\(c\)-functions and Macdonald polynomials (Q6565649)

In this paper the authors want to give some visibility to the so-called \(c\)-functions and explain how they provide a good understanding of combinatorial formulas in Macdonald's theory of polynomials.\N\NThe authors start presenting different broad perspectives indicating why Macdonald polynomials are fascinating objects for research on theoretical and applied physics; later they introduce introduce the main objects of study such as electronic Macdonald polynomials (known as nonsymmetric in the literature), bosonic Macdonald polynomials and fermionic Macdonald polynomials.\N\NThey also develop some practical tools to deal with Macdonald polynomials. Basically, they consider four primary families of operators such as Hecke algebra operators, promotion operators and their related intertwiner operators, Cherednik-Dunkl operators, and some symmetrizers.\N\NWith all of this the authors provide alternative proofs to several classical results by \textit{I. G. Macdonald} [Affine Hecke algebras and orthogonal polynomials. Cambridge: Cambridge University Press (2003; Zbl 1024.33001)]. In addition, they prove the boson-fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials.