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.
Categories: math.CV