Schurifying quasi‐hereditary algebras

Abstract

We study new classes of quasi-hereditary and cellular algebras which generalize Turner's double algebras. Turner's algebras provide a local description of blocks of symmetric groups up to derived equivalence. Our general construction allows one to “schurify” any quasi-hereditary algebra A $A$ to obtain a generalized Schur algebra S A ( n , d ) $S^A(n,d)$ which we prove is again quasi-hereditary if d ⩽ n $d\leqslant n$ . We describe decomposition numbers of S A ( n , d ) $S^A(n,d)$ in terms of those of A $A$ and the classical Schur algebra S ( n , d ) $S(n,d)$ . In fact, it is essential to work with quasi-hereditary superalgebras A $A$ , in which case the construction of the schurification involves a non-trivial full rank sub-lattice T a A ( n , d ) ⊆ S A ( n , d ) $T^A_\mathfrak {a}(n,d)\subseteq S^A(n,d)$ .