Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials

Details Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials

2000-03-29

arxiv.org/abs/math/0003200v1

Mathematics

Quantum Algebra

25 pages; Latex

Scientific paper

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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