Finite Gröbner-Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

The Plactic monoid has its origins in work of Schensted (1961) and Knuth (1970) concerned with certain combinatorial problems and operations on Young tableaux. It was later studied in depth by Lascoux and Schützenberger (1981) and has since become an important tool in several aspects of representation theory and algebraic combinatorics. Various aspects of the corresponding semigroup algebras, the Plactic algebras, have been investigated, for instance in Cedo & Okninski (2004), and Lascoux & Schützenberger (1990). In this talk I will discuss a result in which we show that every Plactic algebra of finite rank admits a finite Gröbner-Shirshov basis. The result is proved by using the combinatorial properties of Young tableaux to construct a finite complete rewriting system for the corresponding Plactic monoid. Also, answering a question of Efim Zelmanov, I will explain how this rewriting system and other techniques may be applied to show that Plactic monoids of finite rank are biautomatic. These results are joint work with A. J. Cain and A. Malheiro.