Modules with Good Filtrations over Generalized Schur Algebras

Abstract

In this dissertation we examine generalized Schur algebras, as defined by Kleshchev and Muth. Given a quasi-hereditary superalgebra $A$, Kleshchev and Muth proved that for $n \geq d$, the generalized Schur algebra $T^A (n,d)$ is again quasi-hereditary.They described the bisuperalgebra struture on $T^A(n) := \bigoplus_d T^A(n,d)$. In particular, there is a coproduct which gives us a way to take a $T^A(n,d)$-module $V$ and $T^A(n,r)$-module $W$ and produce a $T^A(n,d+r)$-module $V \otimes W$. We will prove that if $V$ and $W$ each have standard (resp. costandard) filtrations, then so does $V \otimes W$. In the last chapter we will use this result to prove that in the case that $A$ is the extended zigzag algebra $\EZig$, the extended zigzag Schur algebra $T^\EZig(n,d)$ is Ringel self-dual for all $n \geq d$.