The n-Category Café

The theme for these seminars is representation theory, interpreted broadly. The topics are:

• Young diagrams
• Dynkin diagrams
• q-mathematics
• The three-strand braid group
• Clifford algebras and Bott periodicity
• The threefold and tenfold way
• Exceptional algebras

Seven topics are listed, but there will be 11 seminars, so it’s not a one-to-one correspondence: each topic is likely to take one or two weeks. Here are more detailed descriptions:

Young diagrams

Young diagrams are combinatorial structures that show up in a myriad of applications. Among other things, they classify conjugacy classes in the symmetric groups S_n, irreducible representations of S_n, irreducible representations of the groups SL(n) over any field of characteristic zero, and irreducible unitary representations of the groups SU(n).

Dynkin diagrams

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably Coxeter groups, lattices having such groups as symmetries, and simple Lie algebras. The simply laced Dynkin diagrams also classify the Platonic solids and quivers with finitely many indecomposable representations. This tour of Coxeter and Dynkin diagrams will focus on the connections between these structures.

q-mathematics

A surprisingly large portion of mathematics generalizes to something called $q$-mathematics, involving a parameter $q$. For example, there is a subject called $q$-calculus that reduces to ordinary calculus at $q = 1$. There are important applications of $q$-mathematics to the theory of quantum groups and also to algebraic geometry over $\mathbb{F}_q$, the finite field with $q$ elements. These seminars will give an overview of $q$-mathematics and its applications.

The three-strand braid group

The three-strand braid group has striking connections to the trefoil knot, rational tangles, the modular group PSL(2,$\mathbb{Z}$), and modular forms. This group is also the simplest of the Artin–Brieskorn groups, a class of groups which map surjectively to the Coxeter groups. The three-strand braid group will be used as the starting point for a tour of these topics.

Clifford algebras and Bott periodicity

The Clifford algebra $\mathrm{Cl}_n$ is the associative real algebra freely generated by $n$ anticommuting elements that square to $-1$. Baez will explain their role in geometry and superstring theory, and the origin of Bott periodicity in topology in facts about Clifford algebras.

The threefold and tenfold way

Irreducible real group representations come in three kinds, a fact arising from the three associative normed real division algebras: the real numbers, complex numbers and quaternions. Dyson called this the threefold way. When we generalize to superalgebras this becomes part of a larger classification, the tenfold way. We will examine these topics and their applications to representation theory, geometry and physics.

Exceptional algebras

Besides the three associative normed division algebras over the real numbers, there is a fourth one that is nonassociative: the octonions. They arise naturally from the fact that Spin(8) has three irreducible 8-dimensional representations. We will explain the octonions and sketch how the exceptional Lie algebras and the exceptional Jordan algebra can be constructed using octonions.