arXiv subjects

Chase Vogeli

Publications and source records attributed to Chase Vogeli.

5 recordsLinked to original sources

arXiv2025

Derived induction theory for the K-theory of modular group algebras

We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces calculations to calculations for their $p$-subgroups. We do so by showing that the stable module categories of $kH$ as $H$ ranges over subgroups of $G$ assemble into a categorical Green functor, which results in a spectral Green functor structure on K-theory. By general induction theory, this reduces proving a spectrum-level induction statement to proving an induction statement on $\pi_0$ Green functors, which we accomplish using modular representation theory. For $p$-isolated groups with Sylow $p$-subgroups of order $p$, we produce explicit new calculations of K-groups.

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arXiv2025

$C_p$-Mackey functors in Macaulay2

We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.

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arXiv2024

The Galois-equivariant $K$-theory of finite fields

We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied $RO(G)$-graded coefficient groups of the equivariant Eilenberg--MacLane spectrum $H\underline{\mathbb Z}$. Our comparison between the equivariant $K$-theory spectrum and $H\underline{\mathbb Z}$ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where $G$ has prime order, we provide an explicit presentation of the equivariant $K$-groups.

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arXiv2023

Bredon homological stability for configuration spaces of $G$-manifolds

McDuff and Segal proved that unordered configuration spaces of open manifolds satisfy homological stability: there is a stabilization map $\sigma: C_n(M)\to C_{n+1}(M)$ which is an isomorphism on $H_d(-;\mathbb{Z})$ for $n\gg d$. For a finite group $G$ and an open $G$-manifold $M$, under some hypotheses we define a family of equivariant stabilization maps $\sigma_{G/H}:C_n(M)\to C_{n+|G/H|}(M)$ for $H\leq G$. In general, these do not induce stability for Bredon homology, the equivariant analogue of singular homology. Instead, we show that each $\sigma_{G/H}$ induces isomorphisms on the ordinary homology of the fixed points of $C_n(M)$, and if the group is Dedekind (e.g. abelian), we obtain the following Bredon homological stability statement: $H^G_d(\bigsqcup_{n\geq 0}C_n(M))$ is finitely generated over $\mathbb{Z}[\sigma_{G/H} : H\leq G]$. This reduces to the classical statement when $G=e$.

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arXiv2019

Uniformly vertex-transitive graphs

We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(Γ)$ of a given graph $Γ$ to show that the uniform vertex-transitivity of $Γ$ is equivalent to the existence of cliques of sufficient size in $D(Γ)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-transitivity in the situation of a graph with an imprimitive automorphism group. We classify the non-Cayley uniformly vertex-transitive graphs on less than 30 vertices outside of two complementary pairs of graphs.

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