Paul Terwilliger, Spring 2018:

Previous description:

Paul Terwilliger:

Hopf algebras in Combinatorics

DESCRIPTION: Many types of algebras that you may be familiar with,

have an extra structure that makes them a Hopf algebra. For example

group algebras, Lie algebras, and the ring of symmetric functions.

In this course, we will see how Hopf algebras come up naturally

in combinatorics. We will investigate a number of combinatorial

situations that yield concrete and attractive examples of Hopf algebras.

These examples help to illuminate how Hopf algebras work in general.

The course topics include:

have an extra structure that makes them a Hopf algebra. For example

group algebras, Lie algebras, and the ring of symmetric functions.

In this course, we will see how Hopf algebras come up naturally

in combinatorics. We will investigate a number of combinatorial

situations that yield concrete and attractive examples of Hopf algebras.

These examples help to illuminate how Hopf algebras work in general.

The course topics include:

The definition and basic facts about Hopf algebras

The ring of symmetric functions as a Hopf algebra

The Cauchy product

The Hall inner product

The skew Pieri rule

Positive self dual Hopf algebras

The representation theory of the symmetric group; a Hopf algebra approach

The Hall algebra

Quasisymmetric functions and P-partitions

Shuffles and Lyndon words

The shuffle algebra

Textbook: Darij Grinberg and Victor Reiner. Hopf algebras in Combinatorics.

These notes are freely available on the web at

arXiv:1409.8356v3 25 Aug 2015.

These notes are freely available on the web at

arXiv:1409.8356v3 25 Aug 2015.

semester:

Fall

prereqs:

A good understanding of undergraduate linear algebra. We do not assume prior knowledge of Hopf algebras.