Murnaghan–Nakayama rule

If a permutation has a $k$-cycle and remaining cycle type $\rho$, thenwhere the sum is over removable rim hooks of length $k$.

The parity of the sum of the leg lengths in any sequence that removes all $e$-hooks from $\lambda$ is independent of the sequence. Its sign ${\epsilon }_e(\lambda )$ is therefore well defined and supplies the common sign in repeated applications of the Murnaghan–Nakayama rule.

For even $n$, the virtual charactervanishes on every permutation having an odd cycle. Under the Frobenius characteristic map, the Jacobi–Trudi identity identifies its characteristic with the degree-$n$ part of $H(t)H(-t)$, which contains only products of even-indexed power sums.

At a permutation whose cycle lengths are the principal hook lengths of $\lambda$, the Murnaghan–Nakayama rule has a unique complete removal sequence. Consequently ${\chi }^{\lambda }$ has value $1$ or $-1$ there.

For a partition $\lambda$, the irreducible character ${\chi }^{\lambda }$ vanishes on every cycle type containing an even part exactly when $\lambda$ is a staircase $(m,m-1,\dots ,1)$. A staircase has only odd hook lengths. Conversely, vanishing first forces $\lambda$ to be self-conjugate; applying the Murnaghan–Nakayama rule at the largest even hooks then forces consecutive row lengths.

Conjugating a Young diagram twists its complex Specht module by the sign representation:

For the alternating expression ${\psi }^{\lambda }$ indexed by an integer composition, swapping two adjacent entries of $\lambda -\mathrm{id}$ negates ${\psi }^{\lambda }$. If two entries coincide the expression vanishes; otherwise sorting produces, up to sign, the irreducible character indexed by the resulting partition.