Linear independence of cluster monomials for skew-symmetric cluster algebras

This is joint work with Bernhard Keller, Daniel Labardini-Fragoso and Pierre-Guy Plamondon. We consider a skew-symmetric cluster algebra A. Then A is associated with a quiver Q and its combinatorics is "categorified" by the generalized cluster category associated with Q. By using such categorification we show that the cluster monomials of A are linearly independent, proving an old conjecture of Fomin and Zelevinsky. We also show that the exchange graph of A and its cluster complex are independent on the choice of coefficients, confirming another conjecture of Fomin and Zelevinsky. I will first give an introduction to additive categorifications of cluster algebras and then I will give the proof of these results.