BPS operators in $\mathcal{N}=4$ $SO(N)$ super Yang-Mills theory: plethysms, dominoes and words

Understand

Permutations and associated algebras allow the construction of half and quarter BPS operators in maximally supersymmetric Yang Mills theory with $U(N)$, $SO(N)$ and $Sp(N) $ gauge groups.

• The construction leads to bases for the operators, labelled by Young diagrams and associated group theory data, which have been shown to be orthogonal under the inner product defined by the free field two-point functions.
• In this paper, we study in detail the orientifold projection map between the Young diagram basis for $U(N)$ theories and the Young diagram basis for $ SO(N)$ (and $ Sp(N)$) half-BPS operators.
• We find a simple connection between this map and the plethystic refinement of the Littlewood Richardson coefficients which couple triples of Young diagrams where two of them are identical.