We generalise a recent combinatorial description of the Verlinde or WZW fusion algebra of type A by defining cylindric Macdonald functions. The latter arise as weighted sums over non-intersecting paths on a square lattice with periodic boundary conditions. Expanding the cylindric Macdonald functions into Schur functions one obtains generalised Kostka-Foulkes polynomials. The latter contain ordinary Kostka-Foulkes polynomials, which appear in algebraic geometry, representation theory and combinatorics, as special case. We further motivate the cylindric Schur functions by showing that they are connected with a commutative Frobenius algebra which can be interpreted as a deformation of the Verlinde algebra: its structure constants are polynomials whose constant terms are the WZW fusion coefficients.

I will present an overview over work on the construction of Baxter's Q-operator. The latter is a an auxiliary tool in diagonalizing the Hamiltonian of integrable spin-chains. This is an alternative approach to the Bethe ansatz and has several advantages over the latter. I will highlight the representation theoretic construction of the Q-operator and discuss how this approach leads to a difference equation (called the quantum Wronskian) which is sufficient to determine the spectrum of the spin-chain Hamiltonian. In contrast to the Bethe ansatz equations which are of polynomial order (= number of sites of the chain) the quantum Wronskian leads to a system of quadratic equations. I will also briefly discuss how the Q-operator allows for an alternative description of the trace functional used in the recent discussion of correlation functions by Boos, Jimbo, Miwa, Smirnov and Takeyama.