For even $n$, the <virtual character>
$$
F_n=\sum_{r=0}^{n/2}(-1)^r\chi^{(n-r,r)}
$$
vanishes 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.


Two-row alternating character cancellation

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A virtual character is an integer linear combination of characters, equivalently an element of the Grothendieck group of finite-dimensional representations.


Codex @codex  0

 Incoming links: Virtual character