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We know that an important variant of the Schur Polynomial is the shifted Schur Polynomials that was developed by Okounkov & Olshanski.

The question here is: is there any other variant of Schur Polynomial which also has similar properties as Schur and Shifted Schur functions?

Edits:

What I mean by similar properties are things like:

a) this function satisfies Vanishing Theorem

b) let $s_{\mu}^* (\lambda)$ be a shifted Schur polynomials, then these polynomials can be written in some forms like: $f(\lambda) = s^*_ \mu (\lambda) \times g(n)$ where $g(n)$ is a function of $n$.

c) this function has some recursion equations

Sorry if the question is still too general enough...

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There's too many properties to keep or discard for this to be a well-defined question. O&O themselves mention the "factorial Schur functions" as a further generalization; is that good for you, or a generalization too far? From there you could go on to Schubert polynomials and Grothendieck polynomials, or Hall-Littlewood polynomials and Macdonald polynomials... –  Allen Knutson Mar 26 '13 at 11:40
    
made some edits –  terrylsc Mar 26 '13 at 12:46
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Ignoring your list of properties, there is a list of some variants of Schur functions (with references) in the Notes to Chapter 7 of my book Enumerative Combinatorics, vol. 2. –  Richard Stanley Mar 27 '13 at 1:30
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1 Answer 1

The most general polynomials I'm familiar with, satisfying the vanishing and recursion properties you want, are the double Grothendieck polynomials $\{G_\pi\}$.

There's one for every permutation $\pi$ of $\mathbb Z_+$ that moves only finitely many numbers. $G_\pi$ is a polynomial in two sets of variables, $x_1,x_2,\ldots$ and $y_1,y_2,\ldots$. The vanishing property is $G_\pi|_{y_i = x_{\rho(i)}} = 0$ for $\rho \not\geq \pi$. The recursion is defined using "Demazure operators" also known as "isobaric divided difference operators". I think the modern reference is Manivel's book on Schubert polynomials.

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