September 11, 2022

Seminar on This Week’s Finds

Posted by John Baez

Here’s something new: I’m living in Edinburgh until January! I’ll be working with Tom Leinster at the University of Edinburgh, supported by a Leverhulme Fellowship.

One fun thing I’ll be doing is running seminars on some topics from my column This Week’s Finds. They’ll take place on Thursdays at 3:00 pm UK time in Room 6206 of James Clerk Maxwell Building, home of the Department of Mathematics. The first will be on September 22nd, and the last on December 1st.

We’re planning to

1) make the talks hybrid on Zoom so that people can participate online:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

2) record them and eventually make them publicly available on my YouTube channel.

3) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s here.

More details soon!

I have the topics planned out….

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 Sn, irreducible representations of Sn, 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.

Posted at September 11, 2022 10:09 AM UTC

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Re: Seminar on This Week’s Finds

Great to revisit these topics.

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 $S_n$,…

I’m reminded of my speculative thought

if symmetric groups fall into this picture by being $GL(n,\mathbb{F}_1)$, and if $GL$s are Langlands self-dual, this explains how Young diagrams parameterize conjugacy classes and at the same time irreducible representations,

following on from a comment on the Langlands Program by David Ben-Zvi here

that reps should match CANONICALLY with conjugacy classes in ANOTHER group, the dual group.

Posted by: David Corfield on September 11, 2022 12:31 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

That’s a nice thought! One hard part is figuring out what “canonically” even means here, since there is usually not a very big symmetry group (that I’m aware of) acting on either the set of irreducible representations of a group G or its set of conjugacy classes. The group of outer automorphisms of G acts on both, but in cases of interest it’s often quite small. So if I make up a bijection between irreps and conjugacy classes, the constraint that it’s equivariant isn’t very strong, and we’re left deciding whether this map is “canonical” or “natural” by subjective seat-of-the-pants considerations.

That said, people assure me that there’s a natural map from irreps to conjugacy classes for any finite Coxeter group! These groups are the $\mathbb{F}_1$ versions of simple algebraic groups, so I wonder how this fits into the Langlands dual idea.

Posted by: John Baez on September 11, 2022 4:53 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

I had not previously encountered the claim that (I guess complex) irreps and conjugacy classes were in bijection for finite Coxeter groups. Do you know where to look for a discussion of this? Is the verification case-by-case, or uniform?

Posted by: L Spice on September 11, 2022 7:28 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

That said, people assure me that there’s a natural map from irreps to conjugacy classes for any finite Coxeter group! These groups are the $\mathbb{F}_1$ versions of simple algebraic groups, so I wonder how this fits into the Langlands dual idea.

Having just said that I know nothing about this, permit me further to ponder it. At least for Weyl groups, such a bijection can be viewed as perfectly consistent with the Langlands philosophy: although there are non-self-dual root systems $\mathsf B_n$ and $\mathsf C_n$, their Weyl groups are isomorphic, so we could view a matching of the conjugacy classes and irreps of that Weyl group as a matching of the conjugacy classes of $W(\mathsf B_n)$ with the irreps of $W(\mathsf C_n)$ (or vice versa).

Posted by: L Spice on September 11, 2022 7:32 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

L. Spice wrote:

I had not previously encountered the claim that (I guess complex) irreps and conjugacy classes were in bijection for finite Coxeter groups. Do you know where to look for a discussion of this?

I forget, and I’m having trouble finding a reference. Tantalizingly, on MathOverflow David Jordan wrote:

A related difficulty: you know that there are the same number of conjugacy classes as there are irreducible representations in any finite group, as characters form a basis of class functions. In $S_n$, and also in the Coxeter group examples, one can give explicitly a bijection (so there is both a conjugacy class in $S_n$ and an irrep associated to a Young diagram $\lambda$, for instance). For a general group, there is not a general framework for corresponding conjugacy classes to irreps.

In a comment on this, Alexander Chevrov wrote:

Can you give some details on “and also in the Coxeter group examples, one can give explicitly a bijection?”

But Jordan never replied.

Posted by: John Baez on September 13, 2022 11:27 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Thanks! The general idea makes me think of the Springer correspondence, but a bijection entirely on the level of Weyl groups, and taking in not just Springer representations but all representations, would be so pleasant that it’s a shame not to know about it if it exists.

Posted by: L Spice on September 18, 2022 4:00 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

David Jordan is at Edinburgh, so you can just pop to his office and ask!

Posted by: Tom Leinster on September 13, 2022 11:57 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

We’re planning to

We’re planning to

1) make the talks hybrid on Zoom so that people can participate online:

https://ed-ac-uk.zoom.us/j/82270325098
Meeting ID: 822 7032 5098
Passcode: XXXXXX36

Here the X’s stand for the name of a famous lemma in category theory.

2) record them and eventually make them publicly available on my YouTube channel.

3) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s here.

Posted by: John Baez on September 12, 2022 11:12 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Great! I look forward to them. I am happy they are re taking place in the James Clerk Maxwell building :-)

Posted by: Bruce Bartlett on September 13, 2022 6:26 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Hi Bruce, long time no hear.

Thought you might like this photo of the floor of our building, courtesy of Zoe Wyatt:

Posted by: Tom Leinster on September 13, 2022 8:37 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Hi, Bruce! It’s great to hear from you!

Tom said there’s a hologram of Maxwell in this building. I haven’t seen it yet, but maybe I see its glow in the upper right of Tom’s picture! I’ll try to photograph it.

Posted by: John Baez on September 13, 2022 11:32 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

I think you’re right about the source of the ghostly green glow. It’s in the right place.

Posted by: Tom Leinster on September 13, 2022 11:59 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Good to be back around here.

I suggest a “light field camera” as a gismo to celebrate Maxwell. I see the tech has advanced quite a lot!

Posted by: Bruce Bartlett on September 16, 2022 9:19 AM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Here are the Zoom details:

https://ed-ac-uk.zoom.us/j/82270325098

Meeting ID: 822 7032 5098

The password is the name of the most famous lemma in category theory, followed by the square of the number of letters in that name. For instance, if you think it’s Smith’s lemma then the password is Smith25, because Smith has 5 letters and $5^2 = 25$.

Posted by: Tom Leinster on September 15, 2022 5:33 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

Any clues for TWF fans who aren’t category theorists? I hope not to be guessing lemmata in a frantic attempt to get into the talk!

Posted by: L Spice on September 16, 2022 3:45 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

John has now edited the post above to include the password in plain text anyway. I was being cautious, as I’ll be managing the Zoom calls each week and I wanted to guard against Zoom bombing.

I think it’s not an issue – that even if the password is posted on the web in plain text, I can squash any disruptive behaviour using the various safeguards that Zoom now offers. But early in the pandemic, I was at an online conference that got Zoom-bombed and had to be shut down because of it. So it’s a risk I’m aware of. I’d be happy to hear any advice.

Posted by: Tom Leinster on September 16, 2022 5:03 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

It’s pretty common to make it easy to join math conferences by Zoom; perhaps people got tired of zoom-bombing them or perhaps it worked when mathematicians retaliated en masse by spamming porn sites with videos of Fields medalists.

Nonetheless, I have done what Tom suggested. Anyone who doesn’t know the name of that lemma doesn’t deserve to get in: it indicates not just an ignorance of category theory but a downright resistance to hearing about it (or looking something up).

Posted by: John Baez on September 17, 2022 4:19 PM | Permalink | Reply to this

Re: Seminar on This Week’s Finds

For what it’s worth, I do know the name of that lemma, just wasn’t sure if it was the most famous lemma in category theory (and didn’t want to be stuck guessing other lemmata if I was wrong). It’s quite important to me as an algebraic-group theorist (beyond its importance to schemes in general, it can be used to prove the existence of the Jordan decomposition, for example!), but I didn’t know if there were some other that was perhaps more famous in a purely category-theoretic context.

Posted by: L Spice on September 18, 2022 3:58 PM | Permalink | Reply to this

Re: Seminar on This Weeks Finds

Fortunately for those who aren’t confident about that point, a simple web search (as John implicitly suggested) will point you in the right direction.

Posted by: Mark Meckes on September 19, 2022 5:31 PM | Permalink | Reply to this

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