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(Shifted) Macdonald polynomials: \(q\)-integral representation and combinatorial formula. (English) Zbl 0897.05085

Shifted Macdonald polynomials are related to (ordinary) Macdonald polynomials in an analogous manner as shifted Schur functions are related to (ordinary) Schur functions. Shifted Macdonald polynomials are, up to scalars, defined through certain vanishing conditions. The main result of this paper is a formula which expresses a shifted Macdonald polynomial as a certain weighted sum over semistandard tableaux, which is in analogy with the corresponding known formula for shifted Schur functions. This formula is established through a \(q\)-integral representation for the shifted Macdonald polynomials. A duality theorem for shifted Macdonald polynomials, relating those with parameters \(q,t\) to those with parameters \(1/t,1/q\), is proved as well.

MSC:

05E05 Symmetric functions and generalizations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
18B35 Preorders, orders, domains and lattices (viewed as categories)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22E46 Semisimple Lie groups and their representations
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