Integrality of two variable Kostka functions

Macdonald defined a family of symmetric polynomials which depend on two parameters q and t. The coefficients of the transition matrix from Macdonald polynomials to Schur S-functions are called Kostka functions. Macdonald conjectured that they are polynomials in q and t with non-negative integers as coefficients. In the paper I prove that the Kostka functions are polynomials with integral coefficients. The positivity part remains open.

The proof uses a non-symmetric analogue of Macdonald polynomials (also introduced by Macdonald). I derive a recursion formula for them and a formula relating the symmetric with the non-symmetric Macdonald polynomials. I also define a non-symmetric analogue of Hall-Littlewood polynomials and use them to state and prove an integrality result for the non-symmetric Macdonald polynomials. This implies integrality of Kostka functions.

This paper can be seen as a "quantization" of A recursion and a combinatorial formula for Jack polynomials.

Appeared in: Journal für die reine und angewandte Mathematik 482 (1997) 177-189

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Last updated: October 26, 2000