{"id":"5dc24cbd-f445-4ad8-9a1a-6ad013e85f6d","arxiv_id":"2504.14684","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Character values at principal torsion elements of a compact Lie group equal, up to sign and a constant, the dimension of a representation of a dual centralizer group.","lead":"The paper derives a formula for the values of characters of compact Lie groups at special finite-order elements, expressing them as dimensions of representations of an associated group, up to a sign and a constant. It also proves that, with a few exceptions, the elements with the smallest centralizer are unique, making the formula sharp.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1's constant dm is outsourced to Polo [Polo]; if his classification of heights in Φ_{0,m} (at most one simple root of height >m, then height 2m) is wrong, the dimension interpretation and numerical constant in the main formula fail.","rationale":"The reader's weakest_assumption identifies the same external dependency, so I agree. The internal argument of Theorem 4.1, including the product formula, Corollary 4.2, and the integrality/dominance check for the G_λ(m) weight, is a clean application of the Weyl character formula, and I found no internal inconsistency there. Part B's case-by-case classification is dense but the exceptions are stated explicitly and the Kac-coordinate analyses are plausible. A possible secondary point is that the final if-and-only-if statement needs monotonicity of centralizer dimension among divisors of m; this follows immediately by inclusion of the sets of roots fixed by C_d for d dividing m, so it is not a serious gap. The single most load-bearing concern remains the outsourcing of dm to Polo's companion paper. Since this is an external condition to verify rather than a demonstrated error, the reader's CONDITIONAL verdict is appropriate; I would not upgrade to REJECT or ACCEPT on the evidence available.","tokens_in":40742,"tokens_out":11696,"duration_ms":104597,"concrete_test":"Write a script (e.g. in SageMath) that enumerates positive roots of each irreducible root system and, for every m dividing the Coxeter number h, computes Φ_{0,m} = {α : m | ht(α)}, checks Polo's height claim (at most one simple root of height greater than m, and if so exactly 2m), and recomputes dm = ∏_{α∈Φ_{0,m}^+} ⟨ρ/m, α∨⟩ / ⟨ρ_m, α∨⟩. Compare all values of dm, including E6 with m=4 and E7 with m=9, against the tables in [Polo, arXiv:2504.09204]. Any mismatch, or any case with two such simple roots, invalidates Theorem 4.1 as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula in Theorem 4.1 asserts Θ_λ(C_m) = (−1)^w μ(c(G)) · dλ/dm, with dm the dimension of the minuscule representation of the simply connected cover of G(m)_der of highest weight ρ/m − ρ_m. The proof, in Section 5 and in the theorem statement, explicitly delegates the determination of dm to Patrick Polo's companion paper [Polo]. The integrality and dominance of ρ/m − ρ_m on G(m)_der, and the numerical value of dm, rest on Polo's claim that at most one simple root of Φ_{0,m} has height greater than m, and that if such a root exists its height is 2m. This is a load-bearing parameter, not a cosmetic one: if Polo's height classification fails in some case, then dm is not the stated representation dimension and the right-hand side of the main formula changes. The manuscript itself flags the dependency twice: Section 5 says 'This analysis is completed in [Polo]', and Theorem 4.1 says 'Its complete description is due to Patrick Polo in [Polo]'. No machine-checked proof or independent verification is included, and the companion paper is not reproduced here. Thus the main formula is conditionally established: the internal Weyl-character-formula derivation appears sound, but the final numerical content for arbitrary groups is only as reliable as this external, unverified computation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies characters of irreducible representations of a connected reductive algebraic group G at principal elements C_m = ψ(e^{πi/m}) of the principal SL2(C). Its central result, Theorem 4.1, expresses the character value Θ_λ(C_m), up to an explicit sign, as the ratio d_λ/d_m of two dimensions: d_λ is the dimension of a representation of the derived group of G_λ(m), and d_m is the dimension of a minuscule representation of the derived group of G(m). The proof derives this from the Weyl character formula by comparing orders of vanishing at z = e^{πi/m}, and the nonvanishing criterion is tied to equality of centralizer dimensions of (λ̂+ρ̂)(e^{2πi/m}) and ρ̂(e^{2πi/m}) in Ĝ. Part B contains a case-by-case analysis of conjugacy classes of elements of order m with minimal-dimensional centralizer in classical and exceptional groups, leading to the nonvanishing iff statement for m | h outside a few exceptions. The paper also contains a factorization theorem for the adjoint character restricted to the principal SL2(C), a G2 calculation, and a discussion of when restrictions to the principal SL2(C) determine the representation.","tokens_in":41035,"tokens_out":8845,"duration_ms":83784,"significance":"The paper is a substantial contribution if its claims hold. The derivation of Theorem 4.1 from the Weyl character formula is clean and gives a uniform explanation of earlier product formulas of Prasad and of Ayyer–Kumari, while also recovering and generalizing Kostant's theorem on character values at the Coxeter class. The explicit identification of the constant d_m as the dimension of a minuscule representation gives the formula genuine representation-theoretic content. The Part B classification of minimal-dimensional centralizers is useful in its own right and supports the nonvanishing criterion. The paper is honest in flagging that the complete determination of d_m is outsourced to Polo's companion paper [Polo]; this is a load-bearing dependency that the journal should insist be made fully explicit and verifiable.","major_comments":[{"comment":"The constant d_m appearing on the right-hand side of Theorem 4.1 is load-bearing for the numerical content of the main formula, and its determination is explicitly delegated to [Polo]. The manuscript says in Section 5 that 'This analysis is completed in [Polo]' and in Theorem 4.1 that 'Its complete description is due to Patrick Polo in [Polo]', but it does not state Polo's classification or provide any verification of the needed facts: that ρ/m − ρ_m is an integral dominant weight for G(m)_der, that at most one simple root of Φ_{0,m} has height greater than m, and that if such a root exists its height is 2m. If any of these statements is incorrect, the dimension interpretation of d_m and hence the right-hand side of Theorem 4.1 changes. I request that the paper either include a precise statement of Polo's relevant theorem (with enough detail to check the dimension computation) or make the theorem formally conditional on [Polo] and ensure that the companion paper is available and independently verifiable. As it stands, the main formula for arbitrary groups is not self-contained at exactly the point where its numerical content is determined.","section":"Section 5 and Theorem 4.1"}],"minor_comments":[{"comment":"The lemma states 'let m = n/(2d)' for an odd integer d dividing 2n, but the surrounding dimension counts force m = n/d. For instance, if n = 9 and d = 3, the printed formula gives m = 3/2, which is impossible for the dimension of an orthogonal group. Please correct this typo, since the lemma is used to describe the exceptional conjugacy class.","section":"Lemma 16.1"},{"comment":"The proof of Proposition 10.1 is omitted with the explanation that it is a direct computation. Since the proposition supports the interesting claim that rank ≤ 4 groups are determined by restriction to the principal SL2(C), the paper would be stronger if at least the computational criterion was stated or a reference to a verifiable source was given.","section":"Section 10, Proposition 10.1"},{"comment":"For the exceptional groups E6, E7, F4 and G2, the proof is sketched only through a 'flavor' of the E8 argument; the reader is asked to accept the height-function cancellations. A table of the nonzero pairs (ht(α∨), ht(α∨)+⟨α0,α∨⟩) for each type would make the theorem checkable without redoing the case analysis.","section":"Section 9, Theorem 9.1"},{"comment":"The last paragraph of the theorem is easy to misread: the condition 'when G is of type D_{n+1}, m is even and 2n/m is odd' is the safe case where the uniqueness result holds, not an exception. Please rephrase to make clear that these are the cases in which the nonvanishing criterion is being asserted.","section":"Theorem 4.1, statement"}],"recommendation":"major_revision","confidential_remarks":"The central derivation is sound and the paper is clearly written, but the dependency on [Polo] for the constant d_m is the main obstacle to unqualified acceptance. If the editor and authors agree that the companion paper is part of the same submission and will be published or posted in final form, the citation may be acceptable; otherwise the present manuscript should be revised to state the needed classification explicitly. The paper would also benefit from a careful proofreading pass to fix typos such as Lemma 16.1's m = n/(2d)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the main theorem is worth taking seriously. Theorem 4.1 gives a uniform formula for character values at principal torsion elements, generalizing Kostant's theorem to all powers of the Coxeter element. The proof is a clean application of the Weyl character formula, and the technical steps—B_lambda(e^{πi/m})=1 and C_lambda=d_lambda/d_m—check out. Second, the paper is honest about what is new and what is not. The product formula and the minimal-centralizer dimension statement are explicitly attributed to earlier work, and the new content is Theorem 4.1 and the Part B classification.\n\nThe soft spot is real and the stress-test note is right: the constant d_m is not computed in this paper. It is outsourced to Polo's companion paper, and the manuscript says so twice. The height classification (at most one simple root of height > m, then height 2m) is the load-bearing assumption. If it is wrong, d_m is wrong and Theorem 4.1 falls apart. This is not a demonstrated error, just a condition to verify. A referee should check Polo's paper carefully before accepting the theorem in full generality. The exceptional-group enumeration in Part B is terse, but it is case-by-case and reproducible from the Kac coordinates they give. The nonvanishing 'if and only if' is carefully restricted to the cases where uniqueness is proved; the authors do not overclaim.\n\nThe math, as far as it goes in this paper, looks solid. The citation pattern is fine; the self-citations are to relevant prior work by Prasad, not padding. No free parameters are fitted, and the derivation is not circular.\n\nWho this is for: representation theorists who want character values at torsion elements, or who care about Kostant's theorem and centralizers. If you want to use Theorem 4.1, read [Polo] first.\n\nRecommendation: send to peer review. The referee needs to verify the Polo dependency, and if that holds, this is a valuable contribution. If I were editing, I would not desk reject.","headline":"New character formula at principal torsion elements, cleanly proved, but the constant d_m is outsourced to Polo; referee must check that companion paper.","tokens_in":41552,"tokens_out":3666,"would_cite":true,"duration_ms":33327,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70","22E55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a connected reductive group, the character of an irreducible representation at a principal torsion element is, up to a sign and a constant depending only on the group and $m$, the dimension of an explicitly constructed representation…","keywords":["character theory","principal SL2","Coxeter element","torsion elements","Weyl character formula","centralizers","Kac coordinates","branching laws"],"falsifier":"Pick a pair $(G,m,\\lambda)$ with $m\\mid h$ outside the exceptions where the two dual-group elements are conjugate, evaluate the product formula (3.3) at $z=e^{\\pi i/m}+\\epsilon$, and compare the leading coefficient with $(-1)^w\\mu(c(G))d_\\lambda/d_m$; a mismatch for any single such triple would refute Theorem 4.1. For the uniqueness half, compute the Kac coordinates for $D_5$ with $m=4$: the paper predicts exactly two conjugacy classes with minimal centralizer, so finding a third would refute Theorem 12.1.","tokens_in":40558,"feed_emoji":"📐","tokens_out":8315,"duration_ms":71116,"temperature":0.7,"pith_summary":"The paper claims that for a connected reductive group, the character of any irreducible representation at a principal torsion element — one that acts on every simple root by the same root of unity — is, up to a sign and a universal constant, the dimension of an explicitly constructed representation of a related group $G_\\lambda(m)$. The group $G_\\lambda(m)$ is built from the roots on which $\\lambda+\\rho$ pairs with a multiple of $m$, so it is not a subgroup of $G$ but a dual object to a centralizer in the dual group. The paper also proves that principal elements have the smallest possible centralizer among elements of a given order, and that for most groups whose order divides the Coxeter number this minimal-centralizer class is unique. A sympathetic reader would care because this turns a hard evaluation problem in representation theory into a dimension computation, recovers the classical theorem at the Coxeter class, and gives explicit character values for all powers of the Coxeter element.","feed_headline":"At torsion elements, characters become dimensions—up to sign","feed_subtitle":"The paper gives an exact dimension ratio when the relevant dual conjugacy holds, with explicit exceptions.","key_machinery":"The load-bearing object is the principal homomorphism $\\psi: SL_2(\\mathbb{C})\\to G$ with $\\psi=\\rho^\\vee$ on the diagonal torus, so that $C_m=\\psi(e^{\\pi i/m})$ acts on every simple root by $e^{2\\pi i/m}$. Restricted to this $SL_2$, the Weyl character formula collapses to a product over positive roots (Theorem 3.1), and the leading term at $C_m$ after writing $z=e^{\\pi i/m}+\\epsilon$ is controlled by the root subsystem $\\Phi_{\\lambda,m}=\\{\\alpha: m\\mid\\langle\\lambda+\\rho,\\alpha^\\vee\\rangle\\}$. The group $G_\\lambda(m)$ is defined by that root subsystem; $G(m)$ is its $\\lambda=0$ case. The constant $d_m$ is the dimension of the minuscule fundamental representation of the simply connected cover of $G(m)_{\\mathrm{der}}$ with highest weight $\\rho/m-\\rho_m$, a weight whose integrality and dimension come from a companion computation. Kac coordinates are the tool used in Part B to decide when the principal element is the sole minimal-centralizer class.","core_discovery":"On the paper's own terms, the central discovery is Theorem 4.1: for a principal element $C_m$ of order $m$ in the adjoint group, if $(\\hat\\lambda+\\hat\\rho)(e^{2\\pi i/m})$ is conjugate to $\\hat\\rho(e^{2\\pi i/m})$ in the dual group, then $\\Theta_\\lambda(C_m)=(-1)^w\\mu(c(G))\\,d_\\lambda/d_m$, where $d_\\lambda$ is the dimension of the highest-weight module of the simply connected cover of $G_\\lambda(m)_{\\mathrm{der}}$ with weight $(\\lambda+\\rho)/m-\\rho_{\\lambda,m}$, and $d_m$ is the analogous dimension for weight $\\rho/m-\\rho_m$ on $G(m)_{\\mathrm{der}}$. Under the stated restrictions on $(G,m)$, $\\Theta_\\lambda(C_m)\\neq 0$ exactly when that conjugacy holds. The proof runs through the Weyl character formula restricted to the principal $SL_2$, producing a product formula whose numerator and denominator cancel except on the roots of $G_\\lambda(m)$; the remaining leading term at $C_m$ is a ratio of Weyl dimension formulas. Part B shows that for $m\\mid h$, except for explicit cases in $D_n$, $E_6$, $E_7$, the principal element is the unique conjugacy class of order $m$ with minimal-dimensional centralizer.","pith_inferences":["If Question 8.1's proposed generalization holds, the same type of vanishing-and-dimension statement would apply to arbitrary torsion elements, not just principal ones; that would give a broad and purely centralizer-theoretic criterion for zero characters.","The uniqueness of the minimal-centralizer class for $m\\mid h$ can be read as a finite-order analogue of the regular-element theorem; the exceptional pairs $(E_6,4)$ and $(E_7,9)$ suggest a small finite list of obstructions worth checking for all simply laced types.","The factorization of $f_X(z)f_X(z^{-1})$ into irreducibles gives a combinatorial mechanism for constructing distinct representations with identical restrictions to the principal $SL_2$; classifying such factorizations could settle the SL6 counterexample family.","Because $G_\\lambda(m)$ is built from root divisibility rather than from a subgroup, the formula hints that characters at torsion elements should be thought of as dimensions attached to dual-centralizer data; this may extend to other settings where similar ratio-of-cancellation arguments exist."],"forward_implications":["For every divisor $m$ of the Coxeter number, outside the listed exceptions, $\\Theta_\\lambda(C_m)\\neq 0$ exactly when $(\\hat\\lambda+\\hat\\rho)(e^{2\\pi i/m})$ and $\\hat\\rho(e^{2\\pi i/m})$ are conjugate; the nonzero value is then $(-1)^w\\mu(c(G))d_\\lambda/d_m$.","The classical theorem at the Coxeter class follows as $G(h)$ is a torus: the character is $0$ or $\\pm 1$, and nonzero exactly when the shifted cocharacter lies in the Coxeter class of the dual group.","For classical groups the centralizers $Z_G(C_d)$ and the auxiliary groups $G(d)$ are explicit products of general linear, symplectic, and orthogonal groups, so character values become dimensions of representations of those explicit groups.","The adjoint representation restricted to the principal $SL_2$ has a character that factors into at most three terms of the form $1\\pm z^d$ (Theorem 9.1), a simplification not apparent from the usual sum decomposition.","The Weyl dimension formula is recovered as the $z\\to 1$ limit of the product formula."],"supporting_citations":[{"why":"Completes the computation of $d_m$ and the integrality of $\\rho/m-\\rho_m$, on which the constant in Theorem 4.1 depends.","marker":"[Polo]"},{"why":"Supplies the Coxeter-class character theorem that the paper reproves and extends to powers of the Coxeter element.","marker":"[Kos76]"},{"why":"Provides the character relationship on $GL_n$ that motivates the product formula and dimension-ratio shape of the main theorem.","marker":"[Pr1]"},{"why":"Gives twisted classical character product formulas and observes that the auxiliary group need not be a subgroup, a key structural fact used here.","marker":"[AK]"},{"why":"Contains the Serre theorem that principal elements have minimal-dimensional centralizers, used as Proposition 3.8.","marker":"[EKV]"},{"why":"Supplies the Kac-coordinates formalism used throughout Part B to classify minimal-centralizer torsion classes.","marker":"[Ser06]"},{"why":"Gives the precise formulation of the Coxeter-class theorem quoted as Corollary 4.5.","marker":"[Pr2]"}],"fun_headline_variants":["Torsion characters: exact dimension ratio","When torsion characters equal scaled dimensions","Torsion twist: characters become dimensions","Characters at torsion: dimensions up to sign"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The constant $d_m$ in the main formula rests on a computation carried out in the companion paper [Polo] that $\\rho/m-\\rho_m$ is an integral dominant weight of $G(m)_{\\mathrm{der}}$ with the stated dimension; if that computation is wrong, the constant is wrong.","fun_headline_variants_meta":{"raw":{"variants":["Torsion characters: exact dimension ratio","When torsion characters equal scaled dimensions","Torsion twist: characters become dimensions","Characters at torsion: dimensions up to sign"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00051,"raw_usage":{"total_tokens":2455,"prompt_tokens":892,"completion_tokens":1563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":1510}},"tokens_in":508,"tokens_out":1563,"duration_ms":12717,"temperature":1.0,"reasoning_tokens":1510,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:43:21.786194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a pair $(G,m,\\lambda)$ with $m\\mid h$ outside the exceptions where the two dual-group elements are conjugate, evaluate the product formula (3.3) at $z=e^{\\pi i/m}+\\epsilon$, and compare the leading coefficient with $(-1)^w\\mu(c(G))d_\\lambda/d_m$; a mismatch for any single such triple would refute Theorem 4.1. For the uniqueness half, compute the Kac coordinates for $D_5$ with $m=4$: the paper predicts exactly two conjugacy classes with minimal centralizer, so finding a third would refute Theorem 12.1.","supporting_citations":[],"review_version":1}