2008
DOI: 10.48550/arxiv.0802.1454
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Macdonald polynomials at $t=q^k$

Jean-Gabriel Luque

Abstract: We investigate the homogeneous symmetric Macdonald polynomials P λ (X; q, t) for the specialization t = q k . We show an identity relying the polynomials P λ (X; q, q k ) and P λ 1−q 1−q k X; q, q k . As a consequence, we describe an operator whose eigenvalues characterize the polynomials P λ (X; q, q k ).

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“…One possible approach would consist in searching for the latter operator in the double affine Hecke algebra. Indeed, in previous articles, two of the authors gave q-deformations [3,18] which can be written as symmetric Macdonald functions indexed by rectangular or staircase partitions for some specializations of the parameters (which made us think that the Hecke algebra may play a role). We have not identified the operator yet.…”
Section: Discussionmentioning
confidence: 99%
“…One possible approach would consist in searching for the latter operator in the double affine Hecke algebra. Indeed, in previous articles, two of the authors gave q-deformations [3,18] which can be written as symmetric Macdonald functions indexed by rectangular or staircase partitions for some specializations of the parameters (which made us think that the Hecke algebra may play a role). We have not identified the operator yet.…”
Section: Discussionmentioning
confidence: 99%