Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let \text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$. A {\em Leonard system} on $V$ is a sequence \[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\] where $\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$; $\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$ are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and nonzero when $\vert i-j \vert = 1$. % Leonard systems arise in connection with orthogonal polynomials, representations of many nice algebras, and the study of some highly regular combinatorial objects. We shall use the construction of Leonard pairs of classical type from finite-dimensional modules of $\mathit{sl}_2$ and the construction of Leonard pairs of basic type from finite-dimensional modules of $U_q(\mathit{sl}_2)$. Suppose $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ is a Leonard system. For $0 \leq i \leq d$, let \[ U_i = (E^*_0V+E^*_1V+\cdots + E^*_iV)\cap (E_iV+E_{i+1}V+\cdots + E_dV). \] Then $U_0$, $U_1$, \ldots, $U_d$ is the {\em split decomposition of $V$ for $\Phi$}. % The split decomposition of $V$ for $\Phi$ gives rise to canonical matrix representations of $\A$ and $\B$ in terms of useful parameters for the Leonard system. %These canonical matrix representations for $\A$, $\B$ are respectively lower bidiagonal and upper bidiagonal. In this thesis, we consider when certain Leonard systems share a split decomposition. We say that Leonard systems $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ and $\hat{\Phi}:=(\hat{\A} ;\hat{\B}; \lbrace \hat{E}_i\rbrace_{i=0}^d; \lbrace \hat{E^*}_i\rbrace_{i=0}^d)$ are {\em friends} when $\A = \hat{\A}$ and $\Phi$, $\hat{\Phi}$ have the same split decomposition. % We obtain Leonard systems which share a split decomposition by constructing them from closely related module structures for either $\mathit{sl}_2$ or $U_q(\mathit{sl}_2)$ on $V$. We then describe friends by a parametric classification. In this manner we describe all pairs of friends of classical and basic types. In particular, friendship is not entirely a property of isomorphism classes.

Let V denote a vector space of finite positive dimension. An ordered triple of linear operators on V is said to be a Leonard triple whenever for each choice of element of the triple there exists a basis of V with respect to which the matrix representing the chosen element is diagonal and the matrices representing the other two elements are irreducible tridiagonal. A Leonard triple is said to be modular whenever for each choice of element there exists an antiautomorphism of End(V) which fixes the chosen element and swaps the other two elements. We study combinatorial structures associated with Leonard triples and modular Leonard triples. In the first part we construct a simplicial complex of Leonard triples. The simplicial complex of a Leonard triple is the smallest set of linear operators which contains the given Leonard triple with the property that if two elements of the set are part of a Leonard triple, then the third element of the triple is also in the set. In the second part we construct a Hamming association scheme from modular Leonard triples using a method used previously in the context of Grassmanian codes.