Weight projection matrix ($a^{\prime }=Pa$)

A coordinate matrix expressing restriction of weights to the Cartan subalgebra of a Lie subalgebra. With Dynkin labels as column vectors, its rows encode the subalgebra's coroots in terms of the original coroots. It need not be an idempotent projection operator: an equal-rank embedding can give an invertible matrix. Applying it to the full weight system, including weight multiplicities, permits reconstruction of the restricted irreducible representations.