Paths, Tableaux and $q$-characters of Quantum Affine Algebras

The $q$-character $\chi_q(V)$ of finite dimensional representation $V$ is a polynomial such that each monomial corresponds to the weight of $V$. These monomials are able to be written in terms of tableaux.In this talk, the set of tableaux for the $q$-characters of $U_q(A_n^{(1)})$, $U_q(B_n^{(1)})$, $U_q(C_n^{(1)})$ and $U_q(D_n^{(1)})$ for certain representations will be given, starting from the conjecture of the Jacobi-Trudi type formula of the $q$-characters. To obtain these tableaux, we use the path description due to Gessel and Viennot. The main result is to give some explicit examples of $U_q(C_n^{(1)})$ which is not known before.