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I would like to have some order in my head about different version of Macdonald polynomials and positivity statements about them. I understand the following:

1) There is a definition of Macdonald polynomials for any root system. These can be defined, for example as $W$-invariant polynomials on the torus $T$ of a semi-simple group $G$, which are orthogonal polynomials with respect to Macdonald scalar product and normalized in such a way that $$ P_{\lambda}(q,t,x)=e^{\lambda}+\text{lower order terms} $$ where $\lambda$ is a dominant weight and $x\in T$.

2) In type $A$ there is a notion of transformed Macdonald polynomials, which were extensively studied by Haiman. Haiman denotes them by $\tilde{H}_{\lambda}$

(here $\lambda$ is a partition, which can be thought of as a domonant weight of $GL(n)$); he proved the Macdonald positivity conjecture, which says that ${\tilde H}_{\lambda}(q,t,x)$

is a linear combination of Schur functions in $x$ whose coefficients are polynomials in $q$ and $t$ with non-negative integral coefficients. The definition of ${\widetilde H}_{\lambda}(q,t,x)$ appears for example on page 4 of http://math.berkeley.edu/~mhaiman/ftp/nfact/polygraph-jams.pdf

My questions are these:

a) What is the relation between $P_{\lambda}$ and ${\tilde H}_{\lambda}$? It is not clear to me from the definition.

b) Are there positivity statements for $P_{\lambda}$ itself? Or is there a version of the positivity conjecture for any root system?

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It is a bit hard in my opinion to read what you want out of there, but this is in Macdonald's book, Symmetric Functions and Hall polynomials, 2nd edition. (It is not in the 1st edition.) –  Alexander Woo Oct 13 '11 at 23:09
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1 Answer

As far as I know there is no positivity statement for $P_\lambda$ as a linear combination of characters (which are the analog of Schur functions in general type). There is a very general statement that $P_\lambda$ expands positively as a sum of monomial symmetric functions; this is proved in a root-system uniform way by Ram and Yip in

http://arxiv.org/abs/0803.1146

The proof they give is for the "constant parameters" situation, but the technique works in general. There is some inefficiency in their formula (in type A it can be "compressed" to a formula of Haglund, Haiman and Loehr in http://arxiv.org/abs/math/0409538) which is the subject of current work.

Presumably character-positivity of $P_\lambda$ actually fails already in type A, and in general type there is no notion of plethystic substitution to save us!

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Thank you, this is very helpful. –  Alexander Braverman Oct 13 '11 at 4:05
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