EDIT (2018-11-05) I am slowly making a list of symmetric functions, and generalizations available online here. PDFs with overviews are available there for download.
I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of specializations/relations between all these, and which ones have say a Cauchy identity, positive multiplicative structure constants, constitute a basis, have determinant formulation, combinatorial interpretation, etc.
So far, these are (more or less) the ones I know about, and I would like to know what more I have missed.
- Hall–Littlewood polynomials
- Shifted Schur polynomials
- LLT polynomials
- Quasi-symmetric Schur polynomials
- Factorial Schur polynomials
- Flagged Schur polynomials
- Double Schur polynomials
- Schubert polynomials (and double Schubert)
- Stanley symmetric functions (also known as stable Schubert polynomials)
- Key polynomials (also known as Demazure characters)
- Jack polynomials
- Macdonald polynomials (where there are non-symmetric and non-homogeneous variants)
- Schur polynomials for the symplectic and orthogonal group.
- $k$-Schur functions (defined via cores)
- Loop Schur functions
- Grothendieck polynomials ($K$-theoretical analogue of Schur polynomials)
This should be a community wiki, I guess.
EDIT: I have started making an overview but I get the feeling it should be split into two posets, (specializes/is superset of and expands positively in) since these two relations usually go in the opposite order.