The Schur algebra \(S(B^ +)\) and projective resolutions of Weyl modules
Let \(k\) be an infinite field and let \(n\) and \(r\) be positive integers. Denote the natural action of \(G = \text{GL}_ n(k)\) on \(\otimes^ r k^ n\) by \(T_{n,r}\). If \(H\) is a subgroup of \(G\), then the Schur algebra \(S(H) = S_ k(n,r;H)\) for \(H\), \(n\), \(r\) and \(k\) is defined by \(S(H):= T_{n,r}(k[H])\). Let \(\Lambda = \{(\lambda_ 1,\dots,\lambda_ n) \in \mathbb{Z}^ n \mid \lambda_ j \geq 0\), \(\sum \lambda_ j = r\}\) and let \(\Lambda^ +\) be the subset where \(\lambda_ 1 \geq \lambda_ 2 \geq \cdots \lambda_ n\). Let \(B^ + \subset G\) be the Borel subgroup of all upper triangular matrices. For each \(\lambda\in \Lambda\) there is a one dimensional \(B^ +\)-module \(\kappa_ \lambda\). The Weyl module \(K_ \lambda\) is defined to be \(K_ \lambda := S(G) \otimes_{S(B^ +)} k_ \lambda\). Then \(K_ \lambda \neq 0\) if and only if \(\lambda \in \Lambda^ +\). By constructing the first two steps in a minimal projective resolution of \(k_ \lambda\) and then applying the functor \(S(G)\otimes_{S(B^ +)}(\cdot)\) to that resolution the first two steps in a projective resolution of \(K_ \lambda\) are constructed. For \(n = 2\) and \(\text{char }k > 0\) the first three steps of such a resolution are constructed. For \(n \leq 3\) and \(\text{char }k = 0\) a minimal projective resolution of \(K_ \lambda\) is determined.
scientific article; zbMATH DE number 554254