The Stanley decomposition of the harmonic oscillator

Abstract

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 = { ( j , k ) ∈ ℕ n + 1 × ℕ n + 1 | | j | = | k | } into a finite disjoint union of translates of ℕ cones based on certain 2 n simplices in ℝ 2n+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.