# Vector valued Macdonald polynomials

@article{Dunkl2011VectorVM, title={Vector valued Macdonald polynomials}, author={Charles F. Dunkl and Jean-Gabriel Luque}, journal={arXiv: Combinatorics}, year={2011} }

This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type $A_{N-1}$. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several objects and properties are analyzed, such as the canonical bilinear form which pairs polynomials with those arising from reciprocals of the original parameters, and the symmetrization of the Macdonald polynomials. The main tool of the study is the Yang-Baxter…

## 7 Citations

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

- Mathematics
- 2019

We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric…

### Nonsymmetric Macdonald Superpolynomials

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2020

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [Sém. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875]…

### Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

- MathematicsSymmetry
- 2021

The values of a subclass of the polynomials at the special points 1,t,t2,… or 1, t−1,t−2,….

### Some Singular Vector-Valued Jack and Macdonald Polynomials

- MathematicsSymmetry
- 2019

The singular polynomials whose leading term is x 1 m ⊗ S, where S is an arbitrary reverse standard Young tableau of shape τ, depend on the properties of the edge of the Ferrers diagram of τ .

### A positive-definite inner product for vector-valued Macdonald polynomials

- Mathematics
- 2018

In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the…

### Clustering properties of rectangular Macdonald polynomials

- Mathematics
- 2012

The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the…

### Singular Nonsymmetric Macdonald Polynomials and Quasistaircases

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2020

Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase…

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