2020/06/05 by Meha, Uran
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2006.03456
In this article we introduce the ℕ-decorated plactic monoid of type C, denoted Pl^ℕ(Cn), via a finite convergent presentation ACol, with generating set ACol(Cn) consisting of admissible columns, and an element ε. By Squier's coherent completion theorem, this presentation is extended into a coherent presentation by identifying a family of generating confluences, i.e. generating 3-cells. Here the generating 3-cells are critical branchings on words of length 3. We adapt the notions of crystal structure to ACol(Cn)^∗ , and show that the shape of 3-cells is preserved by the action of Kashiwara operators. Thus we reduce the study of the coherent presentation to only describing the generating 3-cells whose source is a word of highest weight. We then introduce combinatorial objects called C-trees which parameterize the words of highest weight in ACol(Cn)^∗. The C-trees allow for simplifying calculations with the insertion algorithm in type C, as introduced in by Lecouvey, and we prove that the generating 3-cells in ACol are of shape at most (4,3). As a consequence, we show that the column presentation of Pl(Cn), as introduced by Hage, has generating 3-cells of shape at most (4,3). This contrasts the situation in type A, where the 3-cells in the column presentation of Pl(An) are of shape at most (3,3).