University of Edinburgh, September 22 - December 1, 2022

John Baez

Seminar on This Week's Finds

I wrote 300 issues of a column called This Week's Finds, where I explained math and physics. In the fall of 2022 I gave ten talks based on these columns. I will continue in the fall of 2023.

Here you can find lecture notes and videos.

Young diagrams and classical groups

Young diagrams are combinatorial structures that show up in a myriad of applications. Among other things, they can be used to classify conjugacy classes in the symmetric groups \(S_n\), irreducible representations of \(S_n\), and irreducible representations of the classical groups \(\mathrm{GL}(n)\), \(\mathrm{SL}(n)\), \(\mathrm{U}(n)\) and \(\mathrm{SU}(n)\).

Young diagrams; the representation theory of monoids.

Classifying representations of the symmetric group \(S_n\) using Young diagrams with \(n\) boxes.

The classical groups; classifying representations of the monoid of \(N \times N\) matrices using Young diagrams with \(\le N\) rows.

Dynkin diagrams

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or 'ADE' Dynkin diagrams also classify finite subgroups of \(\mathrm{SU}(2)\) and quivers with finitely many indecomposable representations. This tour of Coxeter and Dynkin diagrams will focus on the connections between these structures.

How Coxeter diagrams classify finite reflection groups.

How Coxeter diagrams classify root lattices.

How Coxeter diagrams classify compact semisimple Lie groups.

How the Dynkin diagrams correspond to Lie groups of symmetries.

\(\mathrm{E}_8\) and the octonions.

Quaternions and octonions

There are four normed division algebras: the real numbers, complex numbers, quaternions and octonions. The quaternions are a noncommutative algebra of dimension 4, while the octonions are a noncommutative and nonassociative algebra of dimension 8. Here we explain how to multiply quaternions and octonions using the familiar dot product and cross product of vectors. For the proof that octonion multiplication obeys \(|ab| = |a||b|\), go here:

Quaternions and octonions.

The threefold 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. The spin-1/2 representation of SU(2) is a great example: it is quaternionic, since SU(2) is isomorphic to the group of quaternions \(q\) with \(|q| = 1\), and the spin-1/2 representation is isomorphic to the space of all quaternions. For more, go here:

The threefold way.

© 2022 John Baez