What does the notation $\epsilon_{a..d} 1.1.1.1$ mean in the context of permutation matrices?

Question

I recently encountered the following paragraph in chapter 2 of The little Book of Permutation Matrices by Dennis Morris.

Since they are square matrices, permutation matrices always have a determinant... we realize that the determinant of a permutation matrix will always be of the form

$$det\left(\left[ \begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} \right] \right) = \epsilon_{a..d} 1.1.1.1 = \epsilon_{a..d} = ±1 $$

I'm confused what the notation $\epsilon_{a..d} 1.1.1.1$ means in this context, though. Can someone explain it to me?

Answer 1

I believe that in this case $\epsilon$ is viewed as the sign function. It is a homomorphism from $S_n \rightarrow \{-1,1\}$. You can also view it as equal to $(-1)^n$ where $n$ is the number of transpositions in your permutation.

For instance your permutation matrix is the same as $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 3 & 2 \end{pmatrix}$ which is comprised of two transpositions $(1\;2)$ and $(2\;4)$ so $\epsilon(\sigma) = (-1)^2 = 1$