Abstract
Generalized Macdonald polynomials (GMP) are eigenfunctions of specificallydeformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials. They are orthogonal with respect to a modified scalar product, which could be constructed with the help of an increasingly important triangular perturbation theory, showing up in a variety of applications. A peculiar feature of GMP is that denominators in this expansion are fully factorized, which is a consequence of a hidden symmetry resulting from the special choice of the Hamiltonian deformation. We introduce also a simplified but deformed version of GMP, which we call generalized Schur functions. Our basic examples are bilinear in Macdonald polynomials.
Highlights
Introduction to Generalized Macdonald polynomials (GMP)Let us discuss if one can define the GMP using one of the two possibilities that we discussed in the previous section for the ordinary Macdonald polynomials.3.1 Triangular structure of GMPAs reviewed in detail in recent [73], the basic property of all symmetric polynomials naturally emerging in the course of study of non-perturbative quantum field theory, is that they are triangular combinations of Schur functions
A peculiar feature of GMP is that denominators in this expansion are fully factorized, which is a consequence of a hidden symmetry resulting from the special choice of the Hamiltonian deformation
This property is a bonus from existence of the natural lexicographical ordering for Young diagrams, and it breaks down when one goes up from ordinary to plane partitions [74,75,76], but it is still true for finite sets of Young diagrams, and for the Generalized Macdonald and Kerov functions
Summary
Continue it to any polynomials by linearity. From no on, we use the notation {x} :=. Combinatorial factor z∆ is best defined in the dual parametrization of the Young diagram, ∆ = . The ordinary Macdonald polynomials can be defined as a lower triangular combination of the Schur polynomials SR{p}. R > R′ if r1 > r1′ or if r1 = r1′ , but r2 > r2′ , or if r1 = r1′ and r2 = r2′ , but r3 > r3′ , and so on (2.4) This ordering is not consistent with the transposition of Young diagrams:. It does not lead to an ambiguity, since the Kostka-Macdonald coefficients KR,R′(q, t) for this pair of diagrams, and in all similar cases vanishes [1]. An important point is that one can unambiguously calculate the Macdonald polynomials using this definition. We have a good definition that does not appeal to any additional algebraic structures
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