Is there some established short notation for the symmetric outer product of two vectors? By this I mean a matrix $M$ computed from two vectors $v$ and $w$ using $M=u\otimes v+v\otimes u$ or at the component level written as $M_{ij}=u_iv_j+u_jv_i$. Some people might prefer to also have a factor of $\frac12$ in front of this, I don't care as long as it's consistent.
I could just introduce a function name for this, but in my context symbols would be better as they would allow the names of the component vectors to stand out better. Some infix symbol, or some special kind of parentheses or some such would therefore be preferable.
There is notation for the symmetric product of tensors: $\alpha \odot \beta= \tfrac12(\alpha\otimes\beta + \beta\otimes\alpha)$ sometimes written simply as juxtaposition $\alpha\beta$.