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

## TL;DR

This paper derives identities between theta constants for cyclic 3-sheeted covers of the sphere, utilizing Thomae's formula and symmetric group representations to relate theta constants and branch point polynomials.

## Contribution

It introduces new identities between theta constants for specific algebraic curves using symmetric group representations and Thomae's formula.

## Key findings

- Identities between theta constants expressed as polynomials in branch points
- Application of symmetric group representations to relate theta constants and polynomials
- Extension of Thomae's formula to cyclic 3-sheeted covers

## 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.