arXiv subjects

Jesse Keyes

Publications and source records attributed to Jesse Keyes.

5 recordsLinked to original sources

arXiv2026

Composition of bispans of $G$-sets and plethysm

Let $P(G)$ be the Grothendieck ring of the semiring of endomorphisms of the point in the $1$-category of bispans of finite $G$-sets for a finite group $G$. This is the bispan analogue of the Burnside ring of $G$. The ring $P(G)$ admits a third operation from composition of bispans. We produce a character map for $P(G)$ landing in a plethory built out of polynomial rings and the poset of conjugacy classes of subgroups of $G$. We prove that the character map sends composition of bispans to the plethysm operation -- which is a generalization of composition of polynomials.

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arXiv2026

The Zero Slice of Quaternionic Real Bordism

Using the Hill-Hopkins-Ravenel norm, one can produce a $Q_8$-spectrum $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$, where $Q_8$ is the quaternion group. Working towards a computation of the slice spectral sequence for $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$, we compute the zero slice of $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$ and a bigraded subring of the $\text{RO}(Q_8)$-graded homotopy Mackey functors of this slice.

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arXiv2025

Picard Groups in Equivariant Algebra and Stable Homotopy Theory

Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the equivariant stable homotopy category. Though there is a canonical map of abelian groups $\text{RO}(G) \rightarrow \text{Pic}(\text{Ho}(\text{Sp}^G))$ relating the two, this map is neither injective or surjective in general. Fausk, Lewis, and May give an algebraic expression of $\text{Pic}(\text{Ho}(\text{Sp}^G))$ in terms of the Picard group of the Burnside ring $A(G)$, and this work suggests a folklore isomorphism between $\text{Pic}(A(G))$ and $\text{Pic}(\text{Mack}_G)$. We prove the existence of this folklore isomorphism in the setting of finite groups, then leverage our analysis to prove a classification of invertible Mackey functors in the setting of finite abelian groups. As a consequence, we furnish a classification of invertible $A(G)$-modules again for $G$ a finite abelian group.

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arXiv2025

The $RO(\mathcal{K})$-graded homotopy of Klein-four normed Mackey functors

We compute the $RO(\mathcal{K})$-graded coefficients of the equivariant Eilenberg-Mac Lane spectrum associated to various Hill-Hopkins-Ravenel norms of the constant-$\mathbb{F}_2$ Mackey functor, where $\mathcal{K}$ is the Klein-four group. Further, we analyze the multiplicative structure of these $RO(\mathcal{K})$-graded Tambara functors.

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arXiv2025

The $RO(\mathcal{K})$-graded Coefficients of $H\underline{A}$

In $G$-equivariant stable homotopy theory, it is known that the equivariant Eilenberg-Mac Lane spectra representing ordinary equivariant cohomology have nontrivial $RO(G)$-graded homotopy corresponding to the equivariant (co)homology of representation spheres. We will compute the universal case of this ordinary $RO(G)$-graded homotopy in the case of $G=\mathcal{K}$, where $\mathcal{K}$ is the Klein-four group. In particular, we will compute a subring of the $RO(\mathcal{K})$-graded homotopy of $H\underline{A}$ for $\underline{A}$ the Burnside Mackey functor.

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