Abstract
We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of U_q(gl_n). In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.
Highlights
The Macdonald polynomials Pλ(x; q, t) are a two-parameter family of symmetric functions indexed by partitions λ which form an orthogonal basis for the ring of symmetric functions with respect to a (q, t)-deformation of the standard inner product
We show that the resulting summation expression for Pλ(x; q, t) becomes Macdonald’s branching rule after a summation by parts procedure
We extend the definition of Macdonald polynomials to arbitrary signatures by setting
Summary
The Macdonald polynomials Pλ(x; q, t) are a two-parameter family of symmetric functions indexed by partitions λ which form an orthogonal basis for the ring of symmetric functions with respect to a (q, t)-deformation of the standard inner product. Macdonald proved a branching rule for the Pλ(x; q, t) and conjectured three additional symmetry, evaluation, and norm identities collectively known as Macdonald’s conjectures. These conjectures were proven by Cherednik using techniques from double affine Hecke algebras in [6]. Etingof and Kirillov Jr. realized the Macdonald polynomials in [11] in terms of traces of intertwiners of the quantum group Uq(gln); using this interpretation, they gave new proofs of Macdonald’s conjectures in [12]. In the remainder of the introduction, we summarize our motivations, give precise statements of our results, and explain how they relate to other recent work
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