New conditions for equality of decomposable symmetrized tensors (Q1923203)

Let \(V\) be a finite-dimensional vector space over the field \({\mathcal F}\). Let \(\otimes^{m_V}\) denote the \(m\)th tensor power of \(V\). For \(\sigma\in S_m\), \(P(\sigma)\) is the unique linear operator on \(\otimes^{m_V}\) such that \(P(\sigma) (x_1\otimes \cdots \otimes x_m)= x_{\sigma^{-1} (1)} \otimes \cdots\otimes x_{\sigma^{-1} (m)}\) for every \(x_1, \dots,x_m\) in \(V\). Let \(c:S_m\to {\mathcal F}\) be an arbitrary function from \(S_m\) into \({\mathcal F}\), and \(V_c\) be \((\sum_{\sigma \in S_m} c(\sigma) P(\sigma)) (\otimes^mV)\), which is called the symmetry class associated with \(c\). Then the elements of \(V_c\) of the forms \(x_1* \cdots* x_m= (\sum_{\sigma \in S_m} c(\sigma) P(\sigma)) (x_1\otimes \cdots \otimes x_m)\) are called decomposable symmetrized tensors. In \S 2 the author prepares some combinatorial background for a notion of partition, which is a finite nonincreasing sequence of nonnegative integers. In \S 3, he states conditions for equality of decomposable symmetrized tensors in arbitrary symmetry classes. In \S 4, based on these conditions, he derives other results for decomposable symmetrized tensors belonging to the symmetry classes of tensors associated with special characters.