论文标题
倾斜模块，主导尺寸和Brauer-Schur-Weyl二元性
Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality
论文作者
论文摘要
让$ a $是在$ k $上的标准分层代数，而$ t $ a $ t $ a $ a $ a $ a $。令$λ^+$为$ a \ lmod $中所有简单模块的索引集。我们表明，如果有一个整数$ r \ in \ n $使得对于任何$λ\inλ^+$ $ a $ - 模型，然后$ t $是忠实的$ a $ module，$ a $具有相对于$ t $的双中心属性。作为应用程序，我们证明，如果$ a $是准标准的，具有简单的二元性和$ t $ a给定的忠实倾斜$ a $ module，则$ a $具有相对于$ t $的双重centralizer属性。这提供了一个简单而有用的标准，可以在代数谎言理论中的许多情况下应用。我们通过证明存在唯一的最小基本倾斜模块$ t $ a $ a $ a $ a = \ end _ = \ end _ {\ end_a（t）}（t）$来回答Mazorchuk和Stroppel的问题。我们还建立了Sympletic Schur Algebra $ s^{sy}（m，n）$和$ \ bb_ {n}/\ mathfrak {b} _ {n} _ {n}^{（f）on $ v^{ n} \ Mathfrak {b} _ {n}^{（f）} $时，$ \ cha k> \ min \ {n-f+m，n \} $，其中$ v $是$ 2M $ $ -Dimemensional-demensional-demensional-demensional-dementimal sympletic Space，$ k $ k $，$ k $，$ \ \ mathfrak {b} $ {b} $ { Brauer代数$ \ bb_ {n}（-2m）$由$ e_1e_3 \ cdots E_ {2f-1} $带有$ 1 \ leq f \ leq f \ leq [\ frac {n} {2} {2}] $。
Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $Λ^+$ 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 $λ\inΛ^+$, there is an embedding $Δ(λ)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(λ)$ 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 $1\leq f\leq [\frac{n}{2}]$.
