Extensions of modules for \(\text{SL}(2,K)\) (Q1858173)

This paper extends our knowledge of extensions of modules for the algebraic group \(G=\text{SL}(2,K)\), where \(K\) is an algebraically closed field of characteristic \(p>0\). Let \(\nabla(n)\) be the induced module of highest weight \(n\), and \(\Delta(n)\) its dual. If \(M\) is a \(G\)-module, \(M^{F^n}\) denotes its \(n\)-th Frobenius twist. The first result finds all \(\text{Ext}^i_{G_1}(\Delta(s),\nabla(t))\) as \(G\)-modules, where \(G_1\) denotes the first Frobenius kernel. Next \(\text{Ext}^1_G(\nabla(r)^{F^n}\otimes\Delta(s),\nabla(k)^{F^n}\otimes\Delta(t))\) is computed for \(0\leq s,t\leq p^n-1\), \(0\leq k,r\). The answer is at most one dimensional. Then it is determined which of the corresponding classes are explained by the composition series of some \(\nabla(m)\). As a byproduct one gets an explanation of symmetries in the decomposition matrix for the induced modules that had been observed in the thesis of A.~Henke.