Representations and identities of hypoplactic monoids with involution

Abstract

Let $({\mathsf{hypo}}_n,{}^{♯})$ be the hypoplactic monoid of finite rank $n$ with Schützenberger's involution ${}^{♯}$. In this paper, we exhibit a faithful representation of $({\mathsf{hypo}}_n,{}^{♯})$ 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 $({\mathsf{hypo}}_n,{}^{♯})$. Further, we prove that $({\mathsf{hypo}}_n,{}^{♯})$ 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 $({\mathsf{hypo}}_n,{}^{♯})$.