Fixed points in Gieseker spaces and blocks of Ariki-Koike algebras

TL;DR

This paper explores the geometric and combinatorial structures linking fixed points in Gieseker varieties to the block theory of Ariki-Koike algebras, revealing new insights into their interplay via quiver varieties and multipartition cores.

Contribution

It establishes a novel connection between the geometry of Gieseker varieties and the combinatorics of Ariki-Koike algebra blocks, including new methods to compute multicharges and interpret core blocks.

Findings

Describes fixed point loci in terms of Nakajima quiver varieties.

Reinterprets irreducible component dimensions as twice block weights.

Provides a new computational approach for multicharge of charged multipartitions.

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.