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

For the cycle-sum identity for Young–Jucys–Murphy elements, let $C_m$ be the sum of all $m$-cycles on $\{1,\dots ,m\}$. Multiplying such a cycle by $(jm+1)$ inserts $m+1$ immediately after $j$ in that cycle, with our composition convention. Every $(m+1)$-cycle has a unique predecessor $j$ of $m+1$, so deleting $m+1$ recovers a unique term of $C_m{X}_{m+1}$. Since $X_2=(12)$, induction givesFor $n=1$ the empty product is the identity, also the unique one-cycle.

For the product assertion in a non-hook shape, ${\lambda }_2\ge 2$, so cell $(2,2)$ is present. Its Content of a Young-diagram cell is zero and its label is at least four. Since $X_r{w}_T=c_T(r)w_T$, that factor in the product vanishes. Therefore

A hook partition $\lambda =(n-k,1^k)$ has cell contentsIn a standard Young tableau, the unique zero-content cell is $(1,1)$ and contains $1$. The other eigenvalues are precisely the remaining contents, in whatever order the tableau supplies. Their product is independent of that order:The scalar is nonzero over the complex numbers. For the single row or single column, one of the factorials is $0!=1$; the formula still gives $(n-1)!$ or $(-1{)}^{n-1}(n-1)!$, respectively.

The Gelfand–Tsetlin basis spans $V^{\lambda }$, and the scalar computed on its vectors depends only on the shape. Thusfor every $w\in V^{\lambda }$. The product is the sum of the $(n-1)!$ permutations in the conjugacy class of an $n$-cycle. Taking traces therefore gives $(n-1)!{\chi }^{\lambda }((n))$ equal to the displayed scalar times $\dim V^{\lambda }$.

For a hook partition, a standard Young tableau is uniquely determined by the choice of its $k$ entries below the top cell, selected from $\{2,\dots ,n\}$. The column and the remaining row are then forced to increase. Hence $\dim V^{(n-k,1^k)}=\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$, and cancellation of the factorials yieldsThis uses the central character value of a conjugacy-class sum and tableau counting, without a character rule for removing rim hooks.