New approaches to plactic monoid via Gröbner-Shirshov bases

We give two explicit (quadratic) presentations of the plactic monoid in row and column generators correspondingly. Then we give direct independent proofs that these presentations are Gröbner-Shirshov bases of the plactic algebra in deg-lex orderings of generators. From Composition-Diamond lemma for associative algebras it follows that the set of Young tableaux is the Knuth normal form for plactic monoid ([30], see also Ch. 5 in [32]).