Operator
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[*In this section we will study the action of the one loop dilatation operator on restricted Schur polynomials built using two complex adjoint scalars. The main result of this section is the surprisingly simple result (\[final\]) for the action of the dilatation operator.*]{}

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We will consider the action of the one loop dilatation operator in the $SU(2)$ sector[@Beisert:2003tq] of ${\cal N}=4$ super Yang Mills theory $$D = - g_{\rm YM}^2 {\rm Tr}\,\big[ Y,Z\big]\big[ \partial_Y ,\partial_Z\big]$$ on the restricted Schur polynomial $$\chi_{R,(r,s)}(Z^{\otimes \, n},Y^{\otimes \, m})
=
{1\over n!m!}\sum_{\sigma\in S_{n+m}}{\rm Tr}_{(r,s)}(\Gamma_R(\sigma))
Z^{i_1}_{i_{\sigma(1)}}\cdots Z^{i_n}_{i_{\sigma(n)}}Y^{i_{n+1}}_{i_{\sigma(n+1)}}\cdots Y^{i_{n+m}}_{i_{\sigma(n+m)}}\, .$$ The labels of our restricted Schur polynomial $\chi_{(R,(r,s))}$ are (i) $R$, which is a Young diagram with $n+m$ boxes or equivalently an irreducible representation of $S_{n+m}$, (ii) $r$, which is a Young diagram with $n$ boxes or equivalently an irreducible representation of $S_n$ and (iii) $s$ which is a Young diagram with $m$ boxes or equivalently an irreducible representation of $S_m$. The notation ${\rm Tr}_{(r,s)}$ implies that one should only trace over the subspace carrying the irreducible representation[^1] $(r,s)$ of $S_n\times S_m$ inside the carrier space for irreducible representation $R$ of $S_{n+m}$. This trace is most concretely realized by including a projector $P_{R\to 