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<p>Let <i>e</i> be an integer at least two. We define the <i>e</i>-core and the <i>e</i>-weight of a multipartition associated with a multicharge as the <i>e</i>-core and the <i>e</i>-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 <i>e</i>-core and the <i>e</i>-weight of a block of an Ariki–Koike algebra with quantum parameter <i>e</i> and an analogue of Nakayama’s ‘Conjecture’ that classifies these blocks. Our definition of <i>e</i>-weight of such a block coincides with that first defined by Fayers (Adv Math 206(1):112–144, 2006). We further generalise the notion of a [<i>w</i>&#xa0;:&#xa0;<i>k</i>]-pair for Iwahori–Hecke algebra of type <i>A</i> to the Ariki–Koike algebras and obtain a sufficient condition for such a pair to be Scopes equivalent.</p>

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Cores and weights of multipartitions and blocks of Ariki–Koike algebras

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • Yanbo Li,
  • Kai Meng Tan

摘要

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 (Adv Math 206(1):112–144, 2006). 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.