Irreducible representations of rational Cherednik algebras for exceptional Coxeter groups, part II: some decomposition matrices of $H_c(E_8)$ and $H_c(F_4)$

TL;DR

This paper provides detailed decomposition matrices for certain blocks of the rational Cherednik algebra associated with exceptional Coxeter groups E_8 and F_4, advancing understanding of their module structures and classifications.

Contribution

It presents the decomposition matrices for blocks of defect at most 2 in Category O of the rational Cherednik algebra for E_8 and F_4, with implications for classifying module supports.

Findings

Decomposition matrices for blocks of defect ≤ 2 in E_8 and F_4.

Classification of support dimensions of irreducible modules.

Finite-dimensional modules of H_c(E_8) identified for specific parameters.

Abstract

This paper contains the decomposition matrices for blocks of defect at most $2$ in Category $\mathcal{O}_c(W)$ of the rational Cherednik algebra when $W=E_8$ or $F_4$ with equal parameters $c=1/d$, $d>2$ a regular number of $W$. A corollary of the result is a classification of the dimensions of support of the irreducible modules $L(\tau)$ in $\mathcal{O}_{1/d}(W)$ except in the following cases: $W=E_8$, $d=4$ or $6$ and $L(\tau)$ is in the principal block, or $d=2$ or $3$; $W=F_4$, $d=2$. In particular, this classifies the finite-dimensional modules of $H_c(E_8)$ when $d\neq 2,3,4,6$.