Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ element of $v_i$. So $$ \Upsilon(\{\})=n $$ is the dimension of the vectors in $V$, $$ \Upsilon(\{v\})=\text{sum of elements of $v$}, $$ and $$ \Upsilon(\{v,w\})=v\cdot w. $$
Is there a standard name or notation for this product?