Given four vectors $\vec x_0$, $\vec x_1$, $\vec x_2$, and $\vec x_3$, I would like to write the expression $$\det (\vec x_1, \vec x_2, \vec x_3) - \det (\vec x_0, \vec x_2, \vec x_3) + \det (\vec x_0, \vec x_1, \vec x_3)-\det (\vec x_0, \vec x_1, \vec x_2)$$ more compactly, using sigma notation. For example, $$\sum_{k=0}^3 (-1)^k \det (\vec x_0, \vec x_1, \vec x_2, \vec x_3){\rm (but\space k)}.$$
Is there a notation for what I have written as ${\rm (but\space k)}$?
On
An answer and a comment:
There is (to my knowledge) no universally understood notation which does what you want it to do, though there are a couple of things that you could try.
As JMoravitz suggests, one can use a "hat" for the omitted element. That is, write $$ \sum_{k=0}^{3} \det(x_0, \dotsc, \hat{x}_k, \dotsc, x_n). $$ This is the notation used by Hatcher in his text Algebraic Topology (see page 89). Hatcher is widely read, so I would expect that anyone who has spent any significant time thinking about algebraic topology has probably at least seen this notation.
A similar notation would be to write $$\sum_{k=0}^{3} \det(x_0, \dotsc, x_{k-1}, x_{k+1}, \dotsc, x_3). $$
When specifying the terms of a sequence, one often uses the notation $(x_k)_{k\in \mathscr{K}}$, where $\mathscr{K}$ is some index set. Thus one could potentially write $$ \sum_{k=0}^{3} \det\left( (x_j)_{j\in \{0,1,2,3\} \setminus \{k\}} \right). $$ This notation looks quite awkward to me, but it is (I think) entirely unambiguous.
Similar to the above, though significantly more compactly, $$ \sum_{k=0}^{3} \det\left( (x_j)_{j\ne k} \right). $$ This is a lot cleaner than (3), though might be a little ambiguous, as you would need to specify that the indices of $(x_j)$ run from $0$ to $3$—as written above, this is not obvious. (It seems that Especially Lime had the same idea while I was typing.)
Remember that the goal is clear communication. Find a notation that you like, define it clearly in your writing, and use it consistently. If you can't find a commonly used notation which does the job you want, you shouldn't worry about creating your own.
I would write $(\vec x_j)_{j\neq k}$ for the sequence of vectors with the $k$th omitted, and hence $\det((\vec x_j)_{j\neq k})$ for the determinant you want.