# Generalization of Schur polynomials

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.

• There are some listed on page 405 of Enumerative Combinatorics, vol.2, that don't seem to be on your list. Apr 27, 2015 at 21:57
• There are also stable Grothendieck polynomials and dual stable Grothendieck polynomials: see arxiv.org/abs/0705.2189. Apr 27, 2015 at 22:43
• @PerAlexandersson: hmm, sorry, I thought I saw there something like the Cauchy identity. But perhaps I was dreaming :-) Apr 28, 2015 at 15:10
• How about the rep of the elementary Schur polynomials as cycle index polynomials (CIP--oeis.org/A036039) of the symmetric groups? How would the CIP and their generalizations fit into your scheme? Mar 7, 2016 at 20:25
• @TomCopeland: Sure, I think I found a reference for them: ac.els-cdn.com/0097316580900709/… Mar 7, 2016 at 21:07

• See the Carrell ref in A036039. Also the draft edits for A130561, which contains a ref to a paper by Ernst on "The history of the q-calculus and a new method." Pg. 127 of Ernst has the o.g.f. of the elem. Schur poly. : $\exp\left [\sum_{n > 0} a_n x^n \right ] = \sum_{n \geq 0} S_n(a_1,..,a_n) x^n$ Sep 8, 2016 at 8:28