Decomposition numbers of Ariki-Koike algebras

Dell’Arciprete, Alice (2023) Decomposition numbers of Ariki-Koike algebras. Doctoral thesis, University of East Anglia.

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This thesis is concerned with the representation theory of the symmetric groups and related algebras, in particular the combinatorics underlying the representations of Ariki-Koike algebras. The Ariki-Koike algebras generalise Iwahori-Hecke algebras of the symmetric group, and so in turn generalise the symmetric groups themselves.

The representation theory of these algebras is the subject of a great deal of research, with the most important outstanding problem being the determination of the decomposition numbers, i.e. the composition multiplicities of the simple modules Dμ in the Specht modules Sλ. The aim of this thesis is to contribute and make progress on the decomposition number problem.

We shall first develop some combinatorial lemmas related to the abacus display of multipartitions. Then, we will use these to examine blocks of the Ariki-Koike algebras. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of n whose Specht modules belong to a block B and those of n − δi(B) whose Specht modules belong to the block B′, obtained from B applying a Scopes’ equivalence. This bijection gives us an equivalence for the decomposition numbers of the corresponding Ariki-Koike algebras.

We will then define the addition of a runner full of beads for the abacus display of a multipartition and investigate some combinatorial properties of this operation. We focus our attention on the q-decomposition numbers, i.e. the polynomials arising from the Fock space representation of the quantun group Uq(bsle) that coincide with the decomposition numbers for q = 1. Using an LLTtype algorithm for Ariki-Koike algebras, we relate q-decomposition numbers for different values of e for the class of e-multiregular multipartitions, by adding a full runner of beads to each component of the abacus displays for the labelling multipartitions.

Item Type: Thesis (Doctoral)
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Chris White
Date Deposited: 24 Oct 2023 08:42
Last Modified: 24 Oct 2023 08:42

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