External Vertices for Crystals of Affine Type A

Ola Amara-Omari, Mary Schaps

Research output: Contribution to journalArticlepeer-review

Abstract

We demonstrate that for a fixed dominant integral weight and fixed defect d, there are only a finite number of Morita equivalence classes of blocks of cyclotomic Hecke algebras, by combining some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over Kac-Moody algebras of affine type A. This is an extension of a proof for symmetric groups of a conjecture known as Donovan’s conjecture. We fix a dominant integral weight Λ. The blocks of cyclotomic Hecke algebras HnΛ for the given Λ correspond to the weights P(Λ) of a highest weight representation with highest weight Λ. We connect these weights into a graph we call the reduced crystal P̂ (Λ) , in which vertices are connected by i-strings. We define the hub of a weight and show that a vertex is i-external for a residue i if the defect is less than the absolute value of the i-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a given defect d are i-external in at least one i-string, lying at the high degree end of the i-string. For e = 2, we calculate an approximation to this bound.

Original languageEnglish
Pages (from-to)2785-2800
Number of pages16
JournalAlgebras and Representation Theory
Volume26
Issue number6
DOIs
StatePublished - Dec 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

Keywords

  • Affine Lie algebra
  • Cyclotomic Hecke algebras
  • Donovan’s conjecture
  • Highest weight representation

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