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Decomposition matrices for Ariki-Koike algebras and crystal isomorphisms in Fock spaces

Thomas Gerber 1
1 Algèbre
LMPT - Laboratoire de Mathématiques et Physique Théorique
Abstract : This thesis is devoted to the study of modular representations of Ariki-Koike algebras, and of the connections with Kashiwara's crystal and canonical bases theory via Ariki's categorification theorem. First, we study, using combinatorial tools, the decomposition matrices associated to these algebras, generalising the works of Geck and Jacon. We fully classify the cases of existence and non-existence of canonical basic sets, and we explicitly construct these sets when they exist. Next, we make explicit the crystal isomorphisms for Fock spaces representations of the quantum affine algebra of affine type A. We then construct of a particular isomorphism, so-called canonical, which gives, inter alia, a non-recursive description of any connected component of the crystal. We also stress the links with the combinatorics of words underlying the crystal structure of Fock spaces, by describing notably an analogue of the Robinson-Schensted-Knuth correspondence for affine type A.
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Thomas Gerber. Decomposition matrices for Ariki-Koike algebras and crystal isomorphisms in Fock spaces. Representation Theory [math.RT]. Université François Rabelais - Tours, 2014. English. ⟨tel-01057480⟩

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