Let $e\geq 2$ and $0\leq i\leq e-1$.
For a multipartition $\lambda\vdash_\ell n$ of $n $ one can define the notion of an $i$-good box of the Young diagram of $\lambda$. But there are seemingly different definitions being used for when a node is $i$-good. Most of these differences are explained by the fact that some authors prefer $e$-regular partitions and Specht modules and others prefer $e$-restricted partitions and dual Specht modules, so that is fine.
But I do not understand why the definitions in
Ariki: Representations of Quantum Algebras and Combinatorics of Young Tableaux
Ariki&Mathas: On the decompmosition numbers of the Hecke algebra of G(m,1,n)
should yield the same definitions of $i$-good boxes, or at least the same definition up to transposition. In both cases the addable and removable nodes of residue $i$ along the rim are ordered in some fashion, then pairs of addables and removables are removed until this is no longer possible, and then one has to look at the "highest" left over removable box.
The problem I have is with the different orderings defined on the rim: E.g. let $\lambda=(\lambda^1, \lambda^2)$ be a $2$-multipartition.
- In Ariki's book, the rim starts at the top of the young diagram of $\lambda^2$, goes to its bottom, then to the top of $\lambda^1$ and to its bottom. Finally, we iteratively remove pairs of the shape (removable, addable).
- In Ariki&Mathas, we start at the bottom of $\lambda^2$, go to its top, then to the bottom of $\lambda^1$ and from there to its top. We remove pairs of the shape (addable, removable).
These definitions seem similar enough and if we swap the places of $\lambda^1$ and $\lambda^2$ in the definition in Ariki's book then they yield the exact same node, but this is not what transposition does.
So essentially my problem boils down to me not understanding why Ariki defines "being abov another node" the way he does, see Definition 10.7 in his book.