Alternating multilinear forms (Q1895931)

Let \(E\) be a finite dimensional \(K\)-vector space. The authors study the action of the linear group \(GL (E)\) on the space \(\Lambda^p E\). As \(\Lambda^p E^* \cong (\Lambda^pE)^*\), there is no difference between alternating forms and \(p\)-vectors. Some authors have studied the case \(p = 3\) and \(\dim E \leq 9\), over an algebraically closed field in the general case. The present authors are still interested in the case \(n = 7\), they talk about invariants of arithmetical, algebraic and geometric meaning that one may assign to a \(p\)-vector. Then they apply this to the cases \(n = 6\), \(n = 7\) giving a complete analysis of the situation. Moreover they give a complete proof of Schouten's classification for \(n = 3\), \(n = 7\) and algebraically closed \(K\) and a precise description for the finite fields case.