structures do not mix is: [*at large $N$ restricted Schur polynomials $\chi_{R,(r,s)}$ with $R$ a Young diagram with $n$ columns, each of which has length of $O(N)$, do not mix with operators $\chi_{R',(r',s')}$ that have $n'\ne n$ columns.*]{} The fact that the two column restricted Schur polynomials are a decoupled sector at large $N$ is to be expected. Indeed, at large $N$ these operators correspond to a well defined stable semi-classical object in spacetime (the two giant system). We expect that $n$ column restricted Schur polynomials are also a decoupled sector at large $N$ for the same reason.

Simple Projectors
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[*When all Young diagrams labeling the restricted Schur polynomial have at most 2 columns, the projector $P_{R\to (r,s)}$ simplifies dramatically. The goal of this section is to explain the simplification and exploit it to efficiently build the relevant projectors.*]{}

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The projector $P_{R\to (r,s)}$ projects from representation $R$ of $S_{n+m}$ to representation $(r,s)$ of $S_n\times S_m$. One issue, which complicates things considerably, is that representation $(r,s)$ can be subduced more than once when irreducible representation $R$ is decomposed into irreducible representations of the $S_n\times S_m$ subgroup. Giving a general rule to specify precisely how this multiplicity is resolved is 