Fixed points in Gieseker spaces and blocks of Ariki-Koike algebras
Raphaël Paegelow
TL;DR
This work builds a bridge between the geometry of Gieseker spaces, realized as Nakajima quiver varieties, and the (modular) representation theory of Ariki-Koike algebras. By analyzing the fixed point loci ${\mathpzc{G}(n,r)^{\Gamma_{\mathcalb{s}}}}$ through framed McKay quivers and affine type $A_{\ell}$ data, the authors prove that irreducible components correspond to blocks via a dimension-weight correspondence and root-lattice descriptions. They develop a rich combinatorial framework using multipartitions, abaci, and $\ell$-cores to index blocks, and show that geometric cores reflect the core-block notion on the algebraic side, with explicit links between dominant weights and residues. The results provide a geometric route to understanding decomposition matrices and block structures in cyclotomic Hecke algebras, illuminating the deep interplay between quiver varieties, affine Lie algebras, and modular representation theory. This geometric perspective yields concrete criteria for when two Specht modules lie in the same block and clarifies how $\ell$-core data encode block invariants.
Abstract
In this article, we establish combinatorial links between the irreducible components of the fixed point locus of the Gieseker variety and the block theory of Ariki-Koike algebras. First, we describe the fixed point locus in terms of Nakajima quiver varieties over the McKay quiver of type A. We then reinterpret the dimension of an irreducible component as double the weight of a block. Cores of charged multipartitions have been defined by Fayers and further developed by Jacon and Lecouvey. In addition, we give a new way to compute the multicharge associated with the core of a charged multipartition. Finally, we also explain how the notion of core blocks, defined by Fayers, is interpreted on the geometric side using the deep connection between quiver varieties and affine Lie algebras.