Representations and identities of hypoplactic monoids with involution

Let (hypon,#) be the hypoplactic monoid of finite rank n with Schutzenberger's involution #. In this paper, we exhibit a faithful representation of (hypon,#) 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 (hypon,#). Further, we prove that (hypon,#) 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 (hypon,#).Communicated by Scott Chapman