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To deal with root systems of type B C D, one needs to understand symmetric Laurent polynomials $\Lambda$. I am wondering if the naive definition of power sum symmetric Laurent polynomials form a basis in $\Lambda$. More specifically, define $$ p_r = \sum_{i=1}^N x_i^r + x_i^{-r},$$ and define power sum symmetric Laurent polynomial indexed by a partition $\lambda \vdash n$ as $$ p_\lambda = \prod_{j=1}^n p_{\lambda_i}.$$ What are some standard references that talk about the connection of $\Lambda$ with root systems, as well as Macdonald operators? Thanks.

edit: I guess it is obvious that $p_\lambda$ does form a basis, if I restrict to symmetric Laurent polynomials that are also symmetric with respect to $x_i \mapsto x_i^{-1}$. So I would like to change my question to just asking for a reference relating $\Lambda$ with Macdonald operators.

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  • $\begingroup$ Hi! If I am not mistaken your question is related to mine(though I am working with type A instead of B,C,D at this stage): mathoverflow.net/questions/87939/… $\endgroup$
    – Kerry
    Commented Feb 14, 2012 at 23:43
  • $\begingroup$ Yes I think Fulton Harris has some stuff about Laurent polynomials, but not as extensively as Macdonald's book. $\endgroup$
    – John Jiang
    Commented Feb 14, 2012 at 23:54
  • $\begingroup$ You mean this one:amazon.com/…? $\endgroup$
    – Kerry
    Commented Feb 15, 2012 at 1:00
  • $\begingroup$ Yes. I think you will find most of the stuff known about symmetric polynomials in that book. It's surprising that the analogous theory for Laurent symmetric polynomials aren't developed in monograph form. $\endgroup$
    – John Jiang
    Commented Feb 15, 2012 at 4:59
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    $\begingroup$ Hi! I think you may find it interesting to look up resources related to Konstant polynomial. Personally I do not know very much about Macdonald operators/polynomial but it is nice to know. I also found his book. Thank you. $\endgroup$
    – Kerry
    Commented Feb 16, 2012 at 4:53

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