Tropical representations of Chinese monoids with and without involution

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