• Corpus ID: 14713960

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… 

Figures from this paper

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

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

  • C. Dunkl
  • Mathematics
    Symmetry, 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

TLDR
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

TLDR
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

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

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

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

References

SHOWING 1-10 OF 20 REFERENCES

Vector-Valued Jack Polynomials from Scratch

Vector-valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik{Dunkl

Yang-Baxter Graphs, Jack and Macdonald Polynomials

Abstract. We describe properties of the affine graph underlying the recursions between the different varieties of nonsymmetric Macdonald and Jack polynomials. We use an arbitrary function of one

A q-analogue of the type A Dunkl operator and integral kernel

We introduce the $q$-analogue of the type $A$ Dunkl operators, which are a set of degree--lowering operators on the space of polynomials in $n$ variables. This allows the construction of

Highest weight Macdonald and Jack polynomials

Fractional quantum Hall states of particles in the lowest Landau levels are described by multivariate polynomials. The incompressible liquid states when described on a sphere are fully invariant

Singular Polynomials and Modules for the Symmetric Groups

For certain negative rational numbers k0, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when

The character table of the Hecke algebra $H_n(q)$ in terms of traces of products of Murphy operators

The traces of the Murphy operators of the Hecke algebra $H_n(q)$, and of products of sets of Murphy operators with non-consecutive indices, can be evaluated by a straightforward recursive procedure.

A differential ideal of symmetric polynomials spanned by Jack polynomials at rβ = -(r=1)/(k+1)

For each pair of positive integers (k,r) such that k+1,r-1 are coprime, we introduce an ideal $I^{(k,r)}_n$ of the ring of symmetric polynomials. The ideal $I^{(k,r)}_n$ has a basis consisting of

Model fractional quantum Hall states and Jack polynomials.

TLDR
The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.

Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level

The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain

A New Approach to the Representation Theory of the Symmetric Groups. II

The present paper is a revised Russian translation of the paper “A new approach to representation theory of symmetric groups,” Selecta Math., New Series, 2, No. 4, 581–605 (1996). Numerous