#The Murnaghan–Nakayama rule
The expansion of Schur polynomials in the power-sum basis gives the irreducible characters of the symmetric group as coefficients: \[\schurS_\lambda(\xvec) = \sum_{\mu} \frac{\chi^{\lambda}_{\mu}}{z_\mu} \powerSum_{\mu}(\xvec) = \sum_{\mu} \frac{\powerSum_{\mu}(\xvec)}{z_\mu} \sum_{T \in \BST(\lambda,\mu)} (-1)^{\mathrm{ht}(T)}\] where the sum is taken over border strip tableaux of shape \(\lambda\) and weight \(\mu.\)
Another way to phrase the Murnaghan–Nakayama rule is the following: \[\powerSum_r(\xvec) \schurS_\lambda(\xvec) = \sum_{\mu} (-1)^{\mathrm{ht}(\mu/\lambda)} \schurS_\mu(\xvec)\] where the sum ranges over all \(\mu\) such that \(\mu/\lambda\) is a border-strip with \(r\) boxes. The combinatorial rule was first stated in [Mur37, Nak40].
A completely combinatorial proof of the Murnaghan–Nakayama rule is provided in [Men19], using only a sign-reversing involution.
S. Pfannerer gives a descent refinement of the Murnaghan–Nakayama rule for border-strip tableaux [Pfa21]. The refinement keeps track of descent data while recovering the classical rule after summing over the refined statistic.
We can also write the power-sum expansion as the average \[\schurS_\lambda(\xvec) = \frac{1}{n!} \sum_{\sigma \in \symS_n} \chi^{\lambda}(\sigma) \powerSum_{\text{type}(\sigma)}(\xvec).\]
The characters \(\chi^{\lambda}_{\mu}\) also show up in the Schur expansion of power-sum symmetric functions. \[\powerSum_\mu(\xvec) = \sum_{\lambda} \chi^{\lambda}_{\mu} \schurS_{\lambda}(\xvec).\] This is a consequence of the fact that the characters are orthogonal, so inverting the transition matrix is essentially transposition.
K. J. Westrem studies character sums over the multiset \(\operatorname{Ev}(\lambda)\) obtained from a partition \(\lambda\) by replacing each part \(\lambda_i\) either by \(2\lambda_i\) or by two copies of \(\lambda_i\) [Wes24]. If \(\lambda=(\lambda_1,\dotsc,\lambda_r),\) then \[\sum_{\widetilde{\lambda}\in \operatorname{Ev}(\lambda)} (-1)^{\length(\widetilde{\lambda})} \powerSum_{\widetilde{\lambda}}(\xvec) = 2^r \prod_{i=1}^r \monomial_{(\lambda_i,\lambda_i)}(\xvec).\] Together with the identity \(\chi^\mu_\lambda=\langle \powerSum_\lambda,\schurS_\mu\rangle,\) this gives vanishing results for alternating sums of irreducible symmetric-group characters.
S. Peluse and K. Soundararajan prove that for any fixed prime power \(p^a,\) almost all entries in the character table of \(\symS_n\) are divisible by \(p^a\) as \(n \to \infty\) [PS25]. This proves a conjecture of S. J. Miller and extends their earlier result on divisibility by primes.
Yet another way to compute \(\chi^{\lambda}_{\mu}\) is by taking the coefficient of \(\prod_{i=1}^k x_i^{\lambda_i + \ell -i}\) in \[\prod_{i \lt j} (x_i - x_j ) \; \cdot \; \powerSum_{\mu}(x_1,\dotsc,x_\ell),\] where \(\ell\) is at least the number of parts of \(\lambda.\)
#Formulas for \(\chi^{\lambda}_{\mu}\)
#Murnaghan–Nakayama recursion
One can reformulate the Murnaghan–Nakayama rule as a recursive rule, see e.g. [Sec. 2.4.4, JK84].
Let \(\lambda \vdash n\) and \(\mu \vdash n-m.\) \[\chi^{\lambda}(\mu,m) = \sum_{\nu} (-1)^{\mathrm{ht}(\lambda /\nu)} \chi^{\nu}(\mu)\] where the sum ranges over all \(\nu\) such that \(\lambda/\nu\) is a border-strip of size \(m.\)
#Y. Roichman’s formula
Y. Roichman has an alternative way of expressing the coefficients \(\chi^{\lambda}_{\mu}.\) This generalizes to so-called Kazhdan–Lusztig characters; see [Roi97] and [Roi99].
Theorem (Y. Roichman, 1997).
We have that \[\chi^{\lambda}_{\mu} = \sum_{Q \in \SYT(\lambda)} \mathrm{weight}_\mu(Q)\] where \[\mathrm{weight}_\mu(Q) \coloneqq \prod_{\substack{1 \leq i \leq n \\ i \notin B(\mu)}} f_\mu(i,Q), \quad B(\mu) \coloneqq \{ \mu_1 + \dotsb + \mu_r : 1\leq r \leq \length(\mu)\}\] and \[f_\mu(i,Q) \coloneqq \begin{cases} -1 &\text{ if $i+1$ is southwest of $i$} \\ 0 &\text{ if $i+1$ is northeast of $i,$ $i+2$ southwest of $i+1,$ and $i+1 \notin B(\mu)$} \\ 1 & \text{ otherwise}. \end{cases}\]
A related formulation is due to C. Athanasiadis [Prop. 3.2, Ath15].
Theorem
Let \(\lambda\) be a partition of \(n.\) Then the expansion of the Schur function \(\schurS_{\lambda}\) into power-sum symmetric functions is given by \[\begin{aligned} \label{eq:schurRoichmanPexp} \schurS_\lambda(\xvec) = \sum_{\mu \vdash n} % \frac{\powerSum_\mu(\xvec)}{z_\mu} % \sum_{\substack{ T \in \SYT(\lambda) \\ \DES(T) \in U_\mu }} (-1)^{\DES(T) \setminus S_\mu} \,, \end{aligned}\] where \(S_\mu\) and \(U_\mu\) are defined in the power-sum expansion section.
#R. Holmes’ recursion
R. Holmes [Hol17] generalizes a result by G. James and A. Kerber [Sec. 2.4.3, JK84]. This gives a recursive method of computing \(\chi^{\lambda}_{\mu}.\) The interesting aspect of this recursion is that one computes characters of \(\symS_n\) by using only character values for \(\symS_{n-1}\) (and not smaller groups as with the Murnaghan–Nakayama recursion).
The following set of relations is enough to compute all \(\chi^{\lambda}_{\mu}.\) Here, \(\lambda \vdash n\) and we always assume that the arguments to \(\chi\) are partitions of the same size, and \(\epsilon_i\) denotes the unit vector with \(1\) at the \(i^\thsup\) coordinate.
Special case of the Murnaghan–Nakayama rule: \[\chi^{\lambda}_{(n)} = \begin{cases} (-1)^{n-\lambda_1} & \text{ if $\lambda$ is a hook} \\ 0 &\text{ otherwise}. \end{cases}\]
The James–Kerber branching rule: \[\chi^{\lambda}_{(\mu,1)} = \sum_{\substack{ 1 \leq i \leq \length(\lambda) \\ \lambda_i \gt \lambda_{i+1} }} \chi^{\lambda - \epsilon_i}_{\mu}.\]
R. Holmes recursion, \(m \geq 2:\) \[\chi^{\lambda}_{(\mu,m)} = \frac{1}{m-1} \left[ \sum_{\substack{ 1 \leq i \leq \length(\lambda) \\ \lambda_i \gt \lambda_{i+1} }} (\lambda_i - i) \chi^{\lambda - \epsilon_i}_{(\mu,m-1)} - \sum_{\substack{ 1 \leq j \leq \length(\mu) \\ \mu_{j-1} \gt \mu_{j} }} M_j \cdot \mu_j \cdot \chi^{\lambda}_{(\mu+\epsilon_j,m-1)} \right].\] In the second sum, we use the convention that \(\mu_0 = \infty\) and \(M_j\) denotes the number of parts equal to \(\mu_j\) in the partition \((\mu,m-1).\)
To speed up computation, one can also add the following special case.
Hook formula case: \[\chi^{\lambda}_{(1^n)} = f^\lambda\] where \(f^\lambda\) is the number of standard Young tableaux of shape \(\lambda.\) This quantity can be computed using the hook formula.
#Alfred Young’s construction
A construction due to A. Young is presented in [Thm. 1.6, GE20]; see also the MathOverflow discussion [Ros23]. It states \[\chi^{\lambda}_{\mu} = z_\mu \frac{f^{\lambda}}{n!} \sum_{\substack{\pi \in R(\mu) \\ \sigma \in C(\mu) \\ type(\pi \sigma) = \mu }} \sign(\sigma).\]
#Variants of the Murnaghan–Nakayama rule
The following families have a Murnaghan–Nakayama rule.
the canonical stable Grothendieck polynomials, see [Kun25].
\(K\text{-}k\)-Schur functions, see [Ngu22].
Cylindric Schur functions (cylindric Hecke characters), see [Lem. 5.3, Kor20].
the symplectic, orthogonal, and orthosymplectic Schur functions.
N. Kumari and A. Stokke prove Murnaghan–Nakayama rules for symplectic, orthogonal, and orthosymplectic Schur functions [KS26]. The symplectic and orthogonal rules have three types of terms: adding border strips, removing border strips, and a correction term coming from the Weyl-character denominator. The orthosymplectic rule has a similar shape, but its third term mixes symplectic and ordinary Schur functions.
N. Jing, Y. Wu, and N. Liu use vertex-operator methods to study irreducible characters of the \(q\)-rook monoid algebra [JWL25]. Their formulas include an iterative character formula, a Murnaghan–Nakayama rule for the \(q\)-rook monoid, and a bitrace formula generalizing the Iwahori–Hecke algebra case. N. Jing and N. Liu prove a Murnaghan–Nakayama rule for irreducible characters of the cyclotomic Hecke algebra \(\mathscr H_{m,n}(q,u)\) [JL25]. Their formula gives a combinatorial route to character tables and specializes to several known rules for complex reflection groups and Iwahori–Hecke algebras. Z. Hamaker and B. Rhoades give an algorithm for evaluating irreducible symmetric-group characters on partial-permutation group-algebra elements [HR25]. The algorithm combines the classical Murnaghan–Nakayama rule with a path version reflecting the path and cycle decomposition of a partial permutation.
O. Tout defines generalized characters for the wreath product \(\setZ_k \wr \symS_n\) using a symmetric Gelfand pair [Tou22]. These generalized characters satisfy analogues of several standard character properties, and in the hyperoctahedral case this includes a Murnaghan–Nakayama rule.
#Quantum Murnaghan–Nakayama rule
M. z. Konvalinka proves a skew quantum version of the Murnaghan–Nakayama rule [Kon11]. Here, one multiplies the Schur function with a \(q\)-deformation of the power-sum symmetric functions. These deformations are equal to the Hall–Littlewood \(P\)-functions, indexed by one-part partitions. The paper has several conjectured generalizations of Murnaghan–Nakayama rules for Hall–Littlewood \(P\)-functions.
C. B. Velásquez, N. Bergeron, L. Colmenarejo, F. Saliola, and F. Sottile prove a Murnaghan–Nakayama rule for multiplication by a tautological class in the small quantum cohomology ring of the flag manifold [VBCS+25]. Their proof passes through a formula for multiplying by quantum Schur polynomials indexed by hooks and uses detailed properties of the quantum Bruhat order.
#Plethystic Murnaghan–Nakayama rule
A rule for computing the coefficients in the expansion \[\schurS_\mu \cdot (\powerSum_r[\completeH_m]) = \sum_{\lambda \vdash rm+|\mu|} (-1)^{\mathrm{ht}_r(\lambda/\mu)} \schurS_{\lambda}\] is given by M. Wildon [Wil16]. See also [Wil18] for a more general result.
P. Turek gives a short combinatorial proof of this rule using N. Loehr’s labelled abaci [Tur23]. A vertex-operator approach of Y. Cao, N. Jing, and N. Liu gives a determinant-type plethystic Murnaghan–Nakayama rule and formulas for the Schur expansion of \((\powerSum_n \circ \completeH_k)\schurS_\mu\) [CJL25]. The same authors also prove a spin analogue for Schur \(Q\)-functions, generalizing both the Murnaghan–Nakayama and Pieri rules for Schur \(Q\)-functions [CJL24].
N. Jing and N. Liu prove a multiparametric Murnaghan–Nakayama rule for Macdonald polynomials [JL24]. Their framework also gives iterative formulas for Green polynomials and \((q,t)\)-Kostka polynomials, and recovers several Hecke-algebra and Hall–Littlewood formulas.
Z. Hamaker and B. Rhoades introduce path power-sum symmetric functions while studying characters of local and regular permutation statistics [HR22]. Their path Murnaghan–Nakayama formula expands these path power sums into Schur functions, and combines with character polynomials to describe moments of permutation statistics conditional on cycle type.
#Murnaghan–Nakayama rule for Chern classes of Schubert cells
In [FGX22], the authors give a Pieri rule and a Murnaghan–Nakayama rule for Chern classes of Schubert cells. L. C. Mihalcea, H. Naruse, and C. Su derive hook formulae from Segre–MacPherson classes of Schubert cells and varieties [MNS25]. Their work generalizes a cohomological version of Nakada’s colored hook formula and uses weighted paths in a decorated Bruhat graph.
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