Quantum characters of quantum groups and integrable models

首席研究员与中井若子共同研究，首先提出了经典量子仿射代数的表示的q-特征标用Jacobi-Trudy型行列式表示的猜想，其次研究了行列式的Gessel-Viennot路径和Young tableaux的组合描述问题。结果表明，对于A和B类，Young tableaux规则是用相交路径对合的标准方法确定的，概括为垂直规则和水平规则;而对于C和D类，这两个规则是不够的，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之一。然后，问题是确定这些额外的规则。由于这些规则随着Young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，证明了这一额外规则是由相应路径的形式条件给出的，即“路径没有任何奇II-区域”的条件，并将这一条件转化为Young tableaux条件，给出了额外规则的统一描述。这是研究期间的主要结果。埃尔南德斯（2006）对A型和B型证明了该猜想，而对C型和D型的证明则是一个重要的问题。

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.