Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. The indecomposable tilting modules $\{T(\lambda)\}$ for $G$, which are labeled by highest weight, form an important class of self-dual representations over $k$. In this thesis we investigate semisimple filtrations of minimal length (Loewy series) of tilting modules. We first demonstrate a criterion for determining when tilting modules for arbitrary quasi-hereditary algebras are rigid, i.e. have a unique Loewy series. Our criterion involves checking that $T(\lambda)$ does not have certain subquotients whose composition factors extend more than one layer in the radical or socle series. We apply this criterion to show that the restricted tilting modules for $SL_4$ are rigid when $p \geq 5$, something beyond the scope of previous work on this topic by Andersen and Kaneda. Even when $T(\lambda)$ is not rigid, in many cases it has a particularly structured Loewy series which we call a balanced semisimple filtration, whose semisimple subquotients or "layers" are symmetric about some middle layer. Balanced semisimple filtrations also suggest a remarkably straightforward algorithm for calculating tilting characters from the irreducible characters. Applying Lusztig's character formula for the simple modules, we show that the algorithm agrees with Soergel's character formula for the regular indecomposable tilting modules for quantum groups at roots of unity. We then show that these filtrations really do exist for these tilting modules. In the modular case, high weight tilting modules exhibit self-similarity in their characters at $p$-power scales. This is due to what we call higher-order linkage, an old character-theoretic result relating modular tilting characters and quantum tilting characters at $p$-power roots of unity. To better understand this behavior we describe an explicit categorification of higher-order linkage using the language of Soergel bimodules. Along the way we also develop the algebra and combinatorics of higher-order linkage at the de-categorified level. We hope that this will provide a foundation for a tilting character formula valid for all weights in the modular case when $p$ is sufficiently large.

This thesis concerns the use of perverse sheaves with coefficients in commutative rings and in particular, fields of positive characteristic, in the study of modular representation theory. We begin by giving a new geometric interpretation of classical connections between the representation theory of the general linear groups and symmetric groups. We then survey work, joint with D. Juteau and G. Williamson, in which we construct a class of objects, called parity sheaves. These objects share many properties with the intersection cohomology complexes in characteristic zero, including a decomposition theorem and a close relation to representation theory. The final part of this document consists of two computations of IC stalks in the nilpotent cones of sl₃and sl₄. These computations build upon our calculations in sections 3.5 and 3.6 of (31), but utilize slightly more sophisticated techniques and allow us to compute the stalks in the remaining characteristics. / text