## Illinois Journal of Mathematics

### Irreducible vector-valued modular forms of dimension less than six

Christopher Marks

#### 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–Poincaré 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–Poincaré series.

#### Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1267-1297.

Dates
First available in Project Euclid: 12 July 2013

https://projecteuclid.org/euclid.ijm/1373636684

Digital Object Identifier
doi:10.1215/ijm/1373636684

Mathematical Reviews number (MathSciNet)
MR3082869

Zentralblatt MATH identifier
1343.11052

#### Citation

Marks, Christopher. Irreducible vector-valued modular forms of dimension less than six. Illinois J. Math. 55 (2011), no. 4, 1267--1297. doi:10.1215/ijm/1373636684. https://projecteuclid.org/euclid.ijm/1373636684

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