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arXiv · 2606.03649

Composition of bispans of $G$-sets and plethysm

Nathan Cornelius·Evan Franchere·Usman Hafeez·Jesse Keyes·David Mehrle·Lakshay Modi·Nathaniel Stapleton
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Abstract

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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Nathan Cornelius, Evan Franchere, Usman Hafeez, Jesse Keyes, David Mehrle, Lakshay Modi, Nathaniel Stapleton. 2026-06-02. Composition of bispans of $G$-sets and plethysm. https://arxiv.org/abs/2606.03649

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