$E_7$的多项式表示及其组合蕴涵和PDE蕴涵
Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications
摘要：
本文利用偏微分方程求出了$E_7$的基本不可约模上的多项式代数分解为不可约子模之和。此外，我们还得到了一个组合恒等式，即$E_7$的某些不可约模的维数与二项式系数为55有关。进一步证明了两类具有三个积分参数的不可约子模是对应于Cartan唯一四次$E_7$不变量的基本不变微分算子的解.
英文摘要：
In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of $E_7$ into a sum of irreducible submodules. Moreover, we obtain a combinatorial identity, saying that the dimensions of certain irreducible modules of $E_7$ are correlated by the binomial coefficients of fifty-five. Furthermore, we prove that two families of irreducible submodules with three integral parameters are solutions of the fundamental invariant differential operator corresponding to Cartan's unique quartic $E_7$ invariant.
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