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, $Q\mathrm{\ne }-q^k$ for all $k$ satisfying $\frac{4-n}{2}\le k<n$; if $m$ is large and even, $Q\mathrm{\ne }-q^k$ for all $k$ satisfying $-n<k<n\mathrm{.}$ 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 ${\mathcal{L}}^n(m)$.