of the following notation for restricted characters $$\chi_{R,(r,s)}(\sigma )={\rm Tr}_{(r,s)}\Big(\Gamma_R(\sigma)\Big)={\rm Tr}\Big(P_{R\to (r,s)}\Gamma_R(\sigma)\Big)\, .$$ Now, using the identity (bear in mind that $\psi (n+1)=n+1$) $$Z^{i_1}_{i_{\psi (1)}} \cdots Z^{i_{n-1}}_{i_{\psi (n-1)}} (YZ-ZY)_{i_{\psi (n)}}^{i_n}
Y^{i_{n+2}}_{i_{\psi (n+2)}} \cdots Y^{i_{n+m}}_{i_{\psi (n+m)}}
={\rm Tr}\left(\Big((n,n+1)\, \psi -\psi\,(n,n+1)\Big)Z^{\otimes n}Y^{\otimes m} \right)$$ where $${\rm Tr} (\sigma Z^{\otimes n}Y^{\otimes m})=
Z^{i_1}_{i_{\sigma(1)}}\cdots Z^{i_n}_{i_{\sigma(n)}}Y^{i_{n+1}}_{i_{\sigma(n+1)}}\cdots Y^{i_{n+m}}_{i_{\sigma(n+m)}}\, ,$$ and (this identity is proved in [@Bhattacharyya:2008rc]) $${\rm Tr} (\sigma Z^{\otimes n}Y^{\otimes m})=\sum_{T,(t,u)}{d_T n! m!\over d_t d_u (n+m)!}\chi_{T,(t,u)}(\sigma^{-1})\chi_{T,(t,u)}(Z,Y)$$ we obtain $$D\chi_{R,(r,s)}(Z,Y)=\sum_{T,(t,u)} M_{R,(r,s);T,(t,u)}\chi_{T,(t,u)}(Z,Y)\, ,$$ [$$M_{R,(r,s);T,(t,u)}=g_{YM}^2\sum_{\psi\in S_{n+m-1}}\sum_{R'}
{c_{RR'} d_T n m\over d_t d_u (n+m)!}
\Tr_{(r,s)}\Big(\Gamma_R ((n,n+1))\Gamma_{R'}(\psi)-\Gamma_{R'}(\psi) \Gamma_R((n,n+1))\Big)\times$$ $$\times \chi_{T,(t,u)}(\psi^{-1}(n,n+1) - (n,n+1)\psi^{-1})\, .$$ ]{} The sum over $\psi$ can be done by using the fundamental orthogonality relation $$\begin{aligned}
\label{final}
M_{R,(r,s);T,(t,u)} &&= - g_{YM}^2\sum_{R'}{c_{RR'} d_T n m\over d_{R'} d_t d_u (n+m)}
\Tr\Big( \Big[ \Gamma_R((n,n+1)),P_{R\to (r,s)}\Big]I_{R'\, T'}\times \cr
&&\times \Big[\Gamma_T((n,n+1)),P_{T\to (t,u)}\Big] I_{T'\, R'}\Big)  \, .\end{aligned}$$ The reader should consult Appendix B for a definition of the intertwiners $I_{R'\, T'}$. This expression for the one loop dilatation operator is exact in $N$. It is one of the key results of this article.

To obtain the spectrum of anomalous dimensions, we need to consider the action of the dilatation operator on normalized operators. The two point function for restricted Schur polynomials has been computed 