Orthogonal functions generalizing Jack polynomials
By Stephen Griffeth. 2007
48 citations recorded by the source corpus.
The rational Cherednik algebra ℍ is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for ℍ. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible W-modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on ℂ n with values in the representation S λ . For G(1,1, n) with λ = (n) these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all well-defined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of ℍ so that the rational Cherednik algebra for G(r,p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/37667e8bae38bfea909fcf49b38e0f4f782ea159
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Generalized Jack polynomials and the representation theory of rational Cherednik algebras (2010, methodology): “here because we are using the “inverse” definition of standard Young tableau to that in [ 10 ].”