Plactic monoids are objects that model the representation theory of complex semisimple Lie algebras with the tensor product. These objects enjoy a rich combinatorial structure, with their elements parameterized by tableaux, and their product encoded by Schensted's insertion algorithm. Moreover the algebraic structure of representations can be encoded via crystals, a graph theoretic notion introduced by Kashiwara which descends to the plactic monoids. The work of Cain-Gray-Malheiro identifies column presentations of plactic monoids of classical types A, B, C, D, and G_2, that is presentations via generators and 'well-behaved' oriented relations. Their work opened up a rewriting theory approach to the study of plactic monoids. In this talk, we show that the column presentations interact well with the crystal structure of the plactic monoids, and thus their study via rewriting theory is reduced to words and relations of highest weight. We then introduce combinatorial tools in types A and C, called Yamanouchi trees, which parameterize the words of highest weight and allow for computations of the oriented relations. We relate this notion to GT-patterns and the Q-tableaux of the RSK correspondence. Finally, we show applications of Yamanouchi trees to the combinatorial R-matrix, as well as to the identification of the generating relations between relations i.e. syzygies for the plactic monoids of types A and C.
