Proc. Japan Acad. Ser. A Math. Sci. 90 (2), 37-42, (February 2014) DOI: 10.3792/pjaa.90.37
KEYWORDS: Triangulated Riemann surface, combinatorial Hodge theory, associate matrix, 30F99
In~, Wilson defined holomorphic 1-cochains and combinatrial period matrices of triangulated Riemann surfaces by using the combinatorial Hodge star operator, introduced in~. In this paper, we define a matrix and call this matrix the associate matrix. Then, we prove that among the three matrices, which are a period matrix, a combinatorial period matrix which is introduced by Wilson~ and an associate matrix, there exists a matrix equation. Then we also show that an associate matrix is an element of the Siegel upper half space, so this means that a trianguted Riemann surface gives three elements of the Siegel upper half space.