Nonzero decomposable symmetrized tensors
Let \(V\) be a complex vector space of dimension \(n\). Let \({\overset {m}\otimes} V\) be the \(m-th\) tensor power of \(V\) and write \(v_1\,\otimes\dots\otimes v_m\) for the decomposable (or pure) tensor product of the indicated vectors. For each \(\sigma\in S_m\), the full symmetric permutation group, let \(P(\sigma^{-1})\) be the (unique) linear operator on \({\overset {m}\otimes} V\) having the effect \(P(\sigma^{-1})v_1\otimes\dots\otimes v_m=v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(m)}\). If \(\lambda\) is an irreducible character of \(S_m\), define \(T(\lambda)=\frac{\lambda(id)}{m!}\sum\limits_{\sigma\in S_m}\lambda(\sigma)P(\sigma)\). Then \(T(\lambda)\) is a projection onto the symmetry class of tensors \(V_\lambda\). Examples: \(V_\varepsilon=\wedge^mV;V_1=\) the \(m-th\) induced power space of completely symmetric tensors. Let \(\Gamma\) denote the set of functions from \(\{1,\dots,m\}\) into \(\{1,\dots,n\}\). If \(\{e_1,\dots,e_n\}\) is a basis of \(V\), then \(\{e_{\gamma}^\otimes=e_{\gamma(1)}\otimes\dots\otimes e_{\gamma(m)}:\gamma\in\Gamma\}\) is a basis of \({\overset {m}\otimes} V\). It follows that \(\{e_{\gamma}^*=T(\lambda)e_{\gamma}^\otimes:\gamma\in\Gamma\}\) spans \(V_\lambda\). The first problem one encounters in determining a basis for \(V_\lambda\) is deciding when \(e_{\gamma}^*=0\). For each \(\gamma\in\Gamma\) let \(S_\gamma=\{\sigma\in S_m:\gamma\sigma=\gamma\}\), i.e., \(S_\gamma\) is the stabilizer subgroup of \(S_m\) corresponding to \(\gamma\). It is well known that \(e_{\gamma}^*\neq 0\) if and only if the trivial character is a component of the restriction of \(\lambda\) to \(S_\gamma\). In the classical examples, a \`\` global'' criterion is known: If \(\lambda=\varepsilon,e_{\gamma}^*\neq 0\) if and only if \(\gamma\) is one-to-one. If \(\lambda=1,\;e_{\gamma}^*\neq 0\) for all \(\gamma\in\Gamma\). The present paper describes a global criterion for all \(\lambda\): For a given \(\gamma\in\Gamma\), let \(o(\gamma^{-1}(i))\) be the number of elements in \(\{j:1\leq j\leq m\) and \(\gamma(j)=i\}\). We call this number the multiplicity of \(i\) in \(\gamma\). Of these multiplicities, let \(m_1\) be the largest, \(m_2\) the next largest\(,\dots,\) and \(m_r\) the smallest which is still positive. Then \(m_1\geq m_2\geq\dots\geq m_r\) and \(m_1+\dots+ m_r=m\), i.e., \(M=(m_1,\dots,m_r)\) is a descending partition of \(m\). Let \(\lambda=(\lambda_1,\dots,\lambda_k)\) be the descending partition to which \(\lambda\) corresponds. Theorem: \(e_{\gamma}^*\neq 0\) if and only if \(\sum\limits_{t=1}^s\lambda_t\geq\sum\limits_{t=1}^s m_t,1\leq s\leq k\). Examples: \(\varepsilon =(1,\dots,1)\) majorizes \(M\) if and only if \(M=(1,\dots,1)\), if and only if \(\gamma\) is one to one; \(1=(m)\) majorizes everything. Although it escaped his attention at the time, the author's result gives an affirmative answer to a conjecture of \textit{E. Snapper} [J. Algebra 19, 520-535 (1971; Zbl 0226.20008)]. A more recent paper by the author [Linear multilinear Algebra 5, 79-86 (1977)] extends these results to characters of subgroups of \(S_m\) and to the more general situation \(v_1 *\dots *v_m\neq 0\). There are applications of this later work to generalized matrix functions.
scientific article; zbMATH DE number 3557969