Quantum characters of quantum groups and integrable models

首席研究员与Wakako Nakai共同研究，首次提出了与歪斜杨图相关的经典量子仿射代数表示的q-特征用jacobi - trudy型行列式表示的猜想。其次，他们研究了用Gessel-Viennot路径和Young表来组合描述行列式的问题。结果表明，对于A和B两种类型，Young表的规则是使用相交路径对之间对合的标准方法确定的，并将其概括为垂直和水平规则；同时，对于C和D类型，这两条规则是不够的，还需要其他复杂的额外规则。这种区别来自于生成函数的形式。接下来的问题是确定这些额外的规则。由于这些规则随着Young的场景的大小而增加，并且有无限多的变化，因此它们的统一描述似乎相当困难。然而，经过进一步的研究表明，这些额外的规则是由相应路径形式的条件给出的，即“路径不存在奇数ii -区域”的条件；此外，这一条件被转化为Young tableaux的条件，并给出了额外规则的统一描述。这是本研究期间的主要成果。该猜想由Hernandez（2006）对A和B两种类型进行了证明，而对C和D的证明留作重要问题。

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.