During the last 20 years, it was realized that certain combinatorial objects (combinatorial Hopf algebras, to be precise) underly many mathematical theories. The talk will survey some of these developments, and then focus largely on two of the most emblematic and universal such objects, namely the higher algebraic structures that can be constructed out of permutations, and out of surjections.

The link is made through the notion of special poset (equivalent to labelled poset of Stanley): linear extensions of a poset can be seen as bijections, while the generating function of a poset P with respect to Stanley's classical definition of P-partitions associated to a special poset is a quasi-symmetric: in fact, it is a homomorphism between the Hopf algebra of labelled posets and that of quasi-symmetric functions; while linearisation is a homomorphism onto the Hopf algebra of permutations.

The aim is to generalize this frame to preorders, which are in one-to-one correspondence with finite topologies: the objects corresponding to bijections are surjections: they can be seen as linear extensions of a preorder and are encoded by packed words. We can hence define the notion of T-partitions associated to a finite topology T, and deduce a Hopf algebra morphism from a new Hopf algebra on topologies to the Hopf algebra of packed words.

This is joint work with L. Foissy and F. Patras.

Crystals are colored directed graphs encoding information about Lie algebra representations. Kirillov-Reshetikhin (KR) crystals correspond to certain finite-dimensional representations of affine Lie algebras. I will present a combinatorial model which realizes tensor products of (column shape) KR crystals uniformly across affine types. Some computational applications are discussed. A corollary states that the Macdonald polynomials (which generalize the irreducible characters of semisimple Lie algebras), upon a certain specialization, coincide with the graded characters of tensor products of KR modules. The talk is largely self-contained, and is based on a series of papers with A. Lubovsky, S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.