Conditions for a symmetrized decomposable tensor to be zero
Let \(X_ 1*...*X_ m\) be a decomposable symmetrized tensor corresponding to the symmetric group \(S_ m\) and an irreducible character \(\lambda\). Let \(\lambda =(\lambda_ 1,...,\lambda_ t)\) be a partition of m, and \(\rho\) an element of a subgroup G of \(S_ m\). We denote \(D_{\lambda,\rho}\) the corresponding Young diagram. If \(\lambda =(\lambda_ 1,...,\lambda_ t)\) and \(\gamma =(\gamma_ 1,...,\gamma_ s)\) are partitions of m, we say that \(\lambda\) majorizes \(\gamma\) and write \(\lambda\) \(\succ \gamma\) if \(t\leq s\), \(\sum^{\ell}_{j=1}\lambda_ j\geq \sum^{\ell}_{j=1}\gamma_ j,\) \(\ell =1,...,s\), where we assume that \(\lambda_{t+1}=...=\lambda_ s=0\). The partition \(\lambda '=(\lambda_ 1'=t,...,\lambda '_{\lambda_ 1})\), where \(\lambda_ j'\) is the number of elements in the jth column of \(D_{\lambda,id}\), is called the partition conjugate to \(\lambda\). The following is the main result: Theorem. Let \(\lambda\) be an irreducible character of \(S_ m\). Then \(X_ 1*...*X_ m\) is different from zero if and only if there is a partition w associate with the vectors \(X_ 1,...,X_ m\) such that \(w\succ \lambda '\).
scientific article; zbMATH DE number 4063288