# 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. – Richard Stanley Apr 27 '15 at 21:57
• There are also stable Grothendieck polynomials and dual stable Grothendieck polynomials: see arxiv.org/abs/0705.2189. – Sam Hopkins Apr 27 '15 at 22:43
• @PerAlexandersson: hmm, sorry, I thought I saw there something like the Cauchy identity. But perhaps I was dreaming :-) – Dima Pasechnik Apr 28 '15 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? – Tom Copeland Mar 7 '16 at 20:25
• @TomCopeland: Sure, I think I found a reference for them: ac.els-cdn.com/0097316580900709/… – Per Alexandersson Mar 7 '16 at 21:07

## 1 Answer

(Too long for a comment.) The elementary Schur polynomials (edit: cf. Sec. 1.5 of "Combinatorial solutions to integrable hierarchies" by Kazarian and Lando) are also the refined Lah partition polynomials (divided by n!) of OEIS A130561 and so are connected by simple transformations of the defining indeterminates to the celebrated Bell partition polynomials A036040 of the Faa di Bruno formula (refined Stirling numbers of the second kind), and the cycle index polynomials of the symmetric groups (refined Stirling numbers of the first kind) of A036039. All these partition polynomials can be generalized or restricted then generalized in different directions often by the action of differential operators or as reps of iterated differential ops to incorporate a whole slew of important polynomials, e.g., the associated Laguerre polynomials and generalized Stirling numbers and associated polynomials with numerous combinatorial interpretations.

• What are the elementary Schur polynomials? – Per Alexandersson Sep 8 '16 at 2:08
• 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. : $\exp\left [\sum_{n > 0} a_n x^n \right ] = \sum_{n \geq 0} S_n(a_1,..,a_n) x^n$ – Tom Copeland Sep 8 '16 at 8:28
• See my March 7'th comment above. – Tom Copeland Sep 8 '16 at 8:38