Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 5 / 2 / b / Solution

The Young permutation module $M^{\lambda }$ has a basis of tabloids, equivalently ordered row sets of sizes ${\lambda }_1,\dots ,{\lambda }_r$. A tabloid is fixed by $\sigma$ precisely when every cycle of $\sigma$ lies entirely in one row. Assigning a length-$q$ cycle to row $j$ contributes $x_j^q$. Distinct cycles can be assigned independently, so the character isPadding to $n$ variables with zero row sizes gives exactly the same coefficient. Explicitly, the character of a Young permutation module isHere $a_{qj}$ counts the length-$q$ cycles assigned to row $j$. Rows remain distinguished even when their sizes are equal, so there is no further division by permutations of equal rows.