Consider the category of finitedimensional 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 finitedimensional here? Is it easy to write an explicit basis for a characteristic $p$ tilting module?

$\begingroup$ This seems to be about representations of an algebraic group, not the discrete group $\mathrm{SL}(n,\bar{F}_p)$. $\endgroup$ – Wilberd van der Kallen Jun 6 '16 at 7:14

$\begingroup$ I thought they were the same :o Yes, I meant the algebraic group. I guess I don't know what that thing is now. $\endgroup$ – CatO Minor Jun 6 '16 at 12:04

1$\begingroup$ An example that is not a representation of the algebraic group sends any matrix to its entry wise $p$th root and views the resulting matrix as describing the action on an $n$dimensional vector space. $\endgroup$ – Wilberd van der Kallen Jun 6 '16 at 12:59

$\begingroup$ I've changed it to "algebraic group" in my question. Thanks for your correction. $\endgroup$ – CatO Minor Jun 6 '16 at 14:36
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.

$\begingroup$ That is very interesting, thanks! Is it known whether the same phenomenon happens with tiltings arising as the indecomposable direct summands of tensor powers of special representations, in other settings that have parallels to the positive characteristic setup, such as Category O for Lie superalgebras? $\endgroup$ – CatO Minor Jun 6 '16 at 12:18