Journal article

A Summation Formula for Macdonald Polynomials

Jan de Gier, Michael Wheeler



We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases (Formula presented.) and (Formula presented.) , we recover known expressions for the monomial symmetric and Hall–Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q–Whittaker polynomials.


Funding Acknowledgements

We thank the Galileo Galilei Institute and the organizers of the research program Statistical Mechanics, Integrability and Combinatorics for kind hospitality during part of this work. It is a pleasure to thank Luigi Cantini for collaboration on [1], which was very motivational for this work; Philippe Di Francesco for bringing the papers [8,9] to our attention; Ole Warnaar for directing us to the references [4,7] for Equation (18) and for many helpful remarks; and Paul Zinn-Justin for extended discussions on related topics. JdG and MW are generously supported by the Australian Research Council (ARC) and the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).