Bases for fixed points of unipotent elements acting on the tensor square and the spaces of alternating and symmetric 2-tensors
Let \(V\) be a finite-dimensional vector space over a field \(K\) and let \(g\) be a unipotent automorphism of \(V\). In the paper under review, the author investigates the action of \(g\) on \(V\otimes V\) and certain related modules. Let \(G\) be the cyclic group generated by \(g\) and let \(V=V_1\oplus\cdots\oplus V_r\) be a decomposition of \(V\) into indecomposable \(G\)-submodules. If \(\dim V_i=m_i\), then the elementary divisors of \(g\) acting on \(V\) are \((x-1)^{m_1},\dots,(x-1)^{m_r}\). Clearly, \(V\otimes V\) is the direct sum of the \(G\)-submodules \(V_i\otimes V_j\), and to describe the elementary divisors of \(g\) acting on \(V\otimes V\), it suffices to describe the elementary divisors of \(g\) acting on \(V_i\otimes V_j\). In general, the elementary divisors for the action of \(g\) on \(V_i\otimes V_j\) depend on the characteristic of \(K\) and no concise description of them is available or can be expected. The author's main result is that there is a uniform description of the subspace of fixed points of \(g\) in \(V_i\otimes V_j\), and he gives a field-independent basis for the subspace. If we assume that \(\dim V_j\leq\dim V_i\), the dimension of the subspace of fixed points is \(m_j=\dim V_j\). This tells us that \(V_i\otimes V_j\) is the direct sum of \(m_j\) indecomposable \(G\)-submodules, but their dimensions, and hence the elementary divisors for the action of \(g\), depend on the characteristic of \(K\). The author gives similar formulae for the action of \(g\) on the subspaces of alternating and symmetric tensors in \(V\otimes V\). He also notes that no analogous results can be obtained for the action of \(g\) on \(V\otimes V\otimes V\), since the dimension of the subspace of fixed points is now field-dependent.
scientific article; zbMATH DE number 1797479