  We analyze conditions under which a projection from the vector-valued Jack or
Macdonald polynomials to scalar polynomials has useful properties, especially
commuting with the actions of the symmetric group or Hecke algebra,
respectively, and with the Cherednik operators for which these polynomials are
eigenfunctions. In the framework of the representation theory of the symmetric
group and the Hecke algebra, we study the relation between singular
nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack
and Macdonald polynomials. Moreover, we study the quasistaircase partition as a
continuation of our study on the conjectures of Bernevig and Haldane on
clustering properties of symmetric Jack polynomials.
