The plactic monoid is a fundamental algebraic object which captures a natural monoid structure carried by the set of combinatorial objects of semistandard Young tableaux. Other monoids arise in a similar way by considering different combinatorial objects: the hypoplactic monoid (the monoid of quasiribbon tableaux, connected with the theory of quasisymmetric functions), the sylvester monoid (binary search trees), and the Baxter monoid (pairs of twin binary search trees, connected with the theory of Baxter permutations). In this talk we discuss whether the plactic monoid and its placticlike variants satisfy nontrivial identities. (An identity is a formal equality u=v, where u and v are words over some alphabet of variables, and is non trivial if u and v are not identical as words. It is satisfied by a monoid if every substitution of elements for variables yields equality in the monoid.) We present new results showing that the plactic monoid does not satisfy a nontrivial identity, whereas the above mentioned related monoids do satisfy nontrivial identities. The results were obtained in collaboration with A. CAIN, G. KLEIN, L. KUBAT, and J. OKNINSKI. This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações) and the project PTDC/MHCFIL/2583/2014.
