Lie theory is the home of many beautiful connections between geometry, combinatorics, and topology. In this talk, we´ll explore connections between the combinatorics of the symmetric group S_n and the geometry and topology of the group of unitary matrices with determinant one, SU(n). In particular, we´ll see that to each decomposition of the longest permutation in S_n, there corresponds a factorization of SU(n) into a product of two-spheres. This will lead us to flag manifolds and their cell decompositions. Interest and time permitting, we´ll end up at some infinite-dimensional generalizations and open questions.