Séminaire Lotharingien de Combinatoire, 91B.32 (2024), 12 pp.
Houcine Ben Dali and Michele D'Adderio
Macdonald Characters from a New Formula for Macdonald Polynomials
Abstract.
We introduce a new operator Γ on symmetric functions, which enables us to obtain a creation formula for Macdonald polynomials. This formula makes a connection between the theory of Macdonald operators initiated by Bergeron, Garsia, Haiman and Tesler, and shifted Macdonald polynomials introduced by Knop, Okounkov and Sahi.
We use this formula to introduce a two-parameter generalization of Jack characters. Finally, we provide a change of variables in order to formulate several positivity conjectures related to these generalized characters. Our conjectures extend some important problems on Jack polynomials, including some famous conjectures of Goulden and Jackson.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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