Quantum characters of quantum groups and integrable models

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The head investigator made joint research with Wakako Nakai, and firstly presented a conjecture of the expression of the q-characters of the representations of the classical quantum affine algebras associated to the skew Young diagrams in terms of Jacobi-Trudy-type determinant.Secondly, they studied the problem of the combinatorial descriptions of the determinant by the Gessel-Viennot paths and Young tableaux. It turns out that, for A and B types, the rule for Young tableaux is determined using the standard method of the involution between the pair of the intersecting paths, and they are summarized as vertical and horizontal rules; meanwhile, for C and D types, these two rules are insufficient, and other complicated extra rules are necessary.This distinction comes from the one of the form of the generating functions. Then, the problem is to determine these extra rules. Since these rules increases as the size of Young tableaux and there are infinitely many variations, their unified description seemed rather difficult. However, after further studies, it was shown that this extra rules are given by condition for the form of the corresponding paths, namely, the condition that "the paths does not have any odd II-region"; furthermore, this condition is translated to the condition for Young tableaux and gives the unified description of the extra rules. This is the main result during term of the research. The conjecture was proved by Hernandez (2006) for A and B types, and the proof for C and D is left as an important problem.