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

TL;DR

This paper investigates the properties of tilting modules over standardly stratified algebras, establishing conditions for double centralizer properties, and explores Schur-Weyl duality in symplectic and Brauer algebra contexts.

Contribution

It provides a criterion for tilting modules to induce double centralizer properties and proves the existence of a unique minimal tilting module, also establishing a Schur-Weyl duality involving symplectic Schur algebras.

Findings

Criteria for tilting modules to have double centralizer property.

Existence and uniqueness of minimal basic tilting modules.

Schur-Weyl duality between symplectic Schur algebra and Brauer algebra quotient.

Abstract

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 $1\leq f\leq [\frac{n}{2}]$.