{"id":"addfb4eb-ce60-4738-a1a0-639443a876ec","arxiv_id":"2411.11447","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit Murnaghan-Nakayama rules are established for symplectic, orthogonal, and orthosymplectic Schur functions, each with three distinct types of terms.","lead":"This paper proves new versions of the Murnaghan-Nakayama rule for multiplying power sums by symplectic, orthogonal, and orthosymplectic Schur functions. The new rules contain border-strip addition, border-strip removal, and a third mixed term.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the Lemma 4.2 basis-replacement step is terse but valid.","rationale":"The reader accepted with moderate confidence, flagging the replacement step in Lemma 4.2 as the weakest assumption. I examined that step and found it sound: the hook Schur identity (4.7) is an equality in the ring of supersymmetric functions; after expanding the right-hand side in the triangular basis {hs_σ(X/Y)y_m^k}, coefficient comparison is legitimate, and the same coefficients apply to spo because (2.13) is the exact analogue of (2.12). Thus the replacement is not a hidden assumption. I also spot-checked the symplectic and orthosymplectic rules in small cases; the formulas match. The paper has minor typos (e.g., 'p3' in Example 3.8) and the combinatorial proof of (4.2) is sketched, but the algebraic proofs are complete. No load-bearing concern remains.","tokens_in":16606,"tokens_out":33287,"duration_ms":293546,"concrete_test":"Expand both sides of Lemma 4.2, Eq. (4.4), for n=1, m=2, r=2, λ=(1) using the recursion (2.13) for spo and compare coefficients of each distinct term spo_σ(X/(y1)) y_2^k. If they match, the replacement step is confirmed; run the same expansion for λ=(2,1), n=2, r=3, m=2 as a second check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's flagged concern—the basis-replacement step in Lemma 4.2—is the only candidate for a load-bearing gap, but on inspection it does not land. Equation (4.7) is an equality of supersymmetric functions; expanding the right-hand side via (2.12) into the set {hs_σ(X/Y) y_m^k} (which is linearly independent, since it is triangular over the standard hook-Schur basis) shows the coefficient of each such element is identical on both sides. Replacing hs by spo in both sides is legitimate because spo satisfies the identical vertical-strip recursion (2.13), so the same coefficient comparison holds after replacement. The same reasoning covers Lemma 4.1. The proof is terse and the citation of the basis theorem is compressed, but no hidden assumption is needed. No other step in Theorems 3.4, 3.7, 3.10, or 4.3 appears unsound; spot checks of n=1 and n=2 cases match the formulas.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes Murnaghan–Nakayama rules for symplectic, odd orthogonal, even orthogonal, and orthosymplectic Schur functions. For each family, the product of the relevant power-sum function with a character is expressed as a sum of three terms: a border-strip addition term, a border-strip removal term, and a third term described both algebraically and combinatorially. The proofs use determinantal identities (Lemmas 3.1 and 3.9) and, for the orthosymplectic case, an induction on the number of y-variables that transfers identities from ordinary and hook Schur functions to orthosymplectic Schur functions via basis arguments (Lemmas 4.1 and 4.2).","tokens_in":16805,"tokens_out":10966,"duration_ms":99968,"significance":"If correct, these results provide natural and useful extensions of the classical Murnaghan–Nakayama rule to characters of symplectic, orthogonal, and orthosymplectic groups and supergroups. The symplectic and orthogonal rules are new, and the orthosymplectic rule is a nontrivial hybrid involving both symplectic and ordinary Schur functions. The paper is largely self-contained, with explicit examples, and the algebraic proofs are mostly complete. A notable strength is that the key basis-transfer lemmas are valid: the identities are linear, and the replacement of hook Schur functions by orthosymplectic Schur functions is justified because the coefficient comparison is performed in a basis. The combinatorial descriptions add concreteness, though one combinatorial proof is only sketched.","major_comments":[],"minor_comments":[{"comment":"The combinatorial proof ends with the sentence 'One can describe similar cancellations that occur in the left-hand side of (4.2) and, since each of the correspondences is reversible, this completes the proof.' This is a hand-waving clause; since the algebraic proof of Lemma 4.1 is complete, this is not a correctness issue, but the combinatorial proof should either be finished or explicitly labeled as a sketch.","section":"Section 4, combinatorial proof of (4.2)"},{"comment":"There are several typos: 'Murnaghan–Nakayma' in the Introduction, 'Stanely' should be 'Stanley', and the reference [Nak41] misspells 'Nakayama' as 'Nakayma'.","section":"Introduction and References"},{"comment":"In the last sentence of the proof of Lemma 4.2, 'the equality in (4.3) also holds' should refer to (4.4).","section":"Lemma 4.2 proof"},{"comment":"The displayed expansion of A_{\\alpha+r\\epsilon_j} is dense and the indexing is hard to follow; a short explanation of the row-swap argument and the role of each term would improve readability.","section":"Lemma 3.1 proof"},{"comment":"The five symplectic tableaux of shape (1,1) are not clearly typeset, making it difficult to verify the count and the resulting polynomial.","section":"Example 2.5"},{"comment":"The statement says the proof is obtained by replacing δ=(n,...,1) with δ=(n-1/2,...,1/2); it would be helpful to note explicitly that the determinant identity in Lemma 3.1 and the reflection argument extend to half-integer shifts without change.","section":"Theorem 3.7"},{"comment":"When (2.5) is invoked to replace the sum over ν with s_{\\lambda'/\\mu'}(Y), the argument implicitly uses the conjugate version of (2.5), since the relevant strips are vertical rather than horizontal; this should be stated for clarity.","section":"Theorem 4.3 proof"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound in its main claims; the only concerns are presentation-level. The orthosymplectic basis-transfer step is terse but correct, and the combinatorial proof of (4.2) is incomplete but backed by a full algebraic proof. The typos and the hand-waving clause should be addressed before publication. The paper fits the scope of a combinatorics journal well."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper gives the first Murnaghan–Nakayama rules for symplectic, orthogonal, and orthosymplectic Schur functions. The symplectic rule (Theorem 3.4) is a three-term expansion: add border strips, remove border strips, plus a third sum indexed by the auxiliary partitions μ^(q). The orthogonal rules are the same determinant argument with shifted δ, and the orthosymplectic rule (Theorem 4.3) is genuinely hybrid, mixing spo terms with a sum over ordinary sp_μ s_{λ'/μ'}(Y). I spot-checked the examples and the n=1,2 cases; they match.\n\nStrengths. The proofs are mostly complete and self-contained given classical inputs: Weyl character formulas, the classical MN rule, and Stembridge's basis theorem. Lemma 3.1 is a clean induction. The paper states the third term both algebraically and combinatorially, with worked examples, and the literature review supports the claim of novelty. Citation patterns look appropriate.\n\nSoft spots. The basis-replacement arguments in Lemmas 4.1 and 4.2 are compressed. The reader flagged this as the main risk, and I think the concern does not land. Equation (4.7) is an identity in supersymmetric functions; expanding in the triangular basis {hs_ν(X/Y) y_m^k} shows the coefficients match, and replacing hs by spo is just forming the same linear combination. Likewise for (4.3) in the ordinary s-basis. But the paper should spell that out explicitly; as written, \"the partitions involved are the same\" is too quick for a referee to verify without doing this work. The combinatorial proof of (4.2) is long, depends on figures, and ends with a hand-wavy \"similar cancellations\" clause. Since the algebraic proof is already there, this is a presentation issue, not a correctness issue. Minor typos (for example, an index slip in the n=1 case of Lemma 3.1 and the use of r in Definition 3.2 before it is fixed) should be cleaned up.\n\nRecommendation: this is a solid, specialized contribution. It deserves a serious referee. I would send it out, with the expectation of minor revisions: expand the two basis-replacement steps and tighten the combinatorial proof. Not a desk reject.","headline":"Solid new MN rules for symplectic, orthogonal, and orthosymplectic Schur functions; the proofs are essentially right, with two terse basis-replacement steps that need expanding but are not gaps.","tokens_in":17302,"tokens_out":4910,"would_cite":true,"duration_ms":51084,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E05","05E10","05A19"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves Murnaghan–Nakayama rules for symplectic, orthogonal, and orthosymplectic Schur functions, expanding $p_r$ times each character as a signed sum with border-strip additions, removals, and a reflected term.","keywords":["Murnaghan–Nakayama rule","symplectic Schur functions","orthogonal Schur functions","orthosymplectic Schur functions","border strips","determinantal character formula","supersymmetric functions","hook Schur functions"],"falsifier":"Compute a concrete case with a nonzero third term, such as $n=3$, $\\mu=(4,3,1)$, $r=6$ from Example 3.6, by evaluating both sides of Theorem 3.4 as explicit Laurent polynomials using the determinant character formula; if the coefficient of any $\\mathrm{sp}_{\\mu^{(q)}}$ fails to match, the rule is false. Similarly, test the orthosymplectic case in Example 4.4 with $\\lambda=(2,2)$, $n=2$, $m=2$, $r=3$.","tokens_in":16432,"feed_emoji":"🧮","tokens_out":11077,"duration_ms":87352,"temperature":0.7,"pith_summary":"This paper establishes Murnaghan–Nakayama rules for symplectic, orthogonal, and orthosymplectic Schur functions: explicit formulas for the product of a power-sum symmetric function $p_r$ with each family of character polynomials. In the classical setting, the Murnaghan–Nakayama rule expands $p_r s_\\mu$ as a signed sum of Schur functions indexed by border-strip additions of size $r$. The paper shows that the symplectic, orthogonal, and orthosymplectic analogues each contain three parts: a border-strip addition term, a border-strip removal term, and a third term described by a reflection construction on the Young diagram. These results matter because symplectic, orthogonal, and orthosymplectic Schur functions are characters of natural families of Lie groups and Lie superalgebras, so the rules give a direct way to multiply such characters.","feed_headline":"Symplectic and orthogonal characters get new Murnaghan–Nakayama rules","feed_subtitle":"Formulas multiply power sums with characters via border-strip additions, removals, and a reflected term.","key_machinery":"The engine is the determinant form of the characters. For $\\alpha$ with integer or half-integer coordinates, set $A_\\alpha = \\det(x_i^{\\alpha_j} - x_i^{-\\alpha_j})$ and $N_\\alpha = \\det(x_i^{\\alpha_j} + x_i^{-\\alpha_j})$. Lemma 3.1 gives the identity $p_r A_\\alpha = \\sum_j A_{\\alpha+r\\epsilon_j} + \\sum_j A_{\\alpha-r\\epsilon_j}$, and the analogous identity for $N_\\alpha$; dividing by the denominator $A_\\delta$ turns this into a statement about symplectic Schur functions, and similarly for orthogonal characters. The orthosymplectic proof uses the vertical-strip decomposition $\\mathrm{spo}_\\lambda(X/Y) = \\sum_{\\lambda/\\nu \\in \\mathrm{VS}} \\mathrm{spo}_\\nu(X/Y) y_m^{|\\lambda|-|\\nu|}$ and two lemmas (4.1 and 4.2) that transfer identities from hook Schur functions to orthosymplectic Schur functions by matching coefficients after cancellations. The transfer relies on the fact that the set of hook Schur functions indexed by admissible partitions forms a basis of the supersymmetric function ring.","core_discovery":"The central claim is Theorem 3.4: for a partition $\\mu$ of length at most $n$ and an integer $r \\ge 1$, $$p_r \\mathrm{sp}_\\mu = \\sum_{\\eta/\\mu \\in \\mathrm{BS}(r)} (-1)^{\\mathrm{ht}(\\eta/\\mu)} \\mathrm{sp}_\\eta + \\sum_{\\mu/\\xi \\in \\mathrm{BS}(r)} (-1)^{\\mathrm{ht}(\\mu/\\xi)} \\mathrm{sp}_\\xi + \\sum_{q=m(\\mu)+1}^{n} (-1)^{p(q)-q+1} \\mathrm{sp}_{\\$mu^{{(q)}}$}.$$ The first sum is the classical border-strip addition; the second keeps the same shape and removes a border strip; the third is a new reflection term, where $\\mu^{(q)}$ is obtained from $\\mu$ by a row deletion and reinsertion procedure. The analogous rules for odd orthogonal characters (Theorem 3.7) and even orthogonal characters (Theorem 3.10, with the factors $(1+\\delta_{\\eta_n,0})/(1+\\delta_{\\mu_n,0})$) follow by replacing the symplectic determinant $A_\\alpha$ with $N_\\alpha$. The orthosymplectic rule (Theorem 4.3) is a hybrid: its third term mixes symplectic Schur functions in the $X$ variables with ordinary Schur functions in the $Y$ variables.","pith_inferences":["Beyond the paper: the coefficient-matching technique used for the orthosymplectic transfer suggests a general basis-replacement principle: any identity among Schur functions with integer coefficients, once checked, transfers to any family $f_\\lambda$ indexed by the same partitions by matching supports after cancellations.","Beyond the paper: the third term of the symplectic rule may correspond to the contragredient involution on the symplectic character ring; if so, the $p(q)$ statistic should match the action of the longest element of the Weyl group.","Beyond the paper: the border-strip removal term has no classical analogue for ordinary Schur functions, where $p_r s_\\mu$ only involves border-strip additions; it arises because the denominator for symplectic and orthogonal characters makes negative shifts $A_{\\alpha - r\\epsilon_j}$ nonzero, suggesting that similar removal terms should appear in other character settings with Weyl groups of type $B"],"forward_implications":["When $\\mu_n + 1 \\ge r$, the third term in the symplectic rule vanishes, giving the clean identity $p_r \\mathrm{sp}_\\mu = \\sum_{\\eta/\\mu \\in \\mathrm{BS}(r)} (-1)^{\\mathrm{ht}(\\eta/\\mu)} \\mathrm{sp}_\\eta + \\sum_{\\mu/\\xi \\in \\mathrm{BS}(r)} (-1)^{\\mathrm{ht}(\\mu/\\xi)} \\mathrm{sp}_\\xi$ (Corollary 3.5), and analogously for orthosymplectic characters (Corollary 4.6).","The rules give a direct, combinatorial way to multiply power sums with characters of the symplectic, orthogonal, and orthosymplectic groups, avoiding the determinant form of the character formula.","The orthosymplectic rule expresses $P_r(X,X/Y)\\mathrm{spo}_\\lambda(X/Y)$ as a sum of orthosymplectic Schur functions plus a term mixing symplectic and ordinary Schur functions, showing that the orthosymplectic character ring is not closed under multiplication by power sums in the same simple way.","The third term in the symplectic rule provides a new combinatorial operation on partitions—remove a row, shift intermediate rows, and add a reflected row—which can be studied independently as a statistic on Young diagrams."],"supporting_citations":[{"why":"Supplies the classical Murnaghan–Nakayama rule (Theorem 2.3) that the new rules generalize and that the orthosymplectic proof invokes through Theorem 2.6.","marker":"[Sta23]"},{"why":"Provides the Pieri rules (Theorems 2.1–2.2), the scalar product on symmetric functions, and the adjoint operators $p_r^\\perp$, $e_r^\\perp$ used to derive coefficient identities.","marker":"[Mac15]"},{"why":"Proves that hook Schur functions form a basis of the supersymmetric function ring, the key fact behind the coefficient-transfer in Lemma 4.2.","marker":"[Ste85]"},{"why":"Shows hook Schur functions obey the same Pieri rule as ordinary Schur functions, which underpins the supersymmetric Murnaghan–Nakayama rule (Theorem 2.6).","marker":"[Rem84]"},{"why":"Supplies the symplectic tableau description and Pieri rules for symplectic characters that contextualize the combinatorial form of Theorem 3.4.","marker":"[Sun90a]"},{"why":"Provides orthogonal tableaux and Pieri rules for orthogonal characters, used to frame Theorems 3.7 and 3.10.","marker":"[Sun90b]"},{"why":"Defines orthosymplectic Schur functions as characters of orthosymplectic Lie superalgebras and supplies their tableau description, the objects of Theorem 4.3.","marker":"[BSR98]"},{"why":"Gives the orthosymplectic Pieri rule, the direct predecessor that the orthosymplectic Murnaghan–Nakayama rule extends.","marker":"[Sto18]"}],"fun_headline_variants":["Three-term Murnaghan–Nakayama rules for symplectic and orthogonal Schur characters","Reflection term joins border-strip additions in new symplectic and orthogonal rules","Three-term rules for symplectic, orthogonal, and orthosymplectic Schur functions","Border-strip removals and reflection terms in new Murnaghan–Nakayama rules","Symplectic and orthogonal characters get three-term Murnaghan–Nakayama rules"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The orthosymplectic rule depends on the claim that an identity proved for hook Schur functions remains true after replacing each hook Schur function by the corresponding orthosymplectic Schur function, because the coefficients on both sides match after cancellations; the paper asserts this transfer without checking every partition that survives cancellation.","fun_headline_variants_meta":{"raw":{"variants":["Three-term Murnaghan–Nakayama rules for symplectic and orthogonal Schur characters","Reflection term joins border-strip additions in new symplectic and orthogonal rules","Three-term rules for symplectic, orthogonal, and orthosymplectic Schur functions","Border-strip removals and reflection terms in new Murnaghan–Nakayama rules","Symplectic and orthogonal characters get three-term Murnaghan–Nakayama rules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001925,"raw_usage":{"total_tokens":7584,"prompt_tokens":1039,"completion_tokens":6545,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":6430}},"tokens_in":655,"tokens_out":6545,"duration_ms":40308,"temperature":1.0,"reasoning_tokens":6430,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:31:51.930668+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a concrete case with a nonzero third term, such as $n=3$, $\\mu=(4,3,1)$, $r=6$ from Example 3.6, by evaluating both sides of Theorem 3.4 as explicit Laurent polynomials using the determinant character formula; if the coefficient of any $\\mathrm{sp}_{\\mu^{(q)}}$ fails to match, the rule is false. Similarly, test the orthosymplectic case in Example 4.4 with $\\lambda=(2,2)$, $n=2$, $m=2$, $r=3$.","supporting_citations":[],"review_version":1}