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Induced operators on symmetry classes of tensors

Author(s): Chi-Kwong Li; Alexandru Zaharia
Journal: Trans. Amer. Math. Soc. 354 (2002), 807-836.
MSC (2000): Primary 15A69, 15A60, 15A42, 15A45, 15A04, 47B49
Posted: September 19, 2001
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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{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 $H$ 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 $V$ 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{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.

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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
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

DOI: 10.1090/S0002-9947-01-02785-4
PII: S 0002-9947(01)02785-4
Keywords: Symmetry class of tensors, linear operator, induced operator
Received by editor(s): October 6, 1999
Received by editor(s) in revised form: September 11, 2000
Posted: September 19, 2001
Copyright of article: Copyright 2001, American Mathematical Society

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