The Stanley decomposition of the harmonic oscillator

This paper gives a new decomposition for the ring of polynomial functions on the variety of (n + 1) × (n + 1) complex matrices of rank less than or equal to one. This involves decomposing the monoid $M_n=\left\{\right(j,k)\in {\mathbb{N}}^{n+1}\times {\mathbb{N}}^{n+1}|\left|j\right|=\left|k\right|\}$ into a finite disjoint union of translates of ℕ cones based on certain 2n simplices in ℝ²ⁿ⁺². As a consequence we have a method for writing the normal form of a perturbed n+1 dimensional harmonic oscillator in a unique way.