Short eigenvectors and multidimensional theta functions Original Research Article

A certain family of symmetric matrices, with entries ± 1, is known to determine all the quartic relations that hold between multidimensional theta constants. Attention is drawn here to combinatorial properties of the shortest possible quartic relations, corresponding to vectors with minimal support in a certain eigenspace of such a matrix. A lower bound for the size of the support is established, exhibiting a “phase transition” at dimension four. The multiplicity-free eigenvectors with minimal support form an interesting combinatorial design, with a rich group of symmetries.