Knop F (1997)
Publication Language: English
Publication Status: Published
Publication Type: Journal article, Original article
Publication year: 1997
Publisher: Walter de Gruyter
Book Volume: 482
Pages Range: 177-189
DOI: 10.1515/crll.1997.482.177
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.
APA:
Knop, F. (1997). Integrality of two variable Kostka functions. Journal für die reine und angewandte Mathematik, 482, 177-189. https://doi.org/10.1515/crll.1997.482.177
MLA:
Knop, Friedrich. "Integrality of two variable Kostka functions." Journal für die reine und angewandte Mathematik 482 (1997): 177-189.
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