幺模格的Theta级数、组合恒等式与加权对称多项式
Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials
摘要：
我们找到了两个组合恒等式的根格的θ系列的有限维单李代数的$D_{4n}$型和陪集在其积分型，在著名的Essenstein系列$E_4(z)$和Ramanujan系列$\Dlt_{24}(z)$。利用这两个恒等式，我们确定了某些正定偶幺模格的无限族的θ级数，这些无穷族是由$D_{2n}$型有限维单李代数的根格的有限个拷贝胶合而成的.证明了这些θ级数是$E_4(z)$和$\Dlt_{24}(z)$两个固定多项式族的加权对称多项式。
英文摘要：
We find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type $D_{4n}$ and the cosets in their integral duals, in terms of the well-known Essenstein series $E_4(z)$ and Ramanujan series $\Dlt_{24}(z)$. Using these two identities, we determine the theta series of certain infinite families of postive definite even unimodular lattices obtained by gluing finite copies of the root lattices of the finite-dimensional simple Lie algebras of type $D_{2n}$. It turns out that these theta series are weighted symmetric polynomials of two fixed families of polynomials of $E_4(z)$ and $\Dlt_{24}(z)$.
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