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Category: personal

Christine Bessenrodt (1958-2022)

Published February 1, 2022 by lieven

We were pretty close once. It is a shock to read about her passing on Twitter.

I met Christine in the late 80ties at some representation meeting in Oberwolfach. Christine was a regular at such meetings, being in the Michler-clique from Essen. I don’t recall why I was invited.

We had a fun time, and had a sneaky plan to be invited more regularly to the same conferences. All we had to do was to prove a good result, together…

Easier said than done. Christine’s field was modular representation theory (over $\overline{\mathbb{F}_p}$), and I was interested in the geometry of quiver moduli-spaces (over $\mathbb{C}$).

The next year I ran a post-graduate course on rationality problems and emailed the notes weekly to Christine. After all, results of Lenstra, Colliot-Thelene and Sansuc reduced the problem of (stable) rationality of algebraic tori to integral representation theory, a half-way meeting ground for both of us.

Around that time, our youngest daughter was born, and Christine graciously accepted to be her godmother.
Over the next years, she and Klaus visited us in Antwerp and we week-ended in their brand new house in the outskirts of Duisburg, close to a lake.



Christine in Oberwolfach

Christine and I were working on the rationality problem for matrix invariants.

A sufficiently general $n \times n$ matrix is diagonalisable and is therefore determined, up to conjugacy, by $n$ free parameters (the eigenvalues), so the corresponding quotient-variety is rational.

Now consider couples of $n \times n$ matrices under simultaneous conjugation. In the mid 80ties, Formanek proved rationality for $n=3$ and $n=4$, by using the theory of algebraic tori, and that was about all that was known.

We were able to reduce the question of stable rationality for $n=5$ and $n=7$ to modular representation theory, after which Christine performed her magic to crack the problem. The paper appeared a year later in Inventiones.

Thirty years later, it is still the best result on rationality of matrix invariants.

So, we had our joint result, but its intended use never happened, and our contacts gradually watered down as our mathematical interests again diverged.

My thoughts go out to Klaus and all her loved ones.

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phase transition

Published May 7, 2021 by lieven

Today, our youngest daughter (aka PD2 on this blog) gave birth to a little boy, Gust.

I’m in transition, trying to adjust to this new phase in our lives.



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Kasha-eating dragons

Published March 15, 2021 by lieven

This semester I’m teaching a first course in representation theory. On campus, IRL! It’s a bit strange, using a big lecture room for a handful of students, everyone wearing masks, keeping distances, etc.



So far, this is their only course on campus, so it has primarily a social function. The breaks in between are infinitely more important than the lectures themselves. I’d guess breaks take up more than one third of the four hours scheduled.

At first, I hoped to make groups and their representations relevant by connecting to the crisis at hand, whence the the symmetries of Covid-19 post, and the Geometry of Viruses series of posts.

Not a great idea. I guess most of us are by now over-saturated with Corona-related news, and if students are allowed to come to campus just one afternoon per week, the last thing they want to hear about is, right, Covid.

So I need to change tactics. By now we’ve reached the computation of character tables, and googling around I found this MathOverflow-topic: Fun applications of representations of finite groups.

The highest rated answer, by Vladimir Dotsenko, suggests a problem attributed to Kirillov:

An example from Kirillov’s book on representation theory: write numbers 1,2,3,4,5,6 on the faces of a cube, and keep replacing (simultaneously) each number by the average of its neighbours. Describe (approximately) the numbers on the faces after many iterations.

A bit further down the list, the Lecture notes on representation theory by Vera Serganova are mentioned. They start off with a variation of Kirillov’s question (and an extension of it to the dodecahedron):

Hungry knights. There are n hungry knights at a round table. Each of them has a plate with certain amount of food. Instead of eating every minute each knight takes one half of his neighbors servings. They start at 10 in the evening. What can you tell about food distribution in the morning?

Breakfast at Mars. It is well known that marsians have four arms, a standard family has 6 persons and a breakfast table has a form of a cube with each person occupying a face on a cube. Do the analog of round table problem for the family of marsians.

Supper at Venus. They have five arms there, 12 persons in a family and sit on the faces of a dodecahedron (a regular polyhedron whose faces are pentagons).

Perhaps the nicest exposition of the problem (and its solution!) is in the paper Dragons eating kasha by Tanya Khovanova.

Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Every minute every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of
kasha, each arm manages to steal one-fourth of what is in the bowl. Thus each
dragon steals one-fourth of the kasha of each of his neighbors, while at the same
time all of his own kasha is stolen. Given the initial amounts of kasha in every
bowl, what is the asymptotic behavior of the amounts of kasha?

I can give them quick hints to reach the solution:

  • the amounts of kasha on each face gives a vector in $\mathbb{R}^6$, which is an $S_4$-representation,
  • calculate the character of this kasha-representation,
  • use the character table of $S_4$ to decompose the representation into irreducibles,
  • identify each of the irreducible factors as instances in the kasha-representation,
  • check that the food-grabbing operation is an $S_4$-morphism,
  • remember Schur’s lemma, and compute the scaling factors on each irreducible component,
  • conclude!

But, I can never explain it better than Khovanova’s treatment of the kasha-eating dragons problem.

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