Let $X_{1},...,X_{N}$ be $N$ matrices. I want to compute an antisymmetrized product of $X_{i}$'s in mathematica:
$X_{[a_{1}...a_{N}]} \equiv \tfrac{1}{N!}\sum_{\sigma}(-1)^{P}X_{\sigma(a_{1})}X_{\sigma(a_{2})}...X_{\sigma(a_{N})},$
where the sum is taken over permutations and the sign factor $(-1)^P$ is $+1$ for even permutations and $-1$ for odd ones.
Eventually, I want to calculate the above equation componentwise by substituting e.g. Dirac $\gamma$-matrices into $X_{i}$'s. How can I make a code for that? Thanks for your help.