Each equivalence class has size $\prod_{j=1}^n a_j!$, because its $a_j$ rows of length $j$ may be permuted freely. Swapping two length-$j$ rows multiplies each of $\langle e(t),\omega\rangle$ and $\langle e(u),\omega\rangle$ by the same sign $(-1)^j$, so their product is constant on the class. Since
$$
\langle e(t),e(u)\rangle
=\sum_{\omega\in\Omega^\lambda}
\langle e(t),\omega\rangle\langle e(u),\omega\rangle,
$$
the contribution of every nonzero class is a multiple of $\prod_j a_j!$. The whole inner product is therefore such a multiple.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 4 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-160/4/b/ii/solution