For the product assertion in a non-hook shape, $\lambda_2\geq2$, so cell $(2,2)$ is present. Its <Content of a Young-diagram cell> is zero and its label is at least four. Since $X_rw_T=c_T(r)w_T$, that factor in the product vanishes. Therefore
$$
\boxed{X_2\cdots X_nw_T=0\quad\text{for a non-hook shape}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 103 / 3 / c / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-103/3/c/ii/solution