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?



Tilting modules in positive characteristic
For the algebraic group $\mathrm{SL}(n)$ over a field of characteristic $p>0$ the (indecomposable) tilting modules are the indecomposable direct summands of
tensor products of tensor powers of the fundamental representations.
The fundamental representations themselves are the exterior powers of the 
defining representations. This is easily said, and may be made explicit in small
examples. The smallest examples that I find instructive are the second and third tensor power of the defining representation of $\mathrm{SL}(2)$ in characteristic two. In general it is not easy to find the dimension of a tilting module, let alone a basis.


