Induced operators on symmetry classes of tensors

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.