Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let ${\mathrm{\Lambda }}^{+}$ be an indexing set of all simple modules in $A\text{lmod}$. We show that if there is an integer $r\in \mathbb{N}$ such that for any $\lambda \in {\mathrm{\Lambda }}^{+}$, there is an embedding $\Delta(\lambda)\hookrightarrow T^{\oplus r}$aswellasanepimorphism$T^{\oplus r}\twoheadrightarrow\overline{\nabla}(\lambda)$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 $A={\text{End}}_{{\text{End}}_A(T)}(T)$. We also establish a Schur-Weyl duality between the symplectic Schur algebra $S^{sy}(m,n)$ and ${\text{bb}}_n\mathrm{/}{\mathfrak{B}}_n^{(f)}$ on $V^{\otimes n}\mathrm{/}{V}^{\otimes n}{\mathfrak{B}}_n^{(f)}$ when $\cha K>\min{n-f+m,n}$,where$V$isa$2m$-dimensionalsymplecticspaceover$K$,$\mathfrak{B}{n}^{(f)}$isthetwo-sidedidealoftheBraueralgebra$\bb{n}(-2m)$generatedby$e_1e_3\cdots e_{2f-1}$with$1\leq f\leq [\frac{n}{2}]$.