Decomposable \(\lambda \)-critical tensors
A partition \(\lambda=(\lambda_1,\dots,\lambda_s)\) of \(m\in \mathbb{N}^*\) and a finite dimensional vector space \(V\) over \(\mathbb{C}\) are considered. An irreducible character of the symmetric group of degree \(m\) \(S_m\) is identified with the corresponding partition \(\lambda\). \(V_{\lambda}\) denotes the symmetry class of tensors associated with \(\lambda\) and \(V\). For \(j\in\{1,\dots,\lambda_1\}\) and \(z\in V_{\lambda}\), the concepts of \(j\)-reach and \(j\)-annihilator of \(z\) are defined and they are used to introduce the notion of \(\lambda\)-critical elements of \(V_{\lambda}\). Some important properties of the \(\lambda\)-critical elements of \(V_{\lambda}\) are emphasized and they are used to characterize the decomposable elements of \(V_{\lambda}\) that are \(\lambda\)-critical. A family of generalized Plücker polynomials is constructed so that the set of their common roots includes the set of the families of components of decomposable \(\lambda\)-critical elements of \(V_{\lambda}\). These results extend the topics of the paper \textit{J. A. Dias da Silva} and \textit{F. Rodrigues} [Linear Algebra Appl. 414, No.~1, 172--198 (2006; Zbl 1090.15023)] to the framework of \(\lambda\)-critical tensors.
scientific article; zbMATH DE number 5710791