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Induced operators on symmetry classes of tensors

Authors:	Chi-Kwong Li Alexandru Zaharia

Institution:	Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795 ; Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3 and Institute of Mathematics of The Romanian Academy, 70700 Bucharest, Romania

Abstract:	Let be an -dimensional Hilbert space. Suppose is a subgroup of the symmetric group of degree , and $\chi: H \rightarrow \mathbb C$ is a character of degree 1 on . Consider the symmetrizer on the tensor space $\bigotimes^m V$ $\begin{displaymath}S(v_1\otimes \cdots \otimes v_m) = {1\over \vert H\vert}\sum... ... v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(m)} \end{displaymath}$ defined by and $\chi$ . The vector space $\begin{displaymath}V_\chi^m(H) = S(\bigotimes^m V) \end{displaymath}$ is a subspace of $\bigotimes^m V$ , called the symmetry class of tensors over associated with and $\chi$ . The elements in $V_\chi^m(H)$ of the form $S(v_1\otimes \cdots \otimes v_m)$ are called decomposable tensors and are denoted by $v_1\cdots v_m$ . For any linear operator acting on , there is a (unique) induced operator acting on $V_\chi^m(H)$ satisfying $\begin{displaymath}K(T) v_1* \dots v_m = Tv_1 \cdots * Tv_m. \end{displaymath}$ In this paper, several basic problems on induced operators are studied.

Keywords:	Symmetry class of tensors linear operator induced operator

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