Paper Abstract

Title

On the Weyl filtration dimension of induced modules for a linear algebraic group

Abstract

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p$ whose corresponding root system is irreducible. In this paper we calculate the Weyl filtration dimension of the induced $G$-modules, $\nabla(\lambda)$ and the simple $G$-modules $L(\lambda)$, for $\lambda$ a regular weight. We use this to calculate some $\Ext$ groups of the form $\Ext^*\bigl(\nabla(\lambda),\Delta(\mu)\bigr)$, $\Ext^*\bigl(L(\lambda),L(\mu)\bigr)$, and $\Ext^*\bigl(\nabla(\lambda), \nabla(\mu)\bigr)$, where $\lambda , \mu$ are regular and $\Delta(\mu)$ is the Weyl module of highest weight $\mu$. We then deduce the projective dimensions and injective dimensions for $L(\lambda)$, $\nabla(\lambda)$ and $\Delta(\lambda)$ for $\lambda$ a regular weight in associated generalised Schur algebras. We also deduce the global dimension of the Schur algebras for $\GL_n$, $S(n,r)$, when $p>n$ and for $S(mp,p)$ with $m$ an integer.