Diagrammatic Representation Theory of the Rank Two Symplectic Group

We study the diagrammatic representation theory of the group $Sp_4$ and the quantum group $U_q(\mathfrak{sp}_4)$, expanding on the previous results of Kuperberg about type $B_2= C_2$ webs. In particular, we construct a basis for an integral form of Kuperberg's web category. Using this basis we prove that the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules $\Tilt(U_q(\mathfrak{sp}_4))$. We also use the basis to give recursive formulas for the idempotent projecting to a top summand in a tensor product of fundamental representations. Finally, using our result about the equivalence between Kuperberg's web category and $\Tilt(U_q(\mathfrak{sp}_4))$, we prove that when $[3]=0$ or $[4] = 0$, the semisimple quotient of $U_q(\mathfrak{sp}_4)$ is equivalent to $\Rep(O(2))$. This dissertation contains previously published material