Min-plus Bruhat decomposition

We study min-plus Bruhat cells $B_{trop}^{(n)} \odot p\odot B_{trop}^{(n)}$, where $B_{trop}^{(n)}$ denotes the set of all upper-triangular matrices over \RR \cup \{\infty\}, and $p$ is a permutation. It is proved that $\dim (B_{trop}^{(n)} \odot p \odot B_{trop}^{(n)})= \frac{n(n+1)}{2} + l(p)$, where $l(p)$ is the length of $p$ in Coxeter generators. Also we provide an explicit description of the cell B_{trop}^{(n)}(\RR) \odot p_0\odot B_{trop}^{(n)}(\RR) (up to a subset of dimension less than $n^2$) for the highest in Bruhat order permutation $p_0$. Finally, it is shown that for a tropically non-singular matrix $A$ for at most one permutation $p_A$ it can hold $A\in B_{trop}^{(n)}\odot p_A\odot B_{trop}^{(n)}$.