Paths and Tableaux Descriptions of Jacobi–Trudi Determinant Associated with Quantum Affine Algebra of Type $C_n$

We study the Jacobi–Trudi-type determinant which is conjectured to be the $q$-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type $C_n$. Like the $D_n$ case studied by the authors recently, applying the Gessel–Viennot path method with an additional involution and a deformation of paths, we obtain an expression by apositive sum over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.