[Paper Review] Finite dimensional modules for rational Cherednik algebras

Stephen Griffeth|arXiv (Cornell University)|Dec 23, 2006

Algebraic structures and combinatorial models12 references6 citations

TL;DR

This paper constructs finite-dimensional modules for rational Cherednik algebras of the groups G(r,p,n) using intertwining operators and a commutative family of Dunkl–Opdam operators. It provides an explicit weight basis via a hyperplane arrangement and constructs eigenfunctions for the coinvariant ring whose leading terms are descent monomials in the G(1,1,n) case, extending results of Garsia and Stanton.

ABSTRACT

We construct and study some finite dimensional modules for rational Cherednik algebras for the groups G(r, p, n) by using intertwining operators and a commutative family of operators introduced by Dunkl and Opdam. The coinvariant ring and an analog of the ring constructed by Gordon in the course of proving Haiman’s conjectures on diagonal coinvariants are special cases. We study a certain hyperplane arrangement that determines a weight basis for the irreducible modules in an explicit fashion. Using the same methods we also construct a basis of eigenfunctions for the coinvariant ring for G(r,p, n) whose leading terms are, in the case of G(1,1, n), the descent monomials studied by Garsia and Stanton.

Motivation & Objective

• To construct finite-dimensional modules for rational Cherednik algebras associated with the complex reflection groups G(r,p,n).
• To generalize the coinvariant ring and Gordon’s analog of the diagonal coinvariant ring to the G(r,p,n) setting.
• To provide an explicit weight basis for irreducible modules using a hyperplane arrangement.
• To construct a basis of eigenfunctions for the coinvariant ring whose leading terms correspond to descent monomials in the G(1,1,n) case.

Proposed method

• Utilizes intertwining operators to construct finite-dimensional modules for rational Cherednik algebras of G(r,p,n).
• Applies a commutative family of operators introduced by Dunkl and Opdam to analyze the structure of these modules.
• Employs a hyperplane arrangement to determine a weight basis for irreducible modules in an explicit, combinatorial fashion.
• Constructs eigenfunctions for the coinvariant ring by leveraging the same operator framework and basis construction.
• Relies on the known structure of descent monomials in the G(1,1,n) case to identify leading terms of the eigenfunction basis.
• Establishes a connection between the module construction and the representation theory of rational Cherednik algebras via commutative operator families.

Experimental results

Research questions

• RQ1 How can finite-dimensional modules for rational Cherednik algebras of G(r,p,n) be systematically constructed?
• RQ2 What is the role of the Dunkl–Opdam commutative family of operators in determining weight bases for irreducible modules?
• RQ3 How does the hyperplane arrangement associated with G(r,p,n) determine a weight basis for irreducible modules?
• RQ4 Can a basis of eigenfunctions for the coinvariant ring be constructed such that its leading terms are descent monomials in the G(1,1,n) case?
• RQ5 To what extent do the coinvariant ring and Gordon’s analog for G(r,p,n) generalize the diagonal coinvariant ring structure?

Key findings

• Finite-dimensional modules for rational Cherednik algebras of G(r,p,n) are constructed using intertwining operators and the Dunkl–Opdam commutative family of operators.
• The hyperplane arrangement associated with G(r,p,n) provides an explicit, combinatorial construction of a weight basis for irreducible modules.
• A basis of eigenfunctions for the coinvariant ring is constructed, with leading terms matching the descent monomials of Garsia and Stanton in the G(1,1,n) case.
• The coinvariant ring and Gordon’s analog for G(r,p,n) are shown to be special cases of the constructed modules.
• The method yields a uniform framework for analyzing the representation theory of rational Cherednik algebras across the G(r,p,n) family.
• The construction establishes a direct link between the algebraic structure of the modules and combinatorial bases in symmetric function theory.