Connections between vector-valued and highest weight Jack and Macdonald polynomials

Abstract

We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, specially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.Key words and phrases. Macdonald and Jack Polynomials, singular polynomials, highest weight polynomials, vector-valued polynomials, representation theory of symmetric group and Hecke algebra.This work is partially supported by the "European Regional Development Fund" (ERDF) via the regional (GRR) project MOUSTIC. 1 VECTOR-VALUED AND HIGHEST WEIGHT JACK AND MACDONALD POLYNOMIALS 2 5.3. The symmetric group case 20 5.3.1. Singularity of quasistaircase nonsymmetric Jack polynomials 20 5.3.2. From singular nonsymmetric Jack polynomials to highest weight symmetric Jack polynomials 23 5.4. The Hecke algebra case 23 5.4.1. Singularity of quasistaircase nonsymmetric Macdonald polynomials 24 5.4.2. From singular nonsymmetric Macdonald polynomials to highest weight symmetric Macdonald polynomials 25 5.4.3. A note on specializations 27 6. Factorizations at special points 27 6.1. The symmetric group case 28 6.2. The Hecke algebra case 29 6.2.1. The general polynomials of isotype τ 34 6.2.2. Application to projections and factorizations 35 7. Conclusion and perspective 38 References 39

Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

“…, 0 7 ). Observe that the location of the two out-of-order entries, [2,2] and [3,2], stays the same. We will show that (α (Θ 2,2 ) , β) is the only (1, 4)critical pair where β = (4, 4, 3, 0, 0, 0, 0, 3, 1 9 ).…”

“…Singular polynomials appear as a tool used to construct projection maps for vector-valued Macdonald polynomials and to find factorizations connected with highest weight symmetric polynomials, [2]. In the most general setting, a polynomial p ∈ P is said to be singular if there exist some specialization of (q, t) for which pξ i = pφ i , for all 1 ≤ i ≤ N. When it comes to Macdonald polynomials, we have the following equivalent definition.…”

“…The condition ρ i = 1 is necessary for the validity of the proof, even though it is always true for quasistaircases. For instance, for α = (0, m, 1 n−1 ), q α 2 −α 1 t rα(1)−rα (2) = q m t n . However, α is not of staircase type.…”

“…It is remarkable that this leads directly to singular polynomials, which are defined to be in the joint kernels of Dunkl operators. We already looked at singular Macdonald polynomials in our work with Jean-Gabriel Luque in [2], where the singular polynomials form the basic ingredient of the projection map described there.…”