Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $\Lambda^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $\lambda\in\Lambda^+$, there is an embedding $\Delta(\lambda)\hookrightarrow T^{ \oplus r}$ as well as an epimorphism $T^{ \oplus r}\twoheadrightarrow\overline{ \nabla}(\lambda)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful tilting $A$-module, then $A$ has the double centralizer property with respect to $T$. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module $T$ over $A$ for which $A=\End_{ \End_A(T)}(T)$. We also establish a Schur-Weyl duality between the symplectic Schur algebra $S^{ sy}(m,n)$ and $\bb_{ n}/\mathfrak{ B}_{ n}^{ (f)}$ on $V^{ \otimes n}/V^{ \otimes n}\mathfrak{ B}_{ n}^{ (f)}$ when $\cha K>\min\{ n-f+m,n\}$, where $V$ is a $2m$-dimensional symplectic space over $K$, $\mathfrak{ B}_{ n}^{ (f)}$ is the two-sided ideal of the Brauer algebra $\bb_{ n}(-2m)$ generated by $e_1e_3\cdots e_{ 2f-1}$ with $1\leq f\leq [\frac{ n}{ 2}]$.
Categories: math.RT