Semisimplicity of exterior powers of semisimple representations of groups (Q1972416)

Let \(G\) be a group and \(K\) a field of characteristic \(p>0\). Let \(V=(V_1,\dots,V_s)\) be a sequence of semisimple representations of \(G\), and let \(m_1,\dots,m_s\) be integers satisfying \(1\leq m_i\leq\dim_KV_i=n_i\) for each \(i\). The author proves that if \(\sum_Im_i(n_i-m_i)<p\), then \(\bigvee^{m_1}V_1\otimes_K\cdots\otimes_K\bigvee^{m_s}V_s\) is semisimple. The proof is based on the reduction to the case when \(G\) is a simply connected and quasisimple linear algebraic group. This result gives an affirmative answer to a problem of J.-P.~Serre.