Path description for $q$-characters of fundamental modules in type $C$

In this paper, we investigate the behavior of monomials in the $q$-characters of the fundamental modules over a quantum affine algebra of untwisted type C. As a result, we give simple closed formulae for the $q$-characters of the fundamental modules in terms of sequences of vertices in ${\mathbb{R}}^2$, so-called paths, with an admissible condition. This may be viewed as a type C analog of the path description of $q$-characters in types A and B due to Mukhin–Young.