{"id":"bd086552-db17-45f2-baaf-f006176164e5","arxiv_id":"2412.17324","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Character values at involutions for GL(n), Sp(2n), SO(2n), SO(2n+1), and G2 are written as products or alternating sums of dimensions of subgroup representations, with vanishing conditions, though only the GL(n) case is actually proven here.","lead":"This paper derives closed-form formulas for the trace of an involution acting on highest weight representations of the classical groups and G2, expressing these traces as products or alternating sums of dimensions of smaller representation pieces. The GL(n) computation is new and fully proven, but the corresponding results for the other families are asserted without proofs, and the G2 theorem contains internal sign and case errors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 8.1 for G2 is internally inconsistent: the Weyl denominator sign is wrong, the stated SL2 weights are half-integral in the case (k,l)=(2,1), and the theorem contradicts Proposition 2; the paper's G2 claim is false as written.","rationale":"The reader's verdict rejects the paper, and my independent read supports that rejection, though for a slightly different primary reason. The reader's weakest_assumption is the 'analogous proof' premise for Sp and SO: Theorems 5.1 and 6.1 do not contain the column-operation or shuffle-sign computation, and Theorem 7.1 explicitly says its proof is omitted. That is a real gap. But the most load-bearing, concretely checkable failure is Theorem 8.1: it is not just unsupported but false as written. A false theorem in the G2 section violates the abstract's assertion that the paper computes G2 character values at order-2 elements, independently of whether the classical sections can be repaired. I give credit where it is due: the GL(n) theorem in Section 4 appears to be a genuine self-contained Weyl-character calculation, and the classical formulas may be salvageable. But the submitted text cannot support its central claim. The correct final verdict remains REJECT; the concern does not move the verdict, so I mark the recommendation as UNCHANGED. If the missing Sp/SO/B_n proofs are supplied and Theorem 8.1 is corrected, a revised manuscript could be reconsidered.","tokens_in":16037,"tokens_out":15546,"duration_ms":140415,"concrete_test":"Fix (k,l)=(2,1) and evaluate the G2 character by the Schur quotient of Proposition 1 at X=(x,-x,-x^{-2}). Expand the quotient around x=1 using the correct denominator -x^2+x^{-2}; the limit is -3, in agreement with Proposition 2 and with Reeder. Then compare with Theorem 8.1's 'k even, l odd' case: the weights 5/4 and 3/2 are not SL2 highest weights, the formal value is -45/8, and the proof's own concluding factors give +3. This single calculation isolates the internal inconsistency and shows that Theorem 8.1, as stated, is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim covers G2 as well as A_n, B_n, C_n, D_n. The GL(n) result in Section 4 is a real computation, but the G2 theorem is not merely unproved: Theorem 8.1 is internally inconsistent. Its denominator is printed as S(1,1,0)-S(1,0,0)=-x^2-x^{-2} at X=(x,-x,-x^{-2}); direct substitution gives -x^2+x^{-2}, so the denominator used in the proof has the wrong sign and the claimed nonvanishing at x=1 is actually a 0/0 limit. The case formula for 'k even, l odd' uses SL2 weights (3l+k)/4 and (k+l)/2, which for (k,l)=(2,1) are 5/4 and 3/2, not highest weights of any SL2(C) representation. The proof's own simplification in that case ends with the different factors Theta_5 and Theta_6, so the theorem and its proof disagree. Directly taking the limit of Proposition 1's Schur quotient for (2,1) gives -3, matching Proposition 2; the theorem's displayed expression gives -45/8 if formally evaluated, while the proof's product gives +3. Since Proposition 2 is correct and already appears in Reeder's work, Theorem 8.1 cannot be right as written. Separately, Section 7 explicitly says the proof of both parts is omitted, and Theorems 5.1 and 6.1 are only one-sentence analogies, so the B_n, C_n, and D_n assertions are unsupported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes explicit formulas, in terms of products or alternating sums of dimensions of highest-weight representations of smaller classical groups, for the values of irreducible characters at diagonal elements of order 2 for GL(n,C), Sp(2n,C), SO(2n,C), SO(2n+1,C), and G2(C). The GL(n,C) result is proved in detail from the Weyl character formula, using a determinant expansion and explicit column operations. The symplectic and orthogonal theorems are each dispatched with a one-sentence statement that the proof is analogous, and the odd orthogonal theorem explicitly omits the proofs of both parts. The G2(C) section derives factorizations from the Fulton-Harris SL3(C)-restriction character formula and compares them with a formula attributed to Reeder.","tokens_in":16188,"tokens_out":14207,"duration_ms":127186,"significance":"If the full set of formulas were correct, the paper would give a uniform and attractive reduction of order-2 character values to dimensions of smaller representations, together with parity-based vanishing criteria. The GL(n) theorem appears to be a genuine and checkable result, and the determinant manipulations in Section 4 are explicit enough to verify. However, the advertised scope is far larger than what is actually demonstrated: the C_n, D_n, and B_n assertions are unsupported as submitted, and the G2 theorem is internally inconsistent and false as stated. The significance of the full paper is therefore not established, although the GL(n) portion has independent value.","major_comments":[{"comment":"The proof of Theorem 8.1 states that S(1,1,0)(X)-S(1,0,0)(X) = -x^2 - x^{-2} for X=(x,-x,-x^{-2}). Direct substitution gives S(1,1,0)(X) = -x^2 and S(1,0,0)(X) = -x^{-2}, so the difference is -x^2 + x^{-2}. This is not a cosmetic sign error: the evaluation at x=1 is a 0/0 limit, and replacing the denominator by the printed expression changes the limiting value of the quotient.","section":"§8, Theorem 8.1"},{"comment":"For k even and l odd, the statement of Theorem 8.1 assigns SL2(C) factors of highest weights (3l+k)/4 and (k+l)/2, whereas the proof of Case II concludes with factors of highest weights (2k+3l+1)/4 and (l-1)/2. For (k,l)=(2,1) these are (5/4,3/2) versus (2,0). Neither 5/4 nor 3/2 is a highest weight of an irreducible algebraic SL2(C) representation, and no definition of characters with half-integral highest weights is supplied. If one formally evaluates the displayed product at x=1 one obtains -45/8, while the proof's own product gives +3 and Proposition 2 gives -3. Thus the theorem and its proof disagree, and the G2 claim is false as written.","section":"§8, Theorem 8.1, Case II"},{"comment":"The theorems for Sp(2n,C), SO(2n,C), and SO(2n+1,C) are central to the abstract's claim, but no computations are provided for them. Theorem 5.1 and Theorem 6.1 each say only that the proof is analogous to the GL(n) proof, and Theorem 7.1 explicitly says that the proofs of both parts are omitted. In particular, the constants 2^{d(k)}, 2^{e(k)}, and 2^{2kn-2k^2+k-1}, the two-determinant SO(2n) numerator, and the half-integer shifted SO(2n+1) numerator are never derived. The B_n, C_n, and D_n results are therefore unsupported as submitted.","section":"§5–§7, Theorems 5.1, 6.1, 7.1"},{"comment":"The sentence introducing Proposition 2 states that its proof is a direct consequence of Theorem 5.1 evaluated at (x,-x,-x^{-2}) for x=1. This cannot be correct as printed, because Theorem 5.1 concerns Sp(2n,C) and its element D_{n-k,k}, not G2(C). Since Proposition 2 is used to interpret the G2 character values in Remark 5, its actual source (Reeder [R]) and its logical relation to Theorem 8.1 should be stated correctly.","section":"§8, Proposition 2"}],"minor_comments":[{"comment":"The text says that the highest weight λ0 of GL(2m,C) is given by the recipe of Theorem 2.17 in [AK], but m is not defined; in this part n=2k, so the group should presumably be GL(2k,C) or GL(n,C).","section":"§7, Theorem 7.1(A)"},{"comment":"The notation S[λ]C_{2n} is used for a representation of SO(2n+1,C); the subscript C_{2n} appears to be a typo for C_{2n+1}, since the ambient group has dimension 2n+1.","section":"§7, Theorem 7.1"},{"comment":"The proposition uses both Π_{k,l} and Θ_{k,l} for the G2 highest weight representation and its character; the notation should be made consistent.","section":"§8, Proposition 1"},{"comment":"The theorem statements rely on the parity sets η_i(λ), but the dependence of the final formulas on the choice of the determinant normalization in the GL(n) case is only described informally; making that normalization explicit would improve readability.","section":"General"}],"recommendation":"reject","confidential_remarks":"The GL(n,C) section is a solid computational proof and could form the basis of a publishable paper on its own. As submitted, however, the manuscript claims results for types B_n, C_n, D_n, and G2 that are either unsupported or, in the G2 case, internally inconsistent. I would not recommend accepting or inviting a minor revision in the current scope; a new submission that restricts the claims to what is actually proved, or that supplies complete corrected proofs for all families, would need to be evaluated afresh."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The GL(n) part is the real thing. Theorem 4.1 covers every involution class of GL(n,C), not just the n = 2k case in [DP] and [AK]. The proof is a complete Weyl character formula computation: denominator expansion, column operations, shuffle signs, and the 2^{c(k)} constant are all written down, with no fitted parameters anywhere. The vanishing condition in terms of #η0(λ), the factorization into dimensions of GL(n−k) × GL(k) representations, and the alternating sum in the middle regime all hang together. That is elegant, reusable, and new in this full range.\n\nNow the soft spots, in proportion. The C_n, D_n, and B_n theorems each get \"the proof is analogous,\" and Section 7 says outright that both parts of its proof are omitted. This is not a routine analogy: the SO(2n) numerator is a sum of two determinants, the SO(2n+1) element has an extra coordinate with half-integer shifts, and the constants 2^{d(k)}, 2^{e(k)}, and 2^{2kn−2k^2+k−1} have to fall out of those computations. For a paper whose abstract claims all five families, that is a real gap.\n\nThe G2 section is worse than unproven: Theorem 8.1 is wrong as written. The denominator is printed as −x^2 − x^{−2}; substituting (x, −x, −x^{−2}) into S(1,1,0) − S(1,0,0) gives −x^2 + x^{−2}. The \"k even, l odd\" case assigns SL2 weights (3l+k)/4 and (k+l)/2, which for (k,l)=(2,1) are 5/4 and 3/2—not highest weights of any SL2 representation—and those factors disagree with the proof's own final product. At x = 1 the theorem contradicts the paper's own Proposition 2, and two of the four cases are simply omitted. Proposition 2 itself is correct, but it is already in Reeder's paper. The citation pattern elsewhere is fine; the paper is honest about its debt to [DP], [AK], and [FH].\n\nWho this is for: representation theorists working on character formulas and anyone building on [DP]/[AK]. The GL(n) theorem is worth having and worth citing. But the abstract overclaims, Theorem 8.1 needs to be re-derived, and the Sp/SO proofs need to actually appear. I'd send this to a referee—the GL(n) core is substantive and formally detailed enough to warrant referee time—but I'd expect heavy revision. If the author fixes the G2 section and writes out the missing classical-group proofs, this becomes a good paper.","headline":"Theorem 4.1 for GL(n) is a genuine, fully-proven new result, but the paper's other pillars—Sp/SO by one-sentence analogy and a G2 theorem that contradicts its own proof—are not yet standing.","tokens_in":16978,"tokens_out":8487,"would_cite":true,"duration_ms":64216,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20G05","05E05","20G20","22E46"],"pacs":[],"model":"deepseek-v4-flash","headline":"Order-2 elements are singular, yet this paper proves that the character of any highest-weight representation of a classical group or G₂ at such an element is always zero or, up to sign and an explicit power of 2, a product or alternating…","keywords":["Weyl character formula","character values","order-2 elements","highest weight representations","classical groups","factorization of characters","G2","involutions"],"falsifier":"Check the unproved symplectic analogy on one small case: for $\\mathrm{Sp}(6,\\mathbb{C})$ take $n=3$, $k=1$, the order-2 element $D_{2,1} = (1,1,-1,-1,1,1)$, and the weight $\\lambda = (2,2,1)$, for which $\\lambda + \\rho_3 = (5,4,2)$ and $\\#\\eta_0(\\lambda) = 2 = n-k$. Theorem 5.1B predicts $\\Theta_\\lambda(D_{2,1}) = \\pm 2^1 \\cdot \\dim S\\langle(1,1)\\rangle_{\\mathbb{C}^4} \\cdot \\dim S[(2)]_{\\mathbb{C}^3} = \\pm 2 \\cdot 5 \\cdot 5 = \\pm 50$, where $S\\langle(1,1)\\rangle_{\\mathbb{C}^4}$ is the 5-dimensional irreducible of $\\mathrm{Sp}(4,\\mathbb{C})$ and $S[(2)]_{\\mathbb{C}^3}$ the 5-dimensional irreducible of $\\mathrm{SO}(3,\\mathbb{C})$. Evaluating $\\Theta_\\lambda$ at $D_{2,1}$ directly (via the limiting Weyl character formula of the paper, or with a computer algebra system) and comparing with $\\pm 50$ settles whether the stated constant $2^{(n-2k)^2}$ is correct.","tokens_in":15579,"feed_emoji":"🧮","tokens_out":23409,"duration_ms":170139,"temperature":0.7,"pith_summary":"The paper computes, for every highest-weight representation of $\\mathrm{GL}(n,\\mathbb{C})$, $\\mathrm{Sp}(2n,\\mathbb{C})$, $\\mathrm{SO}(2n,\\mathbb{C})$, $\\mathrm{SO}(2n+1,\\mathbb{C})$, and the exceptional group $\\mathrm{G}_2$, the value of the character (the trace function) at every conjugacy class of order-2 elements. The uniform answer is that such a value is either zero or, up to sign and an explicit power of $2$, the dimension of a tensor product of two smaller highest-weight representations, or an alternating sum of such dimensions over $k$-subsets. Vanishing is decided purely by parity: counting how many coordinates of $\\lambda+\\rho$ are even versus odd, the same count that forces zero for $\\mathrm{GL}$, $\\mathrm{Sp}$, and $\\mathrm{SO}(2n)$ does not do so for $\\mathrm{SO}(2n+1)$. This converts a Weyl-character computation at a singular element into dimension formulas of smaller classical groups and extends the factorization theorems of [DP] and [AK] from special regular elements to all involutions.","feed_headline":"Order-2 character values reduce to dimensions of smaller reps","feed_subtitle":"For GL, Sp, SO, and G₂, the value is either zero or a signed power of 2 times a smaller dimension.","key_machinery":"The machinery is the Weyl character formula taken as a limit: an order-2 element is singular, so the Weyl denominator vanishes, and $\\Theta_\\lambda$ is obtained as the $\\varepsilon \\to 0$ limit of the ratio at the nearby regular element $C_{n-k,k}(\\varepsilon)$ with coordinates $x_j(\\varepsilon) = 1 + j\\varepsilon$. The parity sets $\\eta_i(\\lambda) = \\{a \\in \\lambda + \\rho : a \\equiv i \\bmod 2\\}$ organise the rows of the Weyl numerator, and a fixed sequence of column operations ($C_{n-k+i} \\to C_{n-k+i} - C_{n-2k+i}$, then scaling by $-1/2$) puts that numerator into block form. Three determinant facts decide the outcome: Lemma 1 (expansion into complementary minors with shuffle signs $\\varepsilon_S$), Corollary 1 (a zero submatrix with $a+b > n$ forces the determinant to vanish, giving the vanishing theorem), and Corollary 2 (a zero submatrix with $a+b = n$ splits the determinant into two factors, giving the factorization theorem). In the limit, each surviving factor becomes a character of a smaller classical group at the identity — hence a dimension — and the cross-product $\\prod_{s,t}(x_s(\\varepsilon) + x_t(\\varepsilon))$ becomes the stated power of $2$.","core_discovery":"The central claim, proved in detail for $\\mathrm{GL}(n,\\mathbb{C})$ (Theorem 4.1) and stated by analogy for the other groups, is a complete description of order-2 character values. Fix the involution $C_{n-k,k} = (1,\\ldots,1,-1,\\ldots,-1)$ with $n-k$ ones and $k$ minus-ones, and (after a determinant-character twist if needed) suppose $\\#\\eta_0(\\lambda) \\geq \\#\\eta_1(\\lambda)$, where $\\eta_0(\\lambda)$ and $\\eta_1(\\lambda)$ are the even and odd coordinates of $\\lambda + \\rho$. Then $\\Theta_\\lambda(C_{n-k,k})$ vanishes when $\\#\\eta_0(\\lambda) > n-k$; when $\\#\\eta_0(\\lambda) = n-k$ it equals $\\pm 2^{c(k)} \\dim\\bigl(S(\\lambda_0)_{\\mathbb{C}^{n-k}} \\otimes S(\\lambda_1)_{\\mathbb{C}^k}\\bigr)$ with $c(k) = \\binom{n-2k}{2}$, where $\\lambda_0 + \\rho_{n-k} = \\eta_0(\\lambda)/2$ and $\\lambda_1 + \\rho_k = [\\eta_1(\\lambda)-1]/2$; and when $\\#\\eta_0(\\lambda) < n-k$ it is a signed alternating sum, over $k$-subsets of the odd coordinates, of such tensor-product dimensions divided by $2^{k(n-k-1)}$. The symplectic and even-orthogonal analogues (Theorems 5.1 and 6.1) follow the same schema with constants $2^{(n-2k)^2}$ and $2^{2\\binom{n-2k}{2}-k+1}$, the latter built from Spin representations of smaller even orthogonal groups, while $\\mathrm{SO}(2n+1,\\mathbb{C})$ (Theorem 7.1) has no vanishing part. For $\\mathrm{G}_2(\\mathbb{C})$, the character at the unique order-2 class is zero exactly when $k$ and $l$ are both odd, and otherwise factors as a product of two $\\mathrm{SL}_2(\\mathbb{C})$ characters evaluated at $x^2$ and $x^3$, specialising at $x=1$ to explicit quadratic polynomials in $k$ and $l$ (Theorem 8.1, Proposition 2).","pith_inferences":["The three exponents $c(k) = \\binom{n-2k}{2}$, $d(k) = (n-2k)^2$, and $e(k) = 2\\binom{n-2k}{2} - k + 1$ probably share one combinatorial meaning — plausibly the count of even-coordinate pairs forced by the limit — that the deferred 'analogous' proofs would expose; if so, the same exponent should reappear as a multiplicity in the restriction of the representation to the involution's fixed subgroup.","Remark 4 asks what is special about the element $(x,-x,-x^{-2})$ in $\\mathrm{SL}_3(\\mathbb{C})$; the proof identifies it with the quotient of $\\mathrm{SL}_2 \\times \\mathrm{SL}_2$ by $(-1,-1)$ inside $\\mathrm{G}_2$, so a testable extension is that factorization occurs precisely along the image of that homomorphism.","Since $\\mathrm{SO}(2n+1,\\mathbb{C})$ has no vanishing theorem, an explicit vanishing criterion for $B_n$ remains open; the natural route is the same $\\varepsilon \\to 0$ Weyl-limit computation with the half-integer shifts of $\\rho_{2n+1}$, which should yield a parity condition involving the extra fixed coordinate $1$."],"forward_implications":["Every order-2 character value of $\\mathrm{GL}(n,\\mathbb{C})$ is decided by one parity count: comparing $\\#\\eta_0(\\lambda)$ with $n-k$ selects zero, a two-factor dimension product, or an alternating sum over $\\binom{\\#\\eta_1(\\lambda)}{k}$ terms.","The same parity count governs $\\mathrm{Sp}(2n,\\mathbb{C})$ and $\\mathrm{SO}(2n,\\mathbb{C})$, with constants $2^{(n-2k)^2}$ and $2^{2\\binom{n-2k}{2}-k+1}$, the latter involving Spin representations of the smaller even orthogonal groups.","In the balanced cases ($n = 2k$ or $n = 2k+1$ for $\\mathrm{GL}$, $n = 2k$ for $\\mathrm{Sp}$) the power of $2$ is $1$, so the character value is, up to sign, exactly one dimension, recovering the order-2 case of the factorization theorems of [DP] and [AK].","For $\\mathrm{G}_2(\\mathbb{C})$ the order-2 character value is an explicit quadratic polynomial in the highest-weight coefficients $k,l$, vanishing exactly when both are odd.","$\\mathrm{SO}(2n+1,\\mathbb{C})$ is the exception: it has no vanishing theorem, and its order-2 values are signed sums (single dimensions only in the $n = 2k$ case) of dimensions of smaller orthogonal-group representations."],"supporting_citations":[{"why":"Supplies the original factorization theorem for characters of GL(mn,C) at the special elements t·c_n; Theorem 4.1 specialises to order-2 elements and Remark 1 recovers its Theorem 2.","marker":"[DP]"},{"why":"Generalises the factorization to all classical groups and is the direct predecessor of Theorems 5.1-7.1; its Theorem 2.17 is invoked for the n=2k case of SO(2n+1,C), and Remarks 1-2 recover its Theorems 2.5 and 2.11.","marker":"[AK]"},{"why":"Proposition 24.48, expressing G2(C) characters as a quotient of SL3(C) characters, is the stated main tool for the entire G2 section, yielding Theorem 8.1 and Proposition 2.","marker":"[FH]"},{"why":"Already contains the G2(C) character values at the order-2 class C2 that Proposition 2 reproduces, providing the comparison baseline for the G2 results.","marker":"[R]"}],"fun_headline_variants":["Order-2 char values: zero or factors of smaller rep dims","Exact order-2 char values via smaller rep dimensions","Order-2 characters for GL, Sp, SO, G2: reduced formulas","Order-2 char values: zero or product/alternating sum of dims"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Theorems 5.1, 6.1, and 7.1 defer their proofs to the statement that the argument is analogous to the proof for $\\mathrm{GL}(n,\\mathbb{C})$: no column-operation or shuffle-sign computation is written down for the symplectic and orthogonal Weyl numerators, so the stated constants $2^{(n-2k)^2}$, $2^{2\\binom{n-2k}{2}-k+1}$, and $2^{2kn-2k^2+k-1}$ all depend on that analogy holding.","fun_headline_variants_meta":{"raw":{"variants":["Order-2 char values: zero or factors of smaller rep dims","Exact order-2 char values via smaller rep dimensions","Order-2 characters for GL, Sp, SO, G2: reduced formulas","Order-2 char values: zero or product/alternating sum of dims"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001198,"raw_usage":{"total_tokens":5043,"prompt_tokens":1154,"completion_tokens":3889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":770,"completion_tokens_details":{"reasoning_tokens":3809}},"tokens_in":770,"tokens_out":3889,"duration_ms":22675,"temperature":1.0,"reasoning_tokens":3809,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:38:58.505877+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the unproved symplectic analogy on one small case: for $\\mathrm{Sp}(6,\\mathbb{C})$ take $n=3$, $k=1$, the order-2 element $D_{2,1} = (1,1,-1,-1,1,1)$, and the weight $\\lambda = (2,2,1)$, for which $\\lambda + \\rho_3 = (5,4,2)$ and $\\#\\eta_0(\\lambda) = 2 = n-k$. Theorem 5.1B predicts $\\Theta_\\lambda(D_{2,1}) = \\pm 2^1 \\cdot \\dim S\\langle(1,1)\\rangle_{\\mathbb{C}^4} \\cdot \\dim S[(2)]_{\\mathbb{C}^3} = \\pm 2 \\cdot 5 \\cdot 5 = \\pm 50$, where $S\\langle(1,1)\\rangle_{\\mathbb{C}^4}$ is the 5-dimensional irreducible of $\\mathrm{Sp}(4,\\mathbb{C})$ and $S[(2)]_{\\mathbb{C}^3}$ the 5-dimensional irreducible of $\\mathrm{SO}(3,\\mathbb{C})$. Evaluating $\\Theta_\\lambda$ at $D_{2,1}$ directly (via the limiting Weyl character formula of the paper, or with a computer algebra system) and comparing with $\\pm 50$ settles whether the stated constant $2^{(n-2k)^2}$ is correct.","supporting_citations":[],"review_version":1}