Non-induced modular representations of cyclic groups

Abstract

We compute the ring of non-induced representations for a cyclic group, C_n, over an arbitrary field and show that it has rank $\phi (n)$, where $\phi$ is Euler’s totient function—independent of the characteristic of the field. Along the way, we obtain a “pick-a-number” trick; expressing an integer n as a sum of products of p-adic digits of related integers.

KEYWORDS:

• Cyclic group
• Euler totient function
• modular representation theory
• quantum number

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20C20
• Secondary: 05A30

Acknowledgments

The authors are grateful to Dr. Stuart Martin for his guidance and support and to the reviewer for insightful and detailed comments on an earlier draft of this paper.

Additional information

Funding

The work of the first author was supported by the Woolf Fisher Trust and the Cambridge Commonwealth and International Trust. The work of the second author was supported by an EPSRC DTP Scholarship from the University of Cambridge.