Theta constants identities for Jacobians of cyclic 3-sheeted covers of the sphere and representations of the symmetric group

Abstract

We find identities between theta constants with rational characteristics evaluated at period matrix of $R,$ a cyclic 3 sheeted cover of the sphere with $3k$ branch points ${\lambda }_1...{\lambda }_{3k}.$ These identities follow from Thomae formula \cite{BR}. This formula expresses powers of theta constants as polynomials in ${\lambda }_1...{\lambda }_{3k}.$ We apply the representation of the symmetric group to find relations between the polynomials and hence between the associated theta constants.

7 pages