Nonzero decomposable symmetrized tensors
type: Item
instance of: scholarly article
title: Nonzero decomposable symmetrized tensors
zbMATH Open document ID: 0358.15028
DOI: 10.1016/0024-3795(77)90067-2
author: Russell Merris
published in: Linear Algebra and its Applications
publication date: 1977-01-01T00:00:00Z
review text: 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))\…
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 3557969
MaRDI profile type: Publication
full work available at URL: 0024-3795(77)90067-2
OpenAlex ID: W2074273700
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description: scientific article; zbMATH DE number 3557969