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Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.11447 v2 pith:VYQA56ZW submitted 2024-11-18 math.CO

Nishu Kumari , Anna Stokke This is my paper · ORCID
classification math.CO MSC 05E0505E1005A19
keywords Murnaghan–NakayamarulesymplecticSchurfunctionsorthogonalorthosymplecticborderstripsdeterminantalcharacterformulasupersymmetrichook
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $\lambda=(2,2)$, $n=2$, $m=2$, $r=3$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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
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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).

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.

minor comments (7)
  1. [Section 4, combinatorial proof of (4.2)] 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.
  2. [Introduction and References] There are several typos: 'Murnaghan–Nakayma' in the Introduction, 'Stanely' should be 'Stanley', and the reference [Nak41] misspells 'Nakayama' as 'Nakayma'.
  3. [Lemma 4.2 proof] In the last sentence of the proof of Lemma 4.2, 'the equality in (4.3) also holds' should refer to (4.4).
  4. [Lemma 3.1 proof] 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.
  5. [Example 2.5] The five symplectic tableaux of shape (1,1) are not clearly typeset, making it difficult to verify the count and the resulting polynomial.
  6. [Theorem 3.7] 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.
  7. [Theorem 4.3 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation reduces to classical MN rule, Weyl determinant formulas, and external basis theorems, not to its own conclusions.

full rationale

The derivation chain is self-contained against external classical inputs rather than against its own conclusions. The symplectic rule (Theorem 3.4) is proved from the determinant identity pr A_alpha = sum_j A_{alpha+r epsilon_j} + sum_j A_{alpha-r epsilon_j} (Lemma 3.1), which is proved by induction on n, together with Weyl's symplectic character formula (2.9); no coefficient is fitted and the target product identity is never assumed. The orthogonal rules (Theorems 3.7 and 3.10) are obtained by the same determinant argument with shifted half-integer delta and the analogous determinant N_alpha, again without circular reuse. The orthosymplectic rule (Theorem 4.3) is proved by induction on m, using the hook-Schur Murnaghan-Nakayama rule (Theorem 2.6), which is itself derived from the classical MN rule via plethysm, together with Stembridge's external basis theorem for supersymmetric functions and the vertical-strip recurrence (2.13). The basis-replacement steps in Lemmas 4.1 and 4.2 are terse, but they compare coefficients in Stembridge's known basis and do not invoke the orthosymplectic MN rule being proved; any objection to the terseness of that step is a correctness concern, not a circularity. The self-citations ([Sto18], [Kum24], [PPS22], [SV16]) appear as background or prior context and are not load-bearing; no author-specific uniqueness theorem is used to force the choice of rule, and no parameter is fitted to data. No reduction of the claimed formulas to their own inputs was found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. It relies on classical results: the MN rule, Weyl character formulas, and Stembridge's basis theorem. The only implicit assumption is the standard branching identity for Schur polynomials.

assumptions (4)
  • standard math Classical Murnaghan-Nakayama rule (Theorem 2.3)
    Used as a black box to derive the symplectic and orthosymplectic analogues; cited to Stanley, Theorem 7.17.3. Invoked in Section 2 and again in Lemma 4.2.
  • standard math Weyl character formulas for symplectic, odd orthogonal, and even orthogonal characters (equations 2.8-2.10)
    The determinant expressions are the starting point for all three MN rules in Section 3; correctness of the normalization factors is assumed.
  • standard math Stembridge's theorem that {hsλ(X/Y)} is a basis for the ring of supersymmetric functions
    Used in Lemma 4.2 to justify replacing hook Schur functions by orthosymplectic Schur functions in a proven identity; cited to [Ste85].
  • domain assumption The branching identity (2.5) and its conjugate form for vertical strips in the Y variables
    Used in the proof of Theorem 4.3 to pass from sums over λ/ν in VS to skew Schur functions s_{λ'/μ'}(Y); the paper does not explicitly prove the conjugate variant.

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Pith. "Pith review of Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions." pith.science (2026). https://pith.science/paper/VYQA56ZW

@misc{pith2026241111447,
  author       = {Pith},
  title        = {Pith review of: Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYQA56ZW}},
  note         = {Machine review of arXiv:2411.11447}
}
read the original abstract

We establish new Murnaghan--Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions. The classical Murnaghan--Nakayama rule expresses the product of a power sum symmetric function with a Schur function as a linear combination of Schur functions. Symplectic and orthogonal Schur functions correspond to characters of irreducible representations of symplectic and orthogonal groups. Orthosymplectic Schur functions arise as characters of orthosymplectic Lie superalgebras and are hybrids of symplectic and ordinary Schur functions. We derive explicit formulas for the product of the relevant power-sum function with each of these functions, which can partly be described combinatorially using border strip manipulations. Our Murnaghan--Nakayama rules each include three distinct terms: a classical term corresponding to the addition of border strips to the relevant Young diagram, a term involving the removal of border strips, and a third term, which we describe both algebraically and combinatorially.

Figures

Figures reproduced from arXiv: 2411.11447 by the authors.

Figure 1
Figure 1. The Young diagram of λ/µ = (5, 2, 2)/(2, 1) A skew diagram λ/µ is a horizontal m-strip (respectively a vertical m-strip) if |λ/µ| = m and λ/µ has at most one box in each column (respectively row). Two boxes in a skew diagram are adjacent if they share a common side, and a subset of boxes in a skew diagram is connected if every pair of boxes is connected through a sequence of adjacent boxes. A border strip is a skew … view at source ↗
Figure 2
Figure 2. The partitions in the expansion p4s(3,1) Skew Schur polynomials sλ/µ(X) are generalizations of Schur polynomials. The combinato￾rial objects associated with sλ/µ(X) are semistandard tableaux of shape λ/µ. A semistandard tableau of shape λ/µ is a filling of the Young diagram λ/µ with entries in {1, 2, . . . , n} such that entries increase weakly across rows and strictly down columns. Define the skew Schur polynomial … view at source ↗
Figure 3
Figure 3. Cancellation configuration Since the attached signs are opposite, this case corresponds to a cancellation in the right-hand side of (4.2). Case 1 b) If p ∈ {i1, . . . , is} and λq > λq+1 or q ∈ {i1, . . . , is}, consider ν defined as follows: νi =    λi − 1 if i = q λi − 1 if i ∈ {i1, . . . , is} ∩ ([1, q − 1] ∪ [p + 1, n]) λi − 1 if i ∈ {i1 + 1, . . . , is + 1} ∩ [q + 1, p] λi otherwise. Then λ/ν ∈ VS(s), … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Configuration 1 Case 2 a) Suppose p ̸∈ {i1, . . . , is} and q − 1 ̸∈ {i1, . . . , is}. Consider a partition σ as follows, which is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Configuration 2 Here λ/σ ∈ VS(s), Ω ⊂ σ, σ/Ω ∈ BS(r) and ht(σ/Ω) = p − q. Thus Ω also appears in the left-hand side of (4.2) with the same coefficient. Case 2 b) Suppose p ̸∈ {i1, . . . , is} and q − 1 ∈ {i1, . . . , is}. If λq−1 > λq, consider the partition υ below (s…
Figure 6
Figure 6. Figure 6: Configuration 3 If λq−1 = λq and λp − λq − p + q + r = 1, let υ be as follows (see [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Configuration 4 In both of the cases above illustrated by [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Cancellation Configuration [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

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