Restricting representations of classical algebraic groups to maximal subgroups

Fix an algebraically closed field $K$ having characteristic $p\geq 0$ and let $Y$ be a simple algebraic group of classical type over $K.$ Also let $X$ be maximal among closed connected subgroups of $Y$ and consider a non-trivial $p$-restricted irreducible rational $KY$-module $V.$ In this thesis, we investigate the triples $(Y,X,V)$ such that $X$ acts with exactly two composition factors on $V$ and see how it generalizes a question initially investigated by Dynkin in the $1950$s and then studied by numerous mathematicians. In particular, we study the natural embeddings of $\mbox{SO}_{2n}(K)$ in both $\mbox{Spin}_{2n+1}(K)$ and $\mbox{SL}_{2n}(K)$ and obtain results on the structure of certain Weyl modules.