Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E-4(z) and the Ramanujan series Delta(24)(z). A natural question is what kind of polynomials in E-4(z) and Delta(24)(Z) could be the theta series of positive definite even unimodular lattices. In this 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 E-4(z) and Delta(24)(z). Using these two identities, we prove that three families of weighted symmetric polynomials of two fixed families of polynomials of E-4(z) and Delta(24)(z) are the theta series of certain positive definite even unimodular lattices, obtained by gluing finitely many copies of the root lattices of the finite-dimensional simple Lie algebras of type D-2n. The results also show that the full permutation groups are the hidden symmetry of the theta series of certain unimodular lattices.
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