Basic r-symmetric tropical polynomials

We consider tropical polynomials in nr variables, divided into n blocks of r variables, and especially r-symmetric tropical polynomials, which are invariant under the action of the symmetric group Sn on the blocks. We define a set of basic r-symmetric tropical polynomials and show that the basic 2-symmetric tropical polynomials give coordinates on R2n/Sn more efficiently than known polynomials. Moreover, we present special cases for r≥3 where the basic polynomials separate orbits.