Seminars in Ariki-Koike Algebras

Test

Test

Kai Meng Tan: Cores and core blocks of Ariki-Koike algebras

15th October, 2024

This talk will consist of two parts. In the first part, we will see how certain results (such as the Nakayama 'Conjecture') for the symmetric groups and Iwahori-Hecke algebras of type A can be generalised to Ariki-Koike algebras using the map from the set of multipartitions to that of (single) partitions first defined by Uglov. In the second part, we look at Fayers's core blocks, and see how these blocks may be classified using the notation of moving vectors first introduced by Yanbo Li and Xiangyu Qi. If time allows, we will discuss Scopes equivalences between these blocks arising as a consequence of this classification

Ziqing Xiang: Quantum Wreath Products and Their Representations

19th February, 2024

We introduce a new notion called the quantum wreath product, which produces an algebra B ≀Q H(d) from a given associative algebra B, a positive integer d, and a choice Q = (R, S, ρ, σ) of parameters. Important examples include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang’s wreath Hecke algebras, Rosso-Savage’s (affine) Frobenius Hecke algebras, Kleshchev-Muth’s affine zigzag algebras, and the Hu algebra that quantizes the wreath product Σm ≀ Σ2 between symmetric groups. We will discuss the bases of quantum wreath product algebras, and some of their representations.

Sinéad Lyle: Rouquier blocks for Ariki-Koike algebras

21st July, 2022

The Rouquier blocks, also known as the RoCK blocks, are important blocks of the symmetric groups algebras and the Hecke algebras of type A, with the partitions labelling the Specht modules that belong to these blocks having a particular abacus configuration. We generalize the definition of Rouquier blocks to the Ariki-Koike algebras, where the Specht modules are indexed by multipartitions, and explore the properties of these blocks.

Alice Dell’Arciprete: Scopes equivalence for blocks of Ariki-Koike algebras

19th July, 2022

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr ≀ Sn. The representations of this algebra are naturally indexed by multipartitions of n. We examine blocks of the Ariki-Koike algebra, in an attempt to generalise the combinatorial representation theory of the Iwahori-Hecke algebra. 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.

CJ Lai: Wreath products, Schur dualities, and quasi-hereditary covers

28th February, 2022

The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover by the category 𝒪 of a certain rational Cherednik algebra due to Ginzburg-Guay-Opdam-Rouquier. It was later shown in type A that this category 𝒪 can be realized concretely as the module category of Dipper-James's q-Schur algebra, but this realization problem remains open beyond types A and B. An essential step for type D, i.e., the complex reflection group G(2,2,n), is to study Hu's Hecke subalgebra, which deforms a wreath product that is not a Coxeter group. In this talk, I'll introduce a new theory allowing us to take the wreath product of an algebra by a Hecke algebra. Before our work, wreath products related to Hecke algebras were worked out at the degenerate level by Wan-Wang. Our wreath product produces the Ariki-Koike algebras as special cases as well as new 'Hecke algebras' of wreath products between symmetric groups. These are the first steps towards answering the realization problem for complex reflection groups.

Nicolas Jacon: Cores of Ariki-Koike algebras

8th December, 2020

We study a natural generalization of the notion of cores for l-partitions: the (e,s)-cores. We relate this notion with the notion of weight as defined by Fayers and use it to describe the blocks of Ariki-Koike algebras.