Z3] must be removed.","section":"§1, Theorem 1.13 and Proposition 1.16"},{"comment":"Part (iii) is especially strong and is not supported by the evidence given. Injectivity of p^{f-1}_n + q^f_n requires a dimension inequality A_n^f + A_n^{f-1} ≤ B_{n+1}^{f-1} for the relevant isotypic components, and no general argument is supplied. Similarly, part (i) asserts that the (n+1)-support of an element of a single V_M contains only tableaux with f-values f and f-1; Remark 4.10 shows that the f-part can mix several tableaux, so this is a real combinatorial assertion about higher Specht expansions, not a formality. If either assertion fails, f_F may depend on n and the filtrations in Definition 4.14 and Theorem 4.20 do not exist as stated. Please provide a proof or a much more substantial verification.","section":"§4, Conjecture 4.9(iii)"}],"minor_comments":[{"comment":"The abstract contains the typo 'irreducibe' for 'irreducible'.","section":"Abstract"},{"comment":"The sentence 'Note that when going from n = 5 to n = 6 in Example 1.20' should refer to Example 1.21, since the displayed decompositions are given in Example 1.21.","section":"§1, after Example 1.21"},{"comment":"The entries [Z2] and [Z3] are listed as unpublished preprints without arXiv identifiers; if they are available online, stable identifiers should be provided so that the reader can access the results on which the present paper depends.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is largely a sequel to two unpublished preprints [Z2] and [Z3], and the novel Section 4 is conditional on Conjecture 4.9; the author's own concluding paragraph says that a proof of Conjecture 4.9 is left for further research. The journal should consider whether a paper whose main advertised theorem is conjectural and whose unconditional parts depend heavily on unpublished work meets its standards. This is a suitability question for the editor rather than a judgment about the mathematical honesty of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — here’s my read. The genuinely new and valuable part is Sections 2 and 3: the ring \\tilde\\Lambda of eventually symmetric functions, infinite Ferrers diagrams and tableaux, stable higher Specht polynomials, and the two main structural theorems. Theorem 2.19 (irreducibility of the V_{\\hat M} for both S_N and S_8) and Theorem 3.23 (the maximal completely reducible subrepresentations of Q[x_\\infty]_d and \\tilde\\Lambda_d) are argued with explicit bases and finite-to-infinite limits, and the examples check out. I believe that core is solid and citable.\n\nThe soft spot is exactly where you'd expect it. The advertised filtrations in the abstract and Section 4 depend on Conjecture 4.9, and the paper says so plainly after Definition 4.14. The conjecture asserts that an f-isotypic polynomial viewed in n+1 variables has support only at f and f−1, and that the projections p^f_n and q^f_n+p^{f−1}_n are injective. That injectivity is a dimension inequality without a proof; verification is only for d≤4 and selected tableaux. If it fails, the f-parameter is not intrinsic and the filtrations collapse. This is a big deal because it's the final goal of the paper. It is a load-bearing gap, though honestly declared.\n\nSeparate and smaller: the constructors pull Theorem 1.13 and Proposition 1.16 from the author's unpublished preprints [Z2] and [Z3]. As submitted, the paper isn't self-contained at a load-bearing point. That's a practical obstacle for a referee.\n\nWho's this for: representation theorists and combinatorialists working with stable families of Sn-representations, FI-modules, and symmetric functions. I'd bring it to reading group if the group tolerates conditional sections. It deserves a serious referee — Theorem 3.23 alone justifies that. But I would not accept it as is. A solid revision either proves Conjecture 4.9 or restructures Section 4 into a clearly labeled conditional appendix, and makes [Z2] and [Z3] public. That's my recommendation.","headline":"Sections 2–3 are a solid, citable construction; the advertised filtrations are explicitly conditional on Conjecture 4.9, which is unproved.","tokens_in":65709,"tokens_out":3505,"would_cite":true,"duration_ms":33917,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E10","05E05","20C30","20C32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stable Specht polynomial limits yield irreducible representations of the infinite symmetric group","keywords":["higher Specht polynomials","stable representations","infinite symmetric group","eventually symmetric functions","infinite Ferrers diagrams","infinite Young tableaux","complete reducibility","filtration"],"falsifier":"Compute explicitly the n-support (in the sense of Definition 4.1) for a polynomial F in V_M with f_M = 2 and degree d = 5, using the decomposition of Q[x_{n+1}]_5 given by Theorem 1.23. If the n+1-support contains a tableau N with f_N differing from 2 or 1, or if the map $p_n^{2}$ fails to be injective on the relevant component, then Conjecture 4.9 is false and the filtrations of Section 4 collapse. A computer search over all tableaux of degree 5 and n large enough (say n=11) would settle it.","tokens_in":64564,"feed_emoji":"∞","tokens_out":2304,"duration_ms":24559,"temperature":0.7,"pith_summary":"This paper establishes that the natural limits of higher Specht polynomials, taken as the number of variables grows, live in a ring of eventually symmetric functions and form irreducible representations of the infinite symmetric group. The homogeneous parts of this ring and of its polynomial subring are not completely reducible, and the paper identifies the maximal completely reducible subrepresentations as direct sums of these irreducible pieces indexed by infinite Ferrers diagrams. The main structural results describe these subrepresentations explicitly via three equivalent decompositions, and a conjectural filtration governs the full non-completely-reducible quotients.","feed_headline":"Stable Specht limits yield irreducible S∞ modules","feed_subtitle":"Eventually symmetric functions give the right home for the infinite limits of higher Specht polynomials.","key_machinery":"The central mechanism is the operation $\\iota$ on tableaux and its semi-standard counterpart $\\hat{\\iota}$, which increases the shape by adding a box to the first row while preserving the descent set and the sum $\\Sigma(M)$. Iterating these operations produces infinite Ferrers diagrams, infinite standard and semi-standard Young tableaux, and the stable generalized higher Specht polynomials $F_{\\hat{M},\\hat{T}}$ that are the limits of the finite higher Specht polynomials. The parameter $f_{\\hat{M}}$, defined as the number of positive entries in the first row of the tableau (or equivalently the number of positive multiplicities in the infinite version), measures how much of the symmetric-function ring is mixed into the representation; $f=0$ exactly corresponds to stable polynomials that remain in $\\mathbb{Q}[x_\\infty]$. The key machinery also includes the monomial symmetric function $m_{\\hat{M}}$ attached to $\\hat{M}$, which separates the polynomial part from the symmetric-function part in the decompositions of $\\tilde{\\Lambda}_d$.","core_discovery":"The paper defines the ring $\\tilde{\\Lambda}$ of eventually symmetric functions—bounded-degree power series in infinitely many variables that are invariant under permutations of all but finitely many variables—and shows that it is generated by the polynomial ring $\\mathbb{Q}[x_\\infty]$ and the ring $\\Lambda$ of symmetric functions, with these two generating sets algebraically independent (Proposition 2.4). Inside $\\tilde{\\Lambda}$ it constructs stable generalized higher Specht polynomials $F_{\\hat{M},\\hat{T}}$ indexed by infinite standard and semi-standard Young tableaux. Theorem 2.19 establishes that the spaces $V_{\\hat{M}}$ spanned by these stable polynomials are irreducible representations of both $S_\\infty$ and $S_\\mathbb{N}$, with isomorphism type determined by the infinite Ferrers diagram $\\hat{\\lambda}$. The central structural result, Theorem 3.23, identifies the maximal completely reducible subrepresentation of the homogeneous piece $\\tilde{\\Lambda}_d$ as the direct sum of these irreducible $V_{\\hat{M}}$ over all $\\hat{M}$ in $\\mathrm{SSYT}_d(\\hat{\\lambda})$; similarly for the polynomial subring $\\mathbb{Q}[x_\\infty]_d$. The failure of complete reducibility is traced to the fact that elements of the symmetric-function ring $\\Lambda$ become separate from polynomials in the infinite limit.","pith_inferences":["If Conjecture 4.9 holds for all degrees, the parameter $f$ becomes a well-defined invariant of polynomially supported elements in $\\mathbb{Q}[x_\\infty]_d$, and the filtration on $\\tilde{\\Lambda}$ likely coincides with the filtration induced by the degree of the symmetric-function part in the decomposition of $\\tilde{\\Lambda}$ as $\\mathbb{Q}[x_\\infty] \\otimes \\Lambda$; this would provide a complete","The separation phenomenon observed here—where the limit of a sum of polynomial representations loses the symmetric-function components—suggests a general principle for limits of representations of finite symmetric groups: when the index of a representation involves a parameter that grows with $n$, the limit may fail to be completely reducible, and the missing components correspond to symmetric fun","A testable extension is to compute the matrix of basis changes between the three decompositions of Theorem 3.15 (the $V_{\\hat{M}}$, $\\tilde{V}_{\\hat{M}}$, and $V^{\\vec{h}}_{\\hat{C}}$ bases) for small $d$ and compare with the stability predicted in Remark 3.20; if the matrices stabilize as $n$ grows, it would provide an explicit bijection between $\\mathrm{SSYT}_d(\\hat{\\lambda})$ and the set $A_d(\\h","The conjecture itself might be approachable via the explicit formulas for the embeddings $\\mathbb{Q}[x_n]_d \\hookrightarrow \\mathbb{Q}[x_{n+1}]_d$ given in Examples 4.4–4.7, which suggest a general closed form for the projection maps $p_n^f$ and $q_n^f$ in terms of normalized symmetric functions."],"forward_implications":["If the paper is correct, every stable higher Specht polynomial—the limit of the generalized higher Specht polynomials of Ariki–Terasoma–Yamada and their generalizations—is a basis vector of an irreducible representation of the infinite symmetric group indexed by an infinite Ferrers diagram.","The failure of complete reducibility in $\\tilde{\\Lambda}_d$ and $\\mathbb{Q}[x_\\infty]_d$ is fully accounted for by the separation of symmetric functions from polynomials; the maximal completely reducible subrepresentation is precisely the direct sum of the irreducible pieces with $f=0$.","The filtrations constructed in Section 4 (conditional on Conjecture 4.9) provide a composition series for $\\tilde{\\Lambda}_d$, $\\mathbb{Q}[x_\\infty]_d$, and the quotient $R_{8,k,d}$, whose graded pieces are explicit direct sums of irreducible modules with prescribed multiplicities $c_{\\hat{M},f}$.","The isomorphism types of the irreducible representations $S_{\\hat{\\lambda}}$ depend only on the infinite partition $\\hat{\\lambda}$, and distinct $\\hat{\\lambda}$ give non-isomorphic representations, yielding a classification of these limits.","The results yield explicit computations of dimensions and multiplicities of isotypical components (Corollary 3.19), including formulas that extend the hook-length and multinomial identities to the infinite setting."],"supporting_citations":[{"why":"Provides the compatibility of higher Specht polynomials under adding a box to the first row (ι and ˆι operations) and the stability of the representations, which the current paper extends to the infinite limit.","marker":"[Z2]"},{"why":"Supplies the generalized higher Specht polynomials and their homogeneous decompositions, along with the sets A_d(λ) and the representations V^{\\vec{h}}_C, which are the finite building blocks for the infinite versions.","marker":"[Z3]"},{"why":"Introduced the original higher Specht polynomials, whose stable limits are the primary objects of study.","marker":"[ATY]"},{"why":"Offered a seminormal construction of Specht modules, used via Theorem 1.13 to identify the finite representations V_M as Specht modules.","marker":"[M]"},{"why":"Showed that certain generalized Specht polynomials span irreducible representations, a key input for Theorem 2.19's finite analogue.","marker":"[Pe]"},{"why":"Introduced FI-modules and representation stability, which motivates the construction of limits of stable families of representations in this paper.","marker":"[CF]"},{"why":"Developed FI-module theory further, providing background for why the limits of centrally stable representations should be representations of S_∞.","marker":"[CEF]"},{"why":"Discussed stability patterns in representation theory, contextualizing the stable behavior of the representations indexed by tableaux.","marker":"[SS]"},{"why":"Provided higher Specht bases for generalizations of the coinvariant ring, which the paper's Rn,k and Rn,k,s spaces generalize to the infinite setting.","marker":"[GR]"}],"fun_headline_variants":["Stable Specht polynomials yield new S∞ irreducibles","Eventually symmetric functions host stable Specht modules","New S∞ irreps arise from stable Specht limits","Eventually symmetric ring yields S∞ irreducible reps","Irreducible S∞ modules from stable Specht limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Conjecture 4.9, which asserts that for a polynomial lying in a single irreducible component V_M, its decomposition into components in one more variable only involves tableaux whose f-parameter is f or f-1, and that the associated projection maps are injective; all the filtration results after Definition 4.14 are conditional on this conjecture, verified only for degree at most 4 and a few additional tableaux.","fun_headline_variants_meta":{"raw":{"variants":["Stable Specht polynomials yield new S∞ irreducibles","Eventually symmetric functions host stable Specht modules","New S∞ irreps arise from stable Specht limits","Eventually symmetric ring yields S∞ irreducible reps","Irreducible S∞ modules from stable Specht limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00141,"raw_usage":{"total_tokens":5740,"prompt_tokens":1029,"completion_tokens":4711,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":645,"completion_tokens_details":{"reasoning_tokens":4635}},"tokens_in":645,"tokens_out":4711,"duration_ms":30585,"temperature":1.0,"reasoning_tokens":4635,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:25:18.552662+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute explicitly the n-support (in the sense of Definition 4.1) for a polynomial F in V_M with f_M = 2 and degree d = 5, using the decomposition of Q[x_{n+1}]_5 given by Theorem 1.23. If the n+1-support contains a tableau N with f_N differing from 2 or 1, or if the map $p_n^{2}$ fails to be injective on the relevant component, then Conjecture 4.9 is false and the filtrations of Section 4 collapse. A computer search over all tableaux of degree 5 and n large enough (say n=11) would settle it.","supporting_citations":[],"review_version":1}