Applications of representation theory in discrete mathematics

In this thesis we present two new applications of the representation theory of finite groups in discrete mathematics. The first application is in coding theory. We further develop the theory of upper bounds for error-correcting codes and mixed binary/ternary codes, thereby expanding upon the work of Gijswijt et al. and Brouwer et al. We consider semidefinite programs based on quadruples of codewords and apply a symmetry reduction to obtain an optimization problem of polynomial size. This enables us to solve the semidefinite program with the aid of the computer for several values of n (the length of the words), q (the size of the alphabet) and d (the distance).
The second application of representation theory is concerned with the enumeration of local flows in embedded graphs. We extend the domain of the surface Tutte polynomial of Goodall et al. to include non-orientable embedded graphs and show that this invariant for embedded graphs contains for every finite group G the number of nowhere-identity local G-flows and the number of nowhere-identity local G-tensions as evaluations. We prove that for an embedded graph nowhere-identity local tensions correspond with proper colorings of a covering of the embedded graph, thereby generalizing coloring-flow duality for planar graphs. The surface Tutte polynomial also specializes to a trivariate polynomial for signed graphs. We show that this polynomial has a universal property, which we finally use to give combinatorial interpretations of several of its evaluations.