# On the motivic class of the classifying stack of $G_2$ and the spin groups

## TL;DR

This paper calculates the motivic classes of classifying stacks for certain exceptional and spin groups, revealing their inverse relationship with the group classes and reducing complex computations to simpler subgroup classes.

## Contribution

It provides explicit calculations of the motivic classes for the classifying stacks of $G_2$, $ ext{Spin}_7$, and $ ext{Spin}_8$, and simplifies the computation for $ ext{Spin}_n$ stacks.

## Key findings

- Motivic classes of $G_2$, $ ext{Spin}_7$, and $ ext{Spin}_8$ classifying stacks are inverses of the group classes.
- The computation for $ ext{Spin}_n$ stacks reduces to classes of $ ext{Delta}_n$ groups.
- The approach links motivic classes to subgroup structures within orthogonal groups.

## Abstract

We compute the class of the classifying stack of the exceptional algebraic group $G_2$ and of the spin groups $\mathrm{Spin}_7$ and $\mathrm{Spin}_8$ in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks $\mathscr{B}\mathrm{Spin}_n$ can be reduced to the computation of the classes of $\mathscr{B} \Delta_n$, where $\Delta_n\subset \mathrm{Pin}_n$ is the "extraspecial $2$-group", the preimage of the diagonal matrices under the projection $\mathrm{Pin}_n\to \mathrm{O}_n$ to the orthogonal group.