[Paper Review] Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)

Stephen Griffeth|ArXiv.org|Dec 23, 2006

Algebraic structures and combinatorial models21 references14 citations

TL;DR

This paper develops a combinatorial representation theory for the rational Cherednik algebra of type $G(r,p,n)$ using intertwining operators and orthogonal functions, providing a self-contained proof of Haiman's conjecture (Gordon's theorem) on the diagonal coinvariant ring for $G(r,p,n)$ when $r > 1$, without restrictions on $p$. The key contribution is a new, elementary proof of the Hilbert series formula for the diagonal coinvariant ring using module-theoretic and combinatorial techniques in the context of category $\mathcal{O}$.

ABSTRACT

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p

Motivation & Objective

• To establish a combinatorial framework for the representation theory of the rational Cherednik algebra of type $G(r,p,n)$ using intertwining operators and orthogonal functions.
• To provide a self-contained, elementary proof of the analog of Gordon’s theorem on the diagonal coinvariant ring for $G(r,p,n)$, valid for all $p$ when $r > 1$.
• To construct canonical bases for standard modules $M(V)$ and their irreducible quotients $L(V)$, enabling deeper combinatorial study of the module lattice.
• To verify that the radical of the standard module $M(\mathbf{1})$ is generated by specific monomials, leading to a quotient isomorphic to the diagonal coinvariant ring.

Proposed method

• Introduce and analyze intertwining operators for the rational Cherednik algebra $\mathbb{H}$ of type $G(r,p,n)$, deriving their fundamental algebraic relations.
• Construct a basis for the polynomial representation of $G(r,p,n)$ and compute the action of intertwining operators on this basis.
• Use the Drinfeld Hecke algebra formalism to define $\mathbb{H}$ via relations involving skew-symmetric forms $\langle\cdot,\cdot\rangle_w$ on $V = \mathfrak{h}^* \oplus \mathfrak{h}$.
• Leverage the Dunkl-Opdam commutative subalgebra of $\mathbb{H}$ to analyze the structure of the polynomial representation and its submodules.
• Apply the Poincaré-Birkhoff-Witt theorem to establish a filtration and associated graded structure on $\mathbb{H}$-modules, enabling the use of graded module theory.
• Use the determinant formula for matrices of monomials $f_{i,j}$ and $v_j$ to compute the exponents of the $W$-module $V$, leading to the Hilbert series of the coinvariant ring.

Experimental results

Research questions

• RQ1 Can a combinatorial representation theory for the rational Cherednik algebra of type $G(r,p,n)$ be developed using intertwining operators and orthogonal functions?
• RQ2 Does the diagonal coinvariant ring of $G(r,p,n)$ have a Hilbert series matching the formula predicted by Haiman’s conjecture, and can this be proven without KZ functors or DAHA?
• RQ3 What is the submodule structure of the polynomial representation of $G(r,p,n)$, particularly the radical of $M(\mathbf{1})$?
• RQ4 Can the $\mathbb{H}$-module $L(\mathbf{1})$ be shown to have a BGG resolution of the form $0 \to M(\Lambda^n V) \to \cdots \to M(\mathbf{1}) \to L(\mathbf{1}) \to 0$ at certain parameter values?

Key findings

• The radical of the standard module $M(\mathbf{1})$ is generated by the $W$-module $\mathbb{C}\{f_{\mu_1}, \dots, f_{\mu_n}\}$, which is isomorphic to $V = \mathbb{C}\{x_1^{h+1}, \dots, x_n^{h+1}\}$.
• The determinant of the matrix $A$ with rows $f_{i,j}$ and $v_j$ is $(-1)^n (x_1 \cdots x_n)^{m'} \prod_{1\leq i<j\leq n} (x_i^r - x_j^r)$, confirming the correct degree and structure of the coinvariant ring.
• The exponents of the $W$-module $V$ are $e_i(V) = \overline{m} + (i-1)r$ for $1 \leq i \leq n-1$ and $e_n(V) = (n-1)(r - \overline{m}) + n m'$ when $p > 1$, and $e_i(V) = \overline{m} + (i-1)r$ for $p=1$.
• The Hilbert series of the diagonal coinvariant ring $R$ is given by $\sum \text{tr}(w, (L \otimes \Lambda^n V)_i) t^i = \frac{\det(1 - t^{h+1} w_V)}{\det(1 - t w_{\mathfrak{h}^*})}$, confirming the formula of Gordon’s theorem.
• The image of $S(\mathfrak{h}^*)$ in the quotient $L$ is isomorphic to the ordinary coinvariant ring, showing compatibility with classical results.