Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 5 / 2 / a / 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 givesIn 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.