Plane partitions and spin adapted quantum states

We describe an explicit basis for the $\mathrm{SU}(2)$-invariant space of the exterior power ${\wedge }_{2k}{\mathbb{C}}^{2m}$ via the combinatorics of plane partitions. In quantum chemistry, this is the space of spin adapted quantum states of an electronic system with $m$ spin orbitals and $k$ electron pairs. We construct our basis by identifying the invariant space with an Artinian commutative ring called the excitation ring. We compute a Gröbner basis and enumerate its standard monomials via an explicit bijection to Dyck paths counted by the Narayana numbers.