Scopes equivalence for blocks of Ariki-Koike algebras

Abstract: We consider representations of the Ariki-Koike algebra, a $q$-deformation of the group algebra of the complex reflection group $C_r\wr\mathfrak{S}_n$. 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 $n−\delta_i(B)$ whose Specht modules belong to the block $B'$, obtained from $B$ applying a Scopes' equivalence. This gives us an equality of decomposition numbers for the corresponding Ariki-Koike algebras.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.