Plane partitions and spin adapted quantum states
Abigail Price, Ada Stelzer, Svala Sverrisdóttir
TL;DR
This work constructs an explicit combinatorial model for the space of spin-adapted quantum states arising from SU(2) invariance on the exterior power of a spin-orbital space. By defining the excitation ring $S_{m,k}=\mathbb{C}[X]/I_{m,k}$ with cubics $f_{p,q,r}^{a,b,c}$, the authors obtain a Gröbner-basis-driven standard monomial basis indexed by $k\times(m-k)$ matrices of width at most $2$. They then map these standard monomials to plane partitions via a variant of the RSK correspondence, reducing the counting to plane partitions $\,\mathcal{B}(k,m-k,2)$, whose size equals the Narayana numbers $N(m+1,k+1)$; this yields the exact dimension $\dim\mathcal{H}_{m,2k}^{\operatorname{SU}(2)} = N(m+1,k+1)$. The results unify invariant theory, combinatorics of Dyck paths, and quantum-chemical spin adaptation, providing a concrete model for spin-adapted states with potential for further algebraic-geometry analyses.
Abstract
We describe an explicit basis for the $\operatorname{SU}(2)$-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.