@markusheinrich explained #math to me in #benasque how the twirl over a real adjoint representation instead of a complex one plays out in relation to #SchursLemma. The #commutatant needs to be a #divisionRing and there's only one over the complex numbers, the complex numbers. Over real numbers there are two more possibilities, real numbers and quaternions. Something called the Frobenius indicator is a way to know what will be the #rank of the projector appearing in the result of the twirl for a given #irrep.