be any one of $d_{\tiny\yng(2,1,1)}=3$ states, corresponding to the number of ways to complete the labels. When acting in the subspace, the operator which organizes the boxes into representation $s$ is $$P_s={d_s\over m!}\sum_{\sigma\in S_m}\chi_s(\sigma)\Gamma_R(\sigma)\, .$$ All that remains is to supply a formula for the action of $\Gamma_R(\sigma)$ (for $\sigma =1,(12)$) when acting on the subspace spanned by $|1\rangle$ and $|2\rangle$. The action of $\Gamma_R(\sigma)$ on any Young-Yamonouchi state is well known. For the states above $$\Gamma_R\left((12)\right)|1\rangle = -{1\over 3}|1\rangle +{\sqrt{8}\over 3}|2\rangle$$ $$\Gamma_R\left((12)\right)|2\rangle =  {1\over 3}|2\rangle +{\sqrt{8}\over 3}|1\rangle$$ so that $$\Gamma_R\left( (12)\right)=-{1\over 3}|1\rangle\langle 1| +{\sqrt{8}\over 3}|2\rangle\langle 1|
+{1\over 3}|2\rangle\langle 2| +{\sqrt{8}\over 3}|1\rangle\langle 2|$$ and $$\Gamma_R\left( 1\right)= |1\rangle\langle 1| + |2\rangle\langle 2| \, .$$

The Radial Direction
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[*In this section we describe a limit in which the dilatation operator simplifies significantly. There are two columns in the Young diagrams labeling the restricted Schur polynomials. When the first column contains $O(\sqrt{N})$ boxes more than the second, the dilatation operator simplifies to a lattice realization of the second derivative. The Young diagram label itself defines the lattice.*]{}

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Three Impurities
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The three impurity operators are built using many $Z$s and three $Y$s. To specify these operators, we need to give the 