Moving vectors and core blocks of Ariki-Koike algebras
Yanbo Li, Xiangyu Qi, Kai Meng Tan
TL;DR
This work develops a comprehensive combinatorial and representation-theoretic framework for core blocks of Ariki-Koike algebras using moving vectors. It provides a sharp classification: a block is core exactly when its moving vector has a zero in some component, and it then yields a concrete Scopes-equivalence criterion among core blocks, together with a Kostka-number description for the count of simple modules in a core block. Under suitable multicharge constraints, the authors relate the graded decomposition numbers in characteristic zero to those of type $A$ Hecke algebras via the Fock-space canonical basis and a rank-level duality map, enabling transfer of level-one data to higher levels. Collectively, these results bridge Ariki-Koike block theory with classical combinatorics (Kostka numbers, FLOTW/Kleshchev partitions) and with type $A$ Hecke algebra decomposition theory, offering structural and computable insights for the representation theory of these algebras.
Abstract
We classify the core blocks of Ariki-Koike algebras by their moving vectors. Using this classification, we obtain a necessary and sufficient condition for Scopes equivalence between two core blocks, and express the number of simple modules lying in a core block as a classical Kostka number. Under certain conditions on the multicharge and moving vector, we further relate the graded decomposition numbers of these blocks in characteristic zero with the graded decomposition numbers of the Iwahori-Hecke algebras of type $A$.