Some Orthogonal Polynomials in Four Variables

The symmetric group on 4 letters has the reflection group $D_3$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_3$ and $D_3$. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero–Sutherland model of 4 identical particles on the line.