  Integrality properties of partial sums over irreducible representations,
along columns of character tables of finite groups, were recently derived using
combinatorial topological string theories (CTST). These CTST were based on
Dijkgraaf-Witten theories of flat $G$-bundles for finite groups $G$ in two
dimensions, denoted $G$-TQFTs. We define analogous combinatorial topological
strings related to two dimensional TQFTs based on fusion coefficients of finite
groups. These TQFTs are denoted as $R(G)$-TQFTs and allow analogous integrality
results to be derived for partial row sums of characters over conjugacy classes
along fixed rows. This relation between the $G$-TQFTs and $R(G)$-TQFTs defines
a row-column duality for character tables, which provides a physical framework
for exploring the mathematical analogies between rows and columns of character
tables. These constructive proofs of integrality are complemented with the
proof of similar and complementary results using the more traditional Galois
theoretic framework for integrality properties of character tables. The partial
row and column sums are used to define generalised partitions of the integer
row and column sums, which are of interest in combinatorial representation
theory.
