arXiv subjects

Millie Rose

Publications and source records attributed to Millie Rose.

3 recordsLinked to original sources

arXiv2026

Partition functors and universal exponential relations

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.

math.AT↗

arXiv2025

Combinatorics of factorization systems on lattices

We initiate the combinatorial study of factorization systems on finite lattices, paying special attention to the role that reflective and coreflective factorization systems play in partitioning the poset of factorization systems on a fixed lattice. We ultimately uncover an intricate web of relations with such diverse combinatorial structures as submonoids, monads, Moore systems, transfer systems (from stable equivariant homotopy theory), and poly-Bernoulli numbers.

math.CO↗

arXiv2024

On the image of the total power operation for Burnside rings

We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wrΣ_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr Σ_n)$. As $n$ varies, the subrings $\mathring A(G,n)$ assemble into a commutative graded ring $\mathring A(G)$ with a universal property: $\mathring A(G)$ carries the universal family of power operations out of $A(G)$. We construct character maps for $\mathring A(G,n)$ and give a formula for the character of the total power operation. Using $\mathring A(G)$, we extend the Frobenius--Wielandt homomorphism of Dress--Siebeneicher--Yoshida to wreath products compatibly with the total power operation. Finally, we prove a generalization of Burnside's orbit counting lemma that describes the transfer map $A(G \wr Σ_n) \to A(Σ_n)$ on the subring $\mathring A(G,n)$.

math.RA↗