## September 16, 2022

### Young Diagrams and Classical Groups

#### Posted by John Baez

Young diagrams can be used to classify an enormous number of things. My first one or two This Week’s Finds seminars will be on Young diagrams and classical groups. Here are some lecture notes:

I probably won’t cover all this material in the seminar. The most important part is the stuff up to and including the classification of irreducible representations of the “classical monoid” $\mathrm{End}(\mathbb{C}^n)$. (People don’t talk about classical monoids, but they should.)

Just as a reminder: my talks will be on Thursdays at 3:00 pm UK time in Room 6206 of the James Clerk Maxwell Building at the University of Edinburgh. The first will be on September 22nd, and the last on December 1st.

If you’re actually in town, there’s a tea on the fifth floor that starts 15 minutes before my talk. If you’re not, you can 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.

My talks should eventually show up on my YouTube channel.

Also, you can discuss them on the Category Theory Community Server if you go here.

Posted at September 16, 2022 5:49 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3418

### Re: Young Diagrams and Classical Groups

Should that last link be this? The one you gave is to the ACT@UCR Zulip chat, not the Category Theory one.

Posted by: Tom Leinster on September 16, 2022 6:58 PM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

Whoops! I fixed it, thanks.

This is what comes of simultaneously posting about the same things both in the ACT@UCR Zulip and the category theory Zulip.

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

### Re: Young Diagrams and Classical Groups

I’m excited that these are getting up and running! I’ve been anticipating these seminars for so long, and it’s genuinely exciting that they’re actually about to start.

Something tells me that the audience in the room will be far outnumbered by the hordes encapsulated in the little Zoom participants window in front of me.

Posted by: Tom Leinster on September 16, 2022 7:02 PM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

Still no nilpotent orbits, huh? Let’s see if I can get you to take an interest.

Story #1. Fix a partition $\lambda$ of $n$ and let $O$ be the closure of the space of nilpotent matrices whose Jordan blocks are the sizes of the rows of $\lambda$. Define the orbital scheme $O \cap b$ to be its intersection with the upper triangular matrices. Theorem: the smooth part of $latex O$ is (holomorphic) symplectic, and the orbital scheme is Lagrangian therein.

If we consider the NW $k\times k$ submatrix of $M \in O \cap b$, for each $k$, we get another strictly upper triangular matrix… hence nilpotent. This chain of nilpotents gives a chain of partitions, hence, a standard Young tableau of shape $\lambda$.

This function $O \cap b \to SYT_\lambda$ is “constructible”, and consequently, is constant on a dense open set of each irreducible component of $O\cap b$. So we get a function from the set of components to $latex SYT_\lambda$. Theorem (Spaltenstein): that map is bijective.

Okay, how do we define an action of $S_n$ on the top Borel-Moore homology of this space, now that we know it has the right dimension to be an $S_n$ irrep? Of course it’s enough to determine the action of the simple reflections.

Given a component $C$, sweep it out using the $i$th $SL_2$ along the diagonal. The sweep will be one dimension larger, or stay $C$. In the first case, intersect the sweep with $b$ again, at which point, it will be a schemy union of other components $C'$. Put the multiplicities of those components into the $C$ column of $r_i$. There’s a little more fiddling (like subtracting twice the identity matrix) but this is the geometric origin of the representation.

In particular this produces each irrep of $S_n$ exactly once, each with a canonical basis, in which $r_i$ doesn’t act positively but with predictable signs.

Story #2. A complex vector space with a nilpotent operator is almost the same as a $C[[z]]$-module. Fix three such modules $M_\lambda,M_\nu,M_\mu$, corresponding to the partitions $\lambda,\nu,\mu$. Consider the space of short exact sequences $0 \to M_\lambda \to M_\nu \to M_\mu \to 0$. This time, the components of the space don’t count SYT but instead count Littlewood-Richardson tableaux, i.e. their number is $c_{\lambda\mu}^\nu$.

(Of course it’s interesting to replace $C[[z]]$ with the $p$-adics and to count SES instead of take their top homology.)

Posted by: Allen Knutson on September 17, 2022 2:40 PM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

This looks cool but too complicated for my lectures. I only want to explain stuff that all grad students can easily follow. Maybe I will eventually take my lecture notes and build them into something more mind-blowing, but for now I just need to keep cranking them out at the rate of one a week.

Maybe someone can check to see if this is correct:

While we have studied representations on finite-dimensional vector spaces over $\mathbb{C}$, most of the purely algebraic results hold for any field of characteristic zero! Fields with nonzero characteristic behave very differently, and in fact the irreducible representations of $S_n$ still haven’t been classified over finite fields. But the items here with check marks hold if we replace $\mathbb{C}$ with any field of characteristic zero:

✔ Irreps of $S_n$ are classified by Young diagrams with $n$ boxes.

✔ Polynomial irreps of $\mathrm{End}(\mathbb{C}^N)$ are classified by Young diagrams with $\le N$ rows.

✔ Polynomial irreps of $GL(N,\mathbb{C})$ are classified by Young diagrams with $\le N$ rows.

✔ Algebraic irreps of $GL(N,\mathbb{C})$ are classified by pairs consisting of a Young diagram with $< \;N$ rows and an integer.

✔ Algebraic irreps of $SL(N,\mathbb{C})$ are classified by Young diagrams with $< \;N$ rows.

• Analytic irreps of $SL(N,\mathbb{C})$ are classified by Young diagrams with $< \;N$ rows.

• Analytic irreps of $GL(N,\mathbb{C})$ are classified by pairs consisting of a Young diagram with $< \;N$ rows and an integer.

• Real-algebraic irreps of $\mathrm{U}(N)$ (on complex vector spaces) are classified by pairs consisting of a Young diagram with $< \;N$ rows and an integer.

• Continuous unitary irreps of $\mathrm{U}(N)$ are classified by pairs consisting of a Young diagram with $< \;N$ rows and an integer.

• Real-algebraic irreps (on complex vector spaces) of $SU(N)$ are classified by Young diagram with $< \;N$ rows.

• Continuous unitary irreps of $SU(N)$ are classified by Young diagram with $< \;N$ rows.

There’s probably some way to generalize the algebraic results about $\mathrm{U}(n)$ and $SU(n)$ to other fields of characteristic zero, and I’d be happy to hear about it, but I probably don’t want to talk about it in my lectures notes.

Posted by: John Baez on September 17, 2022 5:01 PM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

Characteristic p is fine for p>n, i.e. you can take the irreps that I defined over ZZ and tensor them with ZZ/p, obtaining exactly the irreps over ZZ/p. Here’s Kleschev’s 2014 ICM talk entitled “Modular Representation Theory of Symmetric Groups” about smaller p.

In your statements about U(N) and SU(N) you can weaken real-algebraic to continuous, as you said, but you can weaken further to measurable if you’d like. (I.e. you pick up no new reps.) This measurable-to-real-algebraic enhancement works with any compact connected Lie group.

Posted by: Allen Knutson on September 17, 2022 9:18 PM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

Hooray! The first This Week’s Finds Seminar is now on YouTube.

In the spirit of sharing, and in case anyone cares, here are the slides for my little introduction.

Posted by: Tom Leinster on September 27, 2022 10:32 PM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

I scrolled through – those aren’t your arms, are they, Tom? The ones with all the tats? (I didn’t think you had tattoos.)

Posted by: Todd Trimble on October 3, 2022 2:19 AM | Permalink | Reply to this

### Re: Young Diagrams and Classical Groups

No, they’re not my arms. It’s just a picture off the web, onto which I unconvincingly pasted a This Week’s Finds screenshot.

Posted by: Tom Leinster on October 3, 2022 3:20 PM | Permalink | Reply to this

Post a New Comment