Let \(\alpha\in \Sigma^k(V^*)\) and \(\beta\in \Sigma^l(V^*)\) be symmetric tensors. The symmetric product that is defined by

\begin{equation*} \alpha\beta(v_1,\ldots, v_{k+l})=\frac{1}{(k+l)!}\sum_{\sigma\in S_k} \alpha(v_{\sigma(1)},\ldots, v_{\sigma(k)})\beta(v_{\sigma(k+1)}, \ldots, v_{\sigma(k+l)}) \end{equation*}

yields another symmetric \(k+l\)-tensor.

Remarks
  • A tensor product of symmetric tensors is not symmetric in general. This is the motivation for the symmetric product.
  • if \(\alpha\) and \(\beta\) are covectors, then \begin{equation*} \alpha\beta = \frac{1}{2}(\alpha\otimes \beta + \beta\otimes \alpha). \end{equation*} [1, Proposition 12.15]

See also Link to heading

References Link to heading

  1. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.