= Solution
Take $m=2$ and $\lambda=(2,1)$, so $\lambda+\delta=(3,1)$. A transposition in $S_3$ has one singleton cycle and one two-cycle. The <Frobenius alternant character formula> therefore gives
$$
\chi^{(2,1)}((12))=[x_1^3x_2]\,(x_1+x_2)(x_1^2+x_2^2)(x_1-x_2)
=[x_1^3x_2]\,(x_1^4-x_2^4)=\boxed{0}.
$$
In the coordinate <permutation representation> on $\mathbb C^3$, a transposition fixes one <basis> vector, so its <character> is $1$. This representation is the direct sum of the invariant line of constant vectors and the <standard representation of the symmetric group>, whose vectors have coordinate sum zero. Subtracting the trivial <character> gives $\chi_V((12))=1-1=0$, as required.
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