Induced operators on symmetry classes of tensors
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Abstract:
Let $V$ be an $n$-dimensional Hilbert space. Suppose $H$ is a subgroup of the symmetric group of degree $m$, and $\chi : H \rightarrow \mathbb C$ is a character of degree 1 on $H$. Consider the symmetrizer on the tensor space $\bigotimes ^m V$ \begin{equation*} S(v_1\otimes \cdots \otimes v_m) = {1\over |H|}\sum _{\sigma \in H} \chi (\sigma ) v_{\sigma ^{-1}(1)} \otimes \cdots \otimes v_{\sigma ^{-1}(m)} \end{equation*} defined by $H$ and $\chi$. The vector space \begin{equation*} V_\chi ^m(H) = S(\bigotimes ^m V) \end{equation*} is a subspace of $\bigotimes ^m V$, called the symmetry class of tensors over $V$ associated with $H$ 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 $T$ acting on $V$, there is a (unique) induced operator $K(T)$ acting on $V_\chi ^m(H)$ satisfying \begin{equation*} K(T) v_1* \dots *v_m = Tv_1* \cdots * Tv_m. \end{equation*} In this paper, several basic problems on induced operators are studied.References
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Additional Information
- Chi-Kwong Li
- Affiliation: Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795
- MR Author ID: 214513
- Email: ckli@math.wm.edu
- Alexandru Zaharia
- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3 and Institute of Mathematics of The Romanian Academy, 70700 Bucharest, Romania
- Email: zaharia@math.toronto.edu
- Received by editor(s): October 6, 1999
- Received by editor(s) in revised form: September 11, 2000
- Published electronically: September 19, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 807-836
- MSC (2000): Primary 15A69, 15A60, 15A42, 15A45, 15A04, 47B49
- DOI: https://doi.org/10.1090/S0002-9947-01-02785-4
- MathSciNet review: 1862569