# Canonical Bases in Affine Type A and Ariki’s Theorem

Ontology type: schema:Chapter

### Chapter Info

DATE

2011

AUTHORS

Meinolf Geck , Nicolas Jacon

ABSTRACT

To complete the main results of the previous chapter, we need to define a new object: the quantum affine algebra $$\mathcal{U}_{q}(\widehat{\mathfrak{sl}}_{e})$$. We give a brief introduction to the theory of canonical bases and crystals for these algebras following the works of Lusztig and Kashiwara. We then state Ariki’s theorem which gives the connection between this theory and the representation theory of Ariki-Koike algebras. As a consequence, we will be able to complete the proof of the main results of the previous chapter. We then go further and survey Uglov’s theory concerning canonical bases of Fock spaces. This will allow us to give an explicit description of the canoncial basic sets for Ariki-Koike algebras in all cases in characteristic 0. We also present an algorithm for computing the decomposition matrices of these algebras. More... »

PAGES

309-361

### Book

TITLE

Representations of Hecke Algebras at Roots of Unity

ISBN

978-0-85729-715-0
978-0-85729-716-7

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-0-85729-716-7_6

DOI

http://dx.doi.org/10.1007/978-0-85729-716-7_6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1052274426

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