The algebra of card shuffling

In 1976 Louis Solomon discovered that when we partition the elements of a finite Coxeter group W in a certain way and form a formal sum from the elements in each partition we obtain an algebra, called the descent algebra of W.

For the most famous family of finite Coxeter groups, the symmetric groups, it was shown in 1989 by Garsia and Reutenauer that these algebras have a nice description which allowed much of their structure to be determined. Moreover, it was shown in 1992 by Bayer and Diaconis that these algebras are closely related to card shuffling.

In this talk we will introduce the descent algebras of the symmetric groups, some of their properties, and where else they arise in mathematics.

This talk will be accessible to graduate students.