From counting to construction of BPS states in \( \mathcal{N} = 4 \) SYM

We describe a universal element \( \mathbb{P} \) in the group algebra of symmetric groups, whose characters provides the counting of quarter and eighth BPS states at weak coupling in \( \mathcal{N} = 4 \) SYM, refined according to representations of the global symmetry group. A related projector \( \mathcal{P} \) acting on the Hilbert space of the free theory is used to construct the matrix of two-point functions of the states annihilated by the one-loop dilatation operator, at finite N or in the large N limit. The matrix is given simply in terms of Clebsch-Gordan coefficients of symmetric groups and dimensions of U(N) representations. It is expected, by non-renormalization theorems, to contain observables at strong coupling. Using the stringy exclusion principle, we interpret a class of its eigenvalues and eigenvectors in terms of giant gravitons. We also give a formula for the action of the one-loop dilatation operator on the orthogonal basis of the free theory, which is manifestly covariant under the global symmetry.