Tropical representations of Chinese monoids with and without involution

Recently, Izhakian and Merlet gave a faithful representation \(\widetilde{\rho }\) $\tilde{\rho }$ of the Chinese monoid \(Ch_{n}\) $C{h}_n$ of every finite rank n as a submonoid of the monoid \(UT_{2\cdot 3^{n-2}}(\mathbb {T})\) $U{T}_{2·3^{n-2}}(\mathbb{T})$ of upper triangular matrices over the tropical semiring \(\mathbb {T}\) $\mathbb{T}$ . We exhibit another faithful representation \(\widetilde{\phi }_n\) ${\tilde{\varphi }}_n$ of \(Ch_{n}\) $C{h}_n$ as a submonoid of the monoid \(UT_{n(n-1)}(\mathbb {T})\) $U{T}_{n(n-1)}(\mathbb{T})$ of upper triangular matrices over \(\mathbb {T}\) $\mathbb{T}$ . The dimension of \(\widetilde{\phi }_n\) ${\tilde{\varphi }}_n$ is smaller than that of \(\widetilde{\rho }\) $\tilde{\rho }$ when \(n\geqslant 4\) $n⩾4$ . Further, we give a faithful representation of the Chinese monoid \((Ch_n,~^\sharp )\) $(C{h}_n,_{}^{♯})$ under Schützenberger’s involution \(^\sharp \) $_{}^{♯}$ .