Schurifying quasi-hereditary algebras

TL;DR

This paper introduces a new method to generalize Schur algebras from quasi-hereditary algebras, expanding their applicability and understanding in representation theory, especially for superalgebras.

Contribution

It defines a 'schurification' process for quasi-hereditary algebras, creating new generalized Schur algebras that retain quasi-hereditary properties and relate to classical decomposition numbers.

Findings

Schurification preserves quasi-heredity for d ≤ n.

Decomposition numbers of generalized Schur algebras relate to original algebra and classical Schur algebra.

Construction involves non-trivial sub-lattices in the superalgebra case.

Abstract

We define and 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 $A$ to obtain a generalized Schur algebra $S^A(n,d)$ which we prove is again quasi-hereditary if $d\leq n$. We describe decomposition numbers of $S^A(n,d)$ in terms of those of $A$ and the classical Schur algebra $S(n,d)$. In fact, it is essential to work with quasi-hereditary superalgebras $A$, in which case the construction of the schurification involves a non-trivial full rank sub-lattice $T^A_\mathfrak{a}(n,d)\subseteq S^A(n,d)$.