Homology, Belyĭfunctions and canonical curves
After a brief review of maps, hypermaps and Belyi function, the author studies a linear representation of the automorphism group of a Riemann surface \(X\) generated by elements \(\alpha_1, \alpha_2, \alpha_3: =(\alpha_1 \alpha_2)^{-1}\). Then he applies the results obtained to the representation of the automorphism group of a Riemann surface on its abelian differentials \(\Omega(X)\) of the first kind, and gets relations among the multiplicities of the eigenvalues of the representation matrix, the degree of the irreducible representation, the number of its occurring times in the representation on \(\Omega(X)\) and the orders of \(\alpha_i\). As examples, the calculations of the representation for several groups are given. The author also mentions the canonical curve \(C(X)\) of \(X\), and the informations at the fixed points on \(C(X)\) of the automorphisms \(\alpha_1, \alpha_2, \alpha_3\) is used to find the equations corresponding to a given map.
scientific article; zbMATH DE number 980710