arXiv subjects

Dinushi Munasinghe

Publications and source records attributed to Dinushi Munasinghe.

3 recordsLinked to original sources

arXiv2026

A Note on Scopes Equivalences for Ariki--Koike Algebras as Categorical Actions

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 Brou\'e'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.

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arXiv2025

Steadied Khovanov-Lauda-Rouquier algebras and local models for blocks

It's known that many different blocks of $\mathbb{F}_pS_n$ for different values of $n$ are equivalent as categories, though the corresponding block algebras are almost never isomorphic. Thus, it is a challenging problem to give one particularly nice representative of this Morita equivalence class of algebras. This has been accomplished for the case of RoCK blocks through work of Chuang--Kessar, Turner, and Evseev--Kleshchev. In this paper, we give a new perspective on this problem, applying not just to RoCK blocks of $S_n$, but also to all blocks of Ariki--Koike algebras. We do this by considering steadied quotients of KLRW algebras: these algebras are a natural generalization of cyclotomic quotients, already related to $S_n$ and Ariki--Koike algebras in work of Brundan--Kleshchev. These algebras are defined by ``tilting'' the cyclotomic relations so that we kill the two-sided ideal defined by certain configurations on the left and right sides of our diagrams. We show a Morita equivalence between these algebras and blocks of Ariki-Koike algebras generalizing the work discussed above.

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arXiv2023

On the representation theory of Schur algebras in type $B$

We study the representation theory of the type B Schur algebra $\mathcal{L}^n(m)$ with unequal parameters introduced in work of Lai, Nakano and Xiang. For generic values of $q,Q$, this algebra is semi-simple and Morita equivalent to the Hecke algebra, but for special values, its category of modules is more complicated. We study this representation theory by comparison with the cyclotomic $q$-Schur algebra of Dipper, James and Mathas, and use this to construct a cellular algebra structure on $\mathcal{L}^n(m)$. This allows us to index the simple $\mathcal{L}^n(m)$-modules as a subset of the set of bipartitions of $n$. For $m$ large, this will be all bipartitions of $n$ if and only if $\mathcal{L}^n(m)$ is quasi-hereditary, in which case, $\mathcal{L}^n(m)$ is Morita equivalent to the cyclotomic $q$-Schur algebra. We prove a modified version of a conjecture of Lai, Nakano and Xiang giving the values of $(q,Q)$ where this holds: if $m$ is large and odd, $Q\neq -q^k$ for all $k$ satisfying $\frac{4-n}{2}\leq k<n$; if $m$ is large and even, $Q\neq -q^k$ for all $k$ satisfying $-n<k<n.$ We also prove two strengthenings of this result: an indexing of the simple modules when $q$ is not a root of unity, and a characterization of the quasi-hereditary blocks of $\mathcal{L}^n(m)$.

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