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

Abstract

For the quantum affine algebra $U_q(\hat{\mathfrak{g}})$ with $\mathfrak{g}$ of classical type, let $\chi_{\lambda/\mu,a}$ be the Jacobi-Trudi type determinant for the generating series of the (supposed) $q$-characters of the fundamental representations. We conjecture that $\chi_{\lambda/\mu,a}$ is the $q$-character of a certain finite dimensional representation of $U_q(\hat{\mathfrak{g}})$. We study the tableaux description of $\chi_{\lambda/\mu,a}$ 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 $\lambda/\mu$ of three rows and of two columns, and give the implicit tableau rule in terms of paths for general $\lambda/\mu$.