Cores and weights of multipartitions and blocks of Ariki-Koike algebras
Yanbo Li, Kai Meng Tan
TL;DR
The paper develops an e-core/e-weight theory for multipartitions associated to Ariki-Koike algebras by transporting multipartition data through the Uglov map. It proves that the e-core is invariant under the extended affine Weyl group action while the e-weight achieves its orbit-minimum in a precise chamber, yielding a Nakayama-type block classification that mirrors the classical type A picture and aligns with Fayers' weight. It extends the obreak[ w:k ]-pair notion to Ariki-Koike algebras and derives a sufficient condition for Scopes equivalence, thereby advancing Morita-type equivalences between blocks. The framework unifies combinatorial abacus machinery with representation-theoretic block structure, offering new tools for block classification and equivalences in cyclotomic Hecke algebras and their higher analogues.
Abstract
Let $e$ be an integer at least two. We define the $e$-core and the $e$-weight of a multipartition associated with a multicharge as the $e$-core and the $e$-weight of its image under the Uglov map. We do not place any restriction on the multicharge for these definitions. We show how these definitions lead to the definition of the $e$-core and the $e$-weight of a block of an Ariki-Koike algebra with quantum parameter $e$, and an analogue of Nakayama's `Conjecture' that classifies these blocks. Our definition of $e$-weight of such a block coincides with that first defined by Fayers. We further generalise the notion of a $[w:k]$-pair for Iwahori-Hecke algebra of type $A$ to the Ariki-Koike algebras, and obtain a sufficient condition for such a pair to be Scopes equivalent.