in [@Bhattacharyya:2008rb] $$\langle\chi_{R,(r,s)}(Z,Y)\chi_{T,(t,u)}(Z,Y)^\dagger\rangle =
\delta_{R,(r,s)\,T,(t,u)}f_R {{\rm hooks}_R\over {\rm hooks}_{r}\, {\rm hooks}_s}\, .$$ In this expression $f_R$ is the product of the weights in Young diagram $R$ and ${\rm hooks}_R$ is the product of the hook lengths of Young diagram $R$. The normalized operators can be obtained from $$\chi_{R,(r,s)}(Z,Y)=\sqrt{f_R \, {\rm hooks}_R\over {\rm hooks}_r\, {\rm hooks}_s}O_{R,(r,s)}(Z,Y)\, .$$ In terms of these normalized operators $$DO_{R,(r,s)}(Z,Y)=\sum_{T,(t,u)} N_{R,(r,s);T,(t,u)}O_{T,(t,u)}(Z,Y)$$ [$$N_{R,(r,s);T,(t,u)}= - g_{YM}^2\sum_{R'}{c_{RR'} d_T n m\over d_{R'} d_t d_u (n+m)}
\sqrt{f_T \, {\rm hooks}_T\, {\rm hooks}_r \, {\rm hooks}_s \over f_R \, {\rm hooks}_R\, {\rm hooks}_t\, {\rm hooks}_u}\times$$ $$\times\Tr\Big(\Big[ \Gamma_R((n,n+1)),P_{R\to (r,s)}\Big]I_{R'\, T'}\Big[\Gamma_T((n,n+1)),P_{T\to (t,u)}\Big]I_{T'\, R'}\Big) \, .$$ ]{} This last expression will be used later when we numerically study the spectrum of the dilatation operator.

Excited Giant Graviton Bound States
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[*The goal of this section is to clearly define the class of operators being considered and to outline the approximations that can be made in the large $N$ limit.*]{}

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In this article we will study restricted Schur polynomials labeled by Young diagrams with at most two columns. The number of $Z$s appearing is $\alpha N$ where $2-\alpha\equiv\zeta\ll 1$. The number of $Y$s appearing is fixed to be $O(1)$. These operators are dual to giant gravitons 