(r,s)}$ (from the carrier space of $R$ to the carrier space of $(r,s)$) and tracing over all of $R$. A simple calculation yields[^2] [$$\begin{aligned}
\label{basicreslt}
D \, \chi_{R,(r,s)}(Z^{\otimes \, n},Y^{\otimes \, m})=&&
{g_{\rm YM}^2\over (n-1)!(m-1)!}\sum_{\psi\in S_{n+m}}
\Tr_{(r,s)}\left(\Gamma_R ((n,n+1) \psi -\psi (n,n+1))\right)\times\cr
\times Z^{i_1}_{i_{\psi (1)}}&&
\cdots
Z^{i_{n-1}}_{i_{\psi (n-1)}}
(YZ-ZY)_{i_{\psi (n)}}^{i_n}\delta^{i_{n+1}}_{i_{\psi (n+1)}}
Y^{i_{n+2}}_{i_{\psi (n+2)}}\cdots Y^{i_{n+m}}_{i_{\psi (n+m)}}
\, .\end{aligned}$$ ]{} The sum over $\psi$ runs only over permutations for which $\psi (n+1)=n+1$. To perform this sum over $\psi$, write the sum over $S_{n+m}$ as a sum over cosets of the $S_{n+m-1}$ subgroup obtained by keeping precisely those permutations that satisfy $\psi (n+1)=n+1$. The result follows immediately from the reduction rule for Schur polynomials (see [@de; @Mello; @Koch:2004ws] and appendix C of [@de; @Mello; @Koch:2007uu]) [$$\begin{aligned}
\nonumber
D \, \chi_{R,(r,s)}=&&
{g_{\rm YM}^2\over (n-1)!(m-1)!}\sum_{\psi\in S_{n+m-1}}\sum_{R'}c_{RR'}\,\Tr_{(r,s)}\Big(\Gamma_R ((n,n+1))\Gamma_{R'}(\psi) \cr
&&-\Gamma_{R'}(\psi) \Gamma_R((n,n+1))\Big)\, Z^{i_1}_{i_{\psi (1)}}
\cdots
Z^{i_{n-1}}_{i_{\psi (n-1)}}
(YZ-ZY)_{i_{\psi (n)}}^{i_n}
Y^{i_{n+2}}_{i_{\psi (n+2)}}\cdots Y^{i_{n+m}}_{i_{\psi (n+m)}}
\, .\end{aligned}$$ ]{} The sum over $R'$ runs over all representations that can be subduced from $R$. Concretely, $R'$ runs over all Young diagrams that can be obtained from $R$ by dropping a single box; $c_{RR'}$ is the weight[^3] of the box that must be removed from $R$ to obtain $R'$. We will make use 