  Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism
of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in
representation theory. More precisely, we construct for all quasi-projective
schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear
triangulated category of motives on $X$. Using work of Ayoub (2007),
Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of
categories has a six functors formalism and computes higher Chow groups.
Indeed, it behaves similarly to other categories of sheaves that one is used
to. We attempt to make its construction also accessible to non-experts. We then
consider the subcategory of stratified mixed Tate motives defined for affinely
stratified varieties $X$, discuss perverse and parity motives and prove
formality results. As an example, we combine these results and Soergel (2000)
to construct a geometric and graded version of Soergel's modular category
$\mathscr O(G)$, consisting of rational representations of a split semisimple
group $G/\mathbb{k}$, and thereby equip it with a full six functor formalism
(see Riche-Soergel-Williamson (2014) and Achar-Riche (2016) for other
approaches). The main idea of using motives in geometric representation theory
in this way as well as many results about stratified mixed Tate motives are
directly borrowed from Soergel and Wendt, who tell the story in characteristic
zero.
