Partition functors and universal exponential relations

@misc{mehrle2026partition,
  title = {{Partition functors and universal exponential relations}},
  author = {David Mehrle and Millie Rose and Nathaniel Stapleton},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28368},
  eprint = {2609.28368},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28368},
}

TY  - UNPB
TI  - Partition functors and universal exponential relations
AU  - David Mehrle
AU  - Millie Rose
AU  - Nathaniel Stapleton
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28368
UR  - https://arxiv.org/abs/2609.28368
AB  - We introduce partition functors: algebraic structures indexed by partitions of finite sets and equipped with restriction and transfer maps along refinements. We construct a monad PD on the category of partition functors and on several categories of partition functors with additional multiplicative structure. For a partition ring, exponential elements in its associated completed ring of symmetric functions acquire canonical logarithms after applying PD. This gives rise to a universal exponential relation between multiplicative and additive power operations. For representation rings this can be used to recover the classical relation between symmetric powers and Adams operations, while for Morava E-theory it can be used to recover Ganter's exponential relation between symmetric powers and Hecke operators. We show that the representation rings of products of symmetric groups form the initial partition ring and that, for symmetric monoidal partition functors, the monad PD is closely related to symmetric invariant tensors and the divided power envelope. We also construct a symmetric monoidal partition ring carrying the universal exponential element, so that its image under PD carries the universal exponential relation.
ER  -