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Description

Let $U$ be the quantum affine algebra of classical type. Let $E_a(z,X)$ and $H_a(z,X)$ be the generating functions of the q-characters $e_{i,a}$ and $h_{i,a}$ of the fundamental and symmetric finite dimensional representations of $U$. Then, it satisfies the basic equality $E_a(z,-X)H_a(z,X)=1$. This equality naturally motivates us to consider the Jacobi-Trudi determinants $\chi_{\lambda,a}$ of $e_{i,a}$ and $h_{i,a}$. We conjecture that, in fact, $\chi_{\lambda,a}$ is the q-character of the irreducible representation $V(\lambda,a)$. This is an affinization of the conjecture by Chari and Kleber. By applying the Gessel-Viennot method to $\chi_{\lambda,a}$, we can get a tableaux description of $\chi_{\lambda,a}$. For $A$ and $B$ types, it immediately reproduces the known results by Bazhanov-Reshetikhin and Kuniba-Ohta-Suzuki. For $C$ and $D$ types, however, it turns out that the situation becomes (much) more complicated, and that explains, at least in our point of view, why a tableaux description has not been known so far except for fundamental and symmetric representations.

In this talk we give a tableaux description for some (non-fundamental, non-symmetric) representations of C type by working out the above method. This is a joint work with W. Nakai.