Séminaire de Géométrie Tropicale
Mercredi 15 janvier, 16h15 salle 1525-502
Discrete convexity
Résumé :
In the talk, I will explain what subsets of the lattice Z^n and what functions on the lattice Z^n could be called convex.
Unimodular set-systems play an important role in the classification of such classes.
The so-called polymatroidal discretely concave functions, most
interesting for applications, are related to the unimodular graphical system A_n := {±e_i, e_i - e_j}.
The dual class is of interest in tropical geometry. Polymatroidal functions naturally
appear in representation theory (Littlwood-Richardson coefficients, Zelevinsky's pictures, RSK algorithm),
play an important role for solution of the Horn problem, for describing submodule invariants over discrete valuation rings, and so on.
We describe several constructions of polymatroidal functions and formulate a conjecture for other unimodular set-systems.