  Isomorphisms are constructed between generalized Schur algebras in different
degrees. The construction covers both the classical case (of general linear
groups over infinite fields of arbitrary characteristic) and the quantized case
(in type $A$, for any non-zero value of the quantum parameter $q$). The
construction does not depend on the characteristic of the underlying field or
the choice of $q \neq 0$. The proof combines a combinatorial construction with
comodule structures and Ringel duality. Applications range from equivalences of
categories to results on the structure and cohomology of Schur algebras to
identities of decomposition numbers and also of $p$-Kostka numbers, in both
cases reproving and generalizing row and column removal rules.
