A Geometric Splitting of the Motive of ${\text{GL}}_n$

A paper by Haynes Miller shows that there is a filtration on the unitary groups that splits in the stable homotopy category, where the stable summands are certain Thom spaces over Grassmannians. We give an algebraic version of this result in the context of Voevodsky’s tensor triangulated category of stable motivic complexes $\mathbf{\text{DM}}(k,R)$, where $k$ is a field. Specifically, we show that there are algebraic analogs of the Thom spaces appearing in Miller’s splitting that give rise to an analogous splitting of the motive $M({\text{GL}}_n)$ in $\mathbf{\text{DM}}(k,R)$, where ${\text{GL}}_n$ is the general linear group scheme over $k$.