Mathematics – Quantum Algebra
Scientific paper
2005-02-02
J. Phys. A: Math. Gen. 39 (2006) 2083-2115
Mathematics
Quantum Algebra
38 pages; final version to appear in J.Phys
Scientific paper
10.1088/0305-4470/39/9/007
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 $\lambda/\mu$.
Nakai Wakako
Nakanishi Tomoki
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