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It's well known that the $A_{n-1}$ Macdonald polynomial $P_\lambda(x;q,t)$ becomes the Schur function when $q=t$. Schur functions are characters of the general linear groups. What about other Macdonald polynomials (at least associated to the non-exceptional root systems) in that specialisation of parameters? I cannot find a reference which contains such information.

Thanks in advance.

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  • $\begingroup$ When you mean other Macdonald polynomials, do you mean the analogue of P in other types? There are also non-symmetric Macdonald polynomials in different types, and there is also the Modified Macdonald polynomial in type A... $\endgroup$
    – Per Alexandersson
    Commented Mar 8, 2017 at 19:41

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You find Macdonald polynomials defined for root systems for example in Macdonald's lecture series "Symmetric functions and orthogonal polynomaials", or as well in various places, like http://en.wikipedia.org/wiki/Macdonald_polynomials for the definition for any root system of finite type.

See also the old MO question universality of Macdonald polynomials and its detailed answers...

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