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I'd like to compute explicitly symmetric Macdonald functions associated to arbitrary (possibly non-reduced) root systems, using Computer Algebra System.

Unfortunately Sage seems to only implement the A-type Macdonald polynomials http://www.sagemath.org/doc/reference/sage/combinat/sf/macdonald.html

  • Is there a nice paper where a combinatorial formula is provided?
  • Has somebody happened to implement it in some programming language?

Of course I can perform the Gram-Schmidt orthogonalization w.r.t. the known measure, but I'll keep it as a last resort.

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you might like to ask developers of this functionality in Sage on groups.google.com/forum/?fromgroups=#!forum/sage-combinat-devel It could be that they have something for your task that haven't made it into Sage proper yet. –  Dima Pasechnik Jan 1 '13 at 4:02

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up vote 3 down vote accepted

There is a better way to compute Macdonald polynomials explicitly than through Gram-Schmidt orthogonalization: using the action of the Macdonald operators. Details can be found here DOI http://dx.doi.org/10.1112/S0010437X03000149 (also avaliable in a somewhat longer version at arXiv:math/0303263)

The first few Macdonald polynomials can be computed in closed form from the Pieri formula. Details can be found here DOI http://dx.doi.org/10.1007/s00209-010-0727-0 (also available at arXiv:1009.4482)

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