Multiple giant gravitons and restricted Schur characters

In $\mathrm{AdS/CFT}$, the open strings ending on multiple giant gravitons in $\mathrm{AdS}_5 \times \mathrm{S}^5$ spacetime are dual to the operators with $O(N_c)$ dimensions in $\mathcal N=4$ super Yang-Mills with $U(N_c)$ gauge group. Under some approximation, the dilatation spectrum of such operators is described by an effective $U(p)$ Hamiltonian where $p$ is the number of giant gravitons.
In the first half of this talk, I argue that the perturbative Hamiltonians at each loop order commute with each other, impllying an all-loop ansatz for the anomalous dimensions with a gapless dispersion. In the second half, I summarize mathematical aspects of the restricted Schur characters which are used to describe the operator mixing problem at finite $N_c$. Then I describe how to compute these characters explicitly particularly when the multiplicity space coming from the restriction of $S_{m+n}$ down to $S_m \times S_n$.
This talk is based on arXiv:2101.05310 and an ongoing work with Sanjaye Ramgoolam and Adrian Padellaro.