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I am looking for quasi-symmetric versions of the classical $e_\lambda$ and $h_\lambda$ (the elementary and complete homogeneous symmetric functions). Is there some reference for this?

I am aware of at least four quasi-symmetric generalizations of Schur polynomials:

  • Extended Schur (Assaf, Searles)
  • Quasi-symmetric Schur (Haglund et al)
  • Young quasi-symmetric Schur
  • Dual Immaculate functions

There are also two quasi-symmetric power-sums which I know of.

I have defined a non-symmetric extension of the elementary symmetric functions, (as a certain specialization of non-symmetric Macdonald polynomials), and this can perhaps be turned into some quasi-symmetric version. However, I have not yet managed to do this.

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I sent an email to Stephanie van Willigenburg, and she suggested the following reference, which contains such generalizations.

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