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:

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: