Irreducible vector-valued modular forms of dimension less than six

TL;DR

This paper classifies holomorphic vector-valued modular forms of low dimension associated with specific representations of the modular group, revealing algebraic structures and explicit series for dimensions less than six.

Contribution

It provides a comprehensive algebraic classification and explicit Hilbert-Poincare series for vector-valued modular forms of dimension under six, extending understanding of their structure.

Findings

For dimensions less than four, spaces are cyclic modules over a skew polynomial ring.

For dimensions four and five, explicit Hilbert-Poincare series are determined.

The space of forms is a free module over classical modular forms in these cases.

Abstract

An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincare series is given, using the fact that the space of vector-valued modular forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert-Poincare series.