Just simple question:
Can anyone provide a list of types of permutation matrices that commute (with the matrices of the same type)?
for one, I can think of rotation matrix… (Oh, wait. it isn't really permutation matrix..)
[Math] Permutation matrices that commute
linear algebramatricespermutation-matrices
Best Answer
Conjugation of one permutation by another leaves the cycle structure unchanged, but permutes the letters in the cycles according to the second permutation. So a permutation $\sigma$ that commutes with $\pi$ must map each cycle of $\pi$ to a cycle of $\pi$ (with the same length, of course). For example, the permutations of $6$ letters that commute with $(123)(456)$ will either map $(123)$ and $(456)$ to themselves or interchange them. They are determined by what they do to $1$ and $4$. Thus there are $6 \times 3 = 18$ possibilities, including the identity (maps $1 \to 1$ and $4 \to 4$) and $(162435)$ (maps $1 \to 6$ and $4 \to 3$).