Consider the category of finite-dimensional representations for $\mathrm{SL}(n,\bar{F}_p)$ in characteristic $p$. I know very little about this but am told there is a highest weight category here with indecomposable tilting modules $T(\lambda)$ which control the category in some way. These have a filtration whose successive quotients are standard objects, and a filtration whose successive quotients are costandard objects. I've never seen a tilting module in any highest weight category up close and personal. Is there any intuition for what they look like? Are they easier to picture in this positive characteristic setup than in characteristic $0$ as they are finite-dimensional here? Is it easy to write an explicit basis for a characteristic $p$ tilting module?

Consider the category of finite-dimensional representations for the algebraic group $\mathrm{SL}(n)$ in characteristic $p$. I know very little about this but am told there is a highest weight category here with indecomposable tilting modules $T(\lambda)$ which control the category in some way. These have a filtration whose successive quotients are standard objects, and a filtration whose successive quotients are costandard objects. I've never seen a tilting module in any highest weight category up close and personal. Is there any intuition for what they look like? Are they easier to picture in this positive characteristic setup than in characteristic $0$ as they are finite-dimensional here? Is it easy to write an explicit basis for a characteristic $p$ tilting module?