On Riemann type relations for theta functions on bounded symmetric domains of type $I$

TL;DR

This paper introduces a practical method to derive algebraic relations among theta functions on type I bounded symmetric domains, linking combinatorial properties of matrices over quadratic and complex fields.

Contribution

It provides a novel framework controlling theta relations via combinatorial properties of matrix pairs over quadratic and complex fields.

Findings

Developed a technique to generate algebraic relations for theta functions.

Connected theta relations to combinatorial properties of matrix pairs.

Enhanced understanding of theta functions on symmetric domains of type I.

Abstract

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type $I$. In our framework, each theta relation is controlled by combinatorial properties of a pair $(T,P)$ of a regular matrix $T$ over an imaginary quadratic field and a positive-definite Hermitian matrix $P$ over the complex number field.