Representations and identities of hypoplactic monoids with involution

Let $({\mathsf{h}\mathsf{y}\mathsf{p}\mathsf{o}}_n,{}^{\mathrm{♯}})$ be the hypoplactic monoid of finite rank $n$ with Schützenberger’s involution ${}^{\mathrm{♯}}$. In this paper, we exhibit a faithful representation of $({\mathsf{h}\mathsf{y}\mathsf{p}\mathsf{o}}_n,{}^{\mathrm{♯}})$ as an involution monoid of upper triangular matrices over any semiring from a large class including the tropical semiring under the skew transposition. We then give a transparent combinatorial characterization of the word identities satisfied by $({\mathsf{h}\mathsf{y}\mathsf{p}\mathsf{o}}_n,{}^{\mathrm{♯}})$. Further, we prove that $({\mathsf{h}\mathsf{y}\mathsf{p}\mathsf{o}}_n,{}^{\mathrm{♯}})$ is non-finitely based if and only if $n=2,3$ and give a polynomial time algorithm to check whether a given word identity holds in $({\mathsf{h}\mathsf{y}\mathsf{p}\mathsf{o}}_n,{}^{\mathrm{♯}})$.