Courses on quasi-symmetric functions

Stephanie van Willigenburg (UBC), who visits us for the LaCIM Thematic Month at CRM has a CRM-Simons Chair, will give next week a mini-course on quasi-symmetric functions on tuesday wednesday and Friday. Please note that the three presentations are independent of each other.

 

Tuesday 9th october 3 PM PK-4323

Title: The combinatorics of quasisymmetric functions with an algebraic viewpoint

Abstract: Quasisymmetric functions were introduced by Gessel in 1984 as refinements of symmetric functions that enumerate chains in posets. Since then they have arisen in a variety of areas from representation theory to category theory. In this first talk we will define the two classical bases of quasisymmetric functions discovered by Gessel, known as the monomial and the fundamental quasisymmetric functions. We will then look at combinatorial tips and tricks for their algebraic properties such as their product and coproduct formulas. No prior knowledge is assumed.

Wednesday 10th october  4 PM PK-4323

Title: The combinatorics of quasisymmetric functions with a geometric viewpoint
Abstract: Quasisymmetric functions were introduced by Gessel in 1984 as refinements of symmetric functions that enumerate chains in posets. Since then they have arisen in a variety of areas from representation theory to category theory. In this second talk we will look at Gessel’s original approach to quasisymmetric functions as weight enumerators of chains in labelled posets, which will encode the labelling on the chains. No prior knowledge is assumed.

Friday 12th october 1.30 PM PK-4323

Title: An introduction to quasisymmetric Schur functions

Abstract: In algebraic combinatorics a central area of study is Schur functions. These functions were introduced early in the last century with respect to representation theory, and since then have become important in other areas such as quantum physics and algebraic geometry.

These functions also form a basis for the algebra of symmetric functions, which in turn forms a subalgebra of the algebra of quasisymmetric functions that itself impacts areas from category theory to card shuffling. Despite this strong connection, the existence of a natural quasisymmetric refinement of Schur functions was considered unlikely for many years.

In this talk we will meet such a natural refinement of Schur functions, called quasisymmetric Schur functions. Furthermore, we will see how these quasisymmetric Schur functions refine many well-known Schur function properties, with combinatorics that strongly reflects the classical case including diagrams, walks in the plane, and pattern avoidance in permutations.

This talk will require no prior knowledge of any of the above terms.