On the representation theory of Schur algebras in type B

We study the representation theory of the type B Schur algebra $L^n\left(m\right)$ with unequal parameters introduced by Lai and Luo. For generic values of $(Q,q)$, this algebra is semi-simple and Morita equivalent to the type B 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 $L^n\left(m\right)$.

This allows us to index the simple $L^n\left(m\right)$-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 $L^n\left(m\right)$ is quasi-hereditary. In this case, the algebra $L^n\left(m\right)$ 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, $Q\ne -q^k$ for all k satisfying $\frac{4-n}{2}\le k<n$; if m is large and even, $Q\ne -q^k$ 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 $L^n\left(m\right)$.