arXiv subjects
Publications and source records attributed to Alice Dell'Arciprete.
For the Iwahori-Hecke algebras of type $A$, James and Mathas proved a theorem which relates $v$-decomposition numbers for different values of $e$, by adding empty runners to the James' abacus display. This result is often referred to as the empty runner removal theorem. In this paper, we extend this theorem to the Ariki-Koike algebras, establishing a similar relationship for the $v$-decomposition numbers.
A categorical action of a Kac--Moody algebra $\mathfrak{g}$ is built on a category $\mathcal{C}$ decomposed according to the weights $P$ of $\mathfrak{g}$, as well as biadjoint endofunctors $\mathcal{E}_i$ and $\mathcal{F}_i$, abstracting $i$-induction and $i$-restriction, which act on the weight spaces of $\mathcal{C}$ in the same way that the Chevalley generators would act on a regular representation. Chuang and Rouquier initially developed these notions for $\mathfrak{sl}_2$-categorical actions, using them to prove Broué's abelian defect group conjecture for symmetric groups by establishing derived equivalences between blocks of the same defect. In the setting of general categorical actions Webster later showed that many of these derived equivalences are, in fact, $t$-exact, and that, as a result, such an action can be used to separate weight spaces of a categorical action into a finite number of Morita equivalence classes, where these equivalences also preserve decomposition numbers. The combinatorics of these powerful abstract results were concretely established in the case of Ariki--Koike algebras by the first author in arxiv:2301.05153v2, and in this short note we discuss how to translate between the two settings.
We show that the extended Khovanov algebra $K^m_n$ is an $(|n-m|-1)$-faithful cover of the Khovanov arc algebra $H^m_n$.
We provide an ${\rm Ext}$-quiver and relations presentation of the Khovanov arc algebras and prove a precise analogue of the Kleshchev--Martin conjecture in this setting.
We consider the representation theory of the Ariki-Koike algebra, a $q$-deformation of the group algebra of the complex reflection group $C_r \wr S_n$. We 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 quantum group $U_q(\widehat{\mathfrak{sl}}_e)$. Using Fayers' LLT-type 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.
We consider the representation theory of the Ariki-Koike algebra, a $q$-deformation of the group algebra of the complex reflection group $C_r \wr \mathfrak{S}_n$. We examine blocks of the Ariki-Koike 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 $n-δ_i(B)$ whose Specht modules belong to the block $B'$, obtained from $B$ applying a Scopes' equivalence. This bijection gives us an equivalence for the $v$-decomposition numbers of the Ariki-Koike algebras.