Interpolating Schur Algebras

We introduce and study a one-parameter family of algebras that naturally generalize the Schur algebras.

We show the Schur algebra is canonically a quotient when the parameter is a nonnegative integer, characterize when they are semisimple, show they are based quasi-hereditary, and that their category of representations is a highest weight category that can be identified as a subcategory of parabolic category $\mathcal{O}$ for the general linear Lie algebra.