Here’s something new: I’ll be living in Edinburgh until January! I’m 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 the James Clerk Maxwell Building, home of the School 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) make lecture notes available on my website.
3) record them and eventually make them publicly available on my YouTube channel.
4) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars: it’s here.
More details soon!
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 Fq, 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, 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 Cln is the associative real algebra freely generated by n anticommuting elements that square to -1. Iwill 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.
Azimuth 
These lectures deserve to be recorded… !
I’ve asked for them to be recorded, so I think it will happen… just not sure yet.
I do wish you’d write GL(n) and U(n) rather than the S versions in the first lecture. Of course there’s the issue that “det^-1” is a rep that doesn’t come from a Young diagram. But two Young diagrams define the same rep of SU(n) if they differ by a bunch of n-columns. There’s only really a proper correspondence if one consider reps of the monoid End(C^n) instead of these groups.
In an unrelated comment, Young diagrams of size n also parametrize nilpotent orbits in GL(n), a connection it is sad to miss out on. (Of course you have an agenda, though, and may have to miss out on it.)
Thanks! I really dislike using the S trick to eliminate the extra representations involving the determinant—or more precisely, I dislike making the main result concern representations of SL and SU instead of GL and U, since that makes the main result seem more technical and unmotivated to beginners, especially since Schur-Weyl duality most naturally involves GL. But I had not spent enough time trying to figure out how to get around these problems! I like the idea of representations of the monoid M(n) = End(Cn). Maybe I’ll talk about representations of all 5 monoids: M, GL, U, SL, and SU.
I’m tempted to talk about various extra uses of Young diagrams, like how they classify abelian p-groups or energy eigenstates of the left-moving massless bosonic free field on the cylinder (which is connected to string theory). But I have to decide how much I can cover while keeping things easy to understand.
For C a category let Rep(C) := functor category from C to Vec. Then we have the one-object category GL_n including into the one-object category M_n including into the many-object category Vec itself. Hence we get restriction maps Rep(Vec) -> Rep(M(n)) -> Rep(GL_n). The irreps of the first (subject to some algebraicity conditions) are the Schur functors, parametrized by all partitions. The irreps of the second are indexed only by partitions of height at most n, or equivalently, decreasing sequences (lambda_1 >= … >= lambda_n >= 0). Then the irreps of the third, by dominant weights (lambda_1 >= … >= lambda_n now potentially negative integers).
You say “Schur-Weyl duality most naturally involves GL” but it’s not true — of these, it most naturally involves M_n.
Something closely related to Schur-Weyl duality is the decomposition of M(a) x M(b) acting on Sym(CC^{ab}), where the M(b) acts on the right by transpose (not inverse). We get a sum over all partitions of height at most min(a,b). If you then pass to the subgroup T^a of the monoid M(a), and ask that it act with weight (1,1,…,1), that picks out (CC^b)^@a, so you still have the GL(b) action but now just the N(T^a)/T^a = S_a action. Now you’re at Schur-Weyl.
If instead you take a = b and extend your functions from Sym(CC^{a^2}) = functions on M(a), to functions on GL(a), then you’re looking at Peter-Weyl.
Thanks! As you know, I have a good highbrow story on Schur functors. But I haven’t updated my lowbrow story accordingly: in this abstract I slipped back to a kind of textbook account that focuses on SL(n) and SU(n).
Your story about M(n) is closer to the highbrow story, and the story about Vec is even closer. So what I should do it start with a good middlebrow story involving the monoid M(n) and then go down to various subgroups and up to the category Vec.
I’m glad you made me think about this; it’s more fun to talk about something when I’m busily trying to assemble a new story about it, rather than trot out an old one.
This is awesome! I would kill to watch a, hopefully introductory, exposition on Young diagrams by you! Here’s to hoping that we get to watch the videos or even better attend via Zoom?
Don’t kill!
In fact we’re planning to
1) make the talks hybrid on Zoom so that people can participate online
2) record them and make them publicly available, and
3) have a Zulip channel on the Category Theory Community Server dedicated to discussion of the seminars.
Thanks. I’ll restrain myself somehow.
BTW while I’m here just wanted to say that you’re a legend. 🙇🏽♂️🙇🏽♂️🙇🏽♂️
You can see the Zoom link on the main blog article here now. I’d do anything to prevent unnecessary deaths!
For an introduction to Young diagrams (there called Young tableaux) is there a better introduction than
Bruce Sagan,
The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions ?
Just asking.
That’s probably a great book, and I should learn all that stuff, but my one or two hours of lectures will talk about the big picture without sinking into the combinatorial algorithms (which I scarcely know).
For me the “tableaux” have numbers in the boxes while the “diagrams” don’t. I’ll mainly be talking about the diagrams.
The terminology Sagan uses is
Ferrers diagram (or shape) (Defn. 2.1.1) for just boxes or dots (no numbers),
Young tableau (Defn. 2.1.3) when appropriate numbers are put in the boxes.
But your terminology seems simpler and clearer, to me at least.
BTW, nlab has a nice page on this:
https://ncatlab.org/nlab/show/Young+diagram
I added a reference to Sagan’s book in my notes!
Here’s how to attend on Zoom:
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.
Remember, meetings are at 3:00 pm UK time on Thursdays from September 22nd to December 1st.
Also, if you join the Category Theory Community Server you can discuss the seminars with me here.
I’ll be posting notes in a while, and also announcing a YouTube link.
Hi, I am very excited to join your seminar in Edinburgh! What are the prerequisites for this seminar? Should I read some stuff until Thursday? I have a bachelors in mathematics (where I took some introductory (algebraic) topology and algebra courses) and a background in programming language theory.
Hi! You can already read lecture notes on the first couple of seminars. That may give you some idea of the prerequisites. But I will probably not be able to cover all the material in those notes, so don’t be intimidated if it seems like a lot. I suggest coming by the first time and seeing if it’s enjoyable.
Is the name of the famous category theory lemma all caps, lowercase or first letter capitalized?
The third!
I think there’s a mistake in your lecture notes. You call the Young symmetrizers central idempotents up to a scalar. But in fact they’re not central! They’re just minimal idempotents. But because the group algebra is a bimodule over itself, the image of a right multiplication operator is still a left submodule. That’s how you get irreducible representations of S_n.
You’re right. I keep getting confused about this point. I’ve fixed the notes.
Thanks, this prevented a lot of embarrassment!