Metrics on Signed Permutations with the Same Peak Set

Let $S_n^B$ be the Coxeter group of type B. We denote the set of indices where $\sigma \in S_n^B$ has a peak as $Peak(\sigma )$ and let $P^B(S;n)=\{\sigma \in S_n^B\mathrm{\mid }Peak(\sigma )=S\}$. In \cite{metrics}, Diaz-Lopez, Haymaker, Keough, Park and White considered metrics for unsigned permutations with the same peak set. In this paper, we generalize their result by studying Hamming, $l_{\mathrm{\infty }}$, and the word metrics on $P^B(S;n)$ for all $S$. We also determine the minimum and maximum possible values that these metrics can achieve in these subsets of $S_n^B$.