Unified viewpoint for hypergeometric functions and the pentagonal number theorem based on representation theory

欧拉著名的五边形数定理被推广为对异常的描述，即两个无限大小矩阵的迹差，它们几乎是共轭的。其中值得注意的例子是q-超几何级数的反演公式。本文从不变量论和表示论的角度探讨了这种现象背后的机理，从而为超几何级数的研究开辟了新的视界。对偶理论是撤销对称性的关键，即使它并不明显。

The famous pentagonal number theorem by Euler has been extended as the description of the anomaly, that is, the difference of the trace of two matrices of infinite size, which are almost conjugate. Notable examples, among them, are inversion formulas for q-hypergeometric series. We investigate the mechanism behind that phenomena from the invariant-theoretic and representation-theoretic points of view, so that some new horizon for the hypergeometric series is opened. Dual pair theory is the key to withdraw the symmetry, even if it is not apparent.