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 $\lambda /\mu$.

38 pages; final version to appear in J.Phys

References in corpus (3)

• Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
• Difference L operators related to q-characters
• Finite-Dimensional Crystals B^{2,s} for Quantum Affine Algebras of type D_{n}^{(1)}

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• Polynomial Relations for q-Characters via the ODE/IM Correspondence
• Yangian characters and classical W-algebras
• Jacobi-Trudi determinants and characters of minimal affinizations
• Eigenvalues of Bethe vectors in the Gaudin model
• Path description for $q$-characters of fundamental modules in type $C$
• On Frenkel-Mukhin algorithm for q-character of quantum affine algebras
• Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type D_n
• Representations of quantum affine algebras of type $B_N$
• Adapted Sequence for Polyhedral Realization of Crystal Bases