Paper detail

Paths, tableaux, and q-characters of quantum affine algebras: the C_n case

For the quantum affine algebra $U_q(\hat{\mathfrak{g}})$ with $\mathfrak{g}$ of classical type, let $χ_{λ/μ,a}$ be the Jacobi-Trudi type determinant for the generating series of the (supposed) $q$-characters of the fundamental representations. We conjecture that $χ_{λ/μ,a}$ is the $q$-character of a certain finite dimensional representation of $U_q(\hat{\mathfrak{g}})$. We study the tableaux description of $χ_{λ/μ,a}$ using the path method due to Gessel-Viennot. It immediately reproduces the tableau rule by Bazhanov-Reshetikhin for $A_n$ and by Kuniba-Ohta-Suzuki for $B_n$. For $C_n$, we derive the explicit tableau rule for skew diagrams $λ/μ$ of three rows and of two columns, and give the implicit tableau rule in terms of paths for general $λ/μ$.