Algebraic Combinatorics Seminar - Matt Szczesny

Title: Combinatorial Hall algebras

Abstract:

The Hall algebra of a finitary category is an associative (and sometimes Hopf) algebra whose structure constants count the number of extensions between objects. Classical examples include categories of quiver representations over a finite field, in which case the Hall algebra contains (half) the corresponding quantum group. When this construction is applied to non-additive categories built from combinatorial objects, such as trees, graphs, matroids, etc. it produces combinatorial Hopf algebras (some previously studied, some new). 

One source of combinatorial examples arises from algebraic geometry over "F_1" - the field of one element. I will discuss joint work with Jaiung Jun which attaches to a smooth projective toric variety a Hall algebra of coherent sheaves in this setting. These can be thought of as gluing together skew shapes.