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I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so:

When P. Gabriel presented the theorem in a conference [sometime around 1970], he said something like this: "OK, this algebra is of finite representation type; thus, the algebra has a linear quiver and you get $n+1$ indecomposable modules (in the case $A_{n}$). In the case $D_{n}$, you have [an exact expression depending on $n$] indecomposable modules. There are the exceptional cases $E_{6}, E_{7}$, and $E_{8}$". Before he mentioned the number of indecomposable modules in the case of $E_{6}$, somebody in his audience said aloud the correct number. Again, before he mentioned the number of indecomposable modules in the case of $E_{7}$, the same individual in the audience gave the correct figure. This situation prompted P. Gabriel to ask: "Who is answering there?" As it turned out, that person was J. Tits. When P. Gabriel asked him if he already knew the theorem, J. Tits replied thus: "No, I don't know it. I was only giving the number of roots of the Lie algebra associated to the corresponding Dynkin diagram."... What was the relationship between the indecomposable modules over algebras and the roots of the Lie algebras? The answer was not very clear back then...

Even though the previous rendering of the story may be a bit inexact (I heard it this way (approximately) in a talk by Professor J. A. de la Peña in 2017), does anybody know whether an interchange of this sort between P. Gabriel and J. Tits actually took place? If so, would you be so kind as to share with us the most precise version of it whereof you are aware, or failing that, a pointer to the literature wherein one can find a detailed account of this story?

Let me thank you in advance for your attention and knowledgeable replies.

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    $\begingroup$ The story might be true, but some things don't add up. For example, the bijection is between the indecomposable representations of the quiver and the simple roots of the Lie algebra corresponding to the Dynkin diagram (in the $A_n$ type the simple roots are $n$ but the number of roots is $n(n+1)/2$). On the other hand, P. Gabriel certainly knew about Lie algebras (he is one of the collaborators for the SGA 3 volumes) and probably knew J. Tits in Paris and had cited some of his work in some paper. $\endgroup$
    – F Zaldivar
    Commented Oct 14, 2021 at 1:30
  • $\begingroup$ Good night! Thank you very much for your observations. Not only am I interested in finding out whether the event actually happened but I am also looking for a more detailed formulation of the story (and of the underlying connection 'twixt the topics). The anecdote was briefly mentioned towards the end of the talk; it was more or less clear to me on that day Professor de la Peña was not worrying too much about the details while telling it... $\endgroup$
    – José Hdz. Stgo.
    Commented Oct 14, 2021 at 4:38
  • $\begingroup$ ... I have reunited the courage to pose this question because I heard it once again (or a slight variation of it) in a talk by Professor R. Bautista that took place last Friday on the ocassion of the 10th anniversary of the CCM UNAM (however, he mentioned that he might have heard it from A. Dress). $\endgroup$
    – José Hdz. Stgo.
    Commented Oct 14, 2021 at 4:38
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    $\begingroup$ @FZaldivar I believe that the bijection is between all positive roots and indecomposable representations (for $A_{n}$, think about the indecomposable representation assigning a $1$-dimensional representation to each vertex.) The mistake in the anecdote is that the $n+1$ should be replaced with $n(n+1)/2.$ $\endgroup$
    – dhy
    Commented Oct 14, 2021 at 4:54
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    $\begingroup$ The number of indecomposables is indeed the number of positive roots of the corresponding root system ( = number of reflections in the corresponding Weyl group). $\endgroup$
    – Sam Hopkins
    Commented Oct 14, 2021 at 12:54

1 Answer 1

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Here is a recent talk by Ringel (in German): Algebra und Kombinatorik.

The related part is:

Besonderes Aufsehen hat ein Ergebnis erregt, das meist Satz von Gabriel genannt wird: Ein zusammenhängender Köcher ist genau dann darstellungsendlich, wenn er vom Typ A_n, D_n, E_6, E_7, E_8 ist. Und es gibt den Zusatz: In diesem Fall sind die Dimensionsvektoren der unzerlegbaren Darstellungen gerade die postitiven Wurzeln des zugehörigen Dynkin-Diagramms. Gabriel selbst nannte den Satz Satz von Yoshii: Yoshii, ein Schüler von Nakayama hatte ein entsprechendes, aber fehlerhaftes Ergebnis publiziert (er behauptete, dass ein weiterer Fall, nämlich Fall E_7~, darstellungsendlich sei). Parallel zu Gabriel, oder sogar etwas früher, haben auch Bäckström (Stockholm) und Kleiner (Kiev) die darstellungsendlichen Köcher bestimmt. Gabriels Fassung erregte vor allem deswegen großes Aufsehen, weil er den Zusammenhang zu Wurzelsystemen, also zur Lie-Theorie herausstellte - aber dieser Aspekt der Theorie stammte gar nicht von ihm selbst, sondern von Tits, der damals auch in Bonn lehrte. Das Auftreten der Dynkin-Diagramme hat viele fasziniert. Die Konstruktion der entsprechenden Hall-Algebren lieferte später einen direkten Zusammenhang zwischen der Darstellungstheorie von Köchern und den Kac-Moody Lie-Algebren: eine "Kategorifizierung" der Wurzelsysteme.

My translation skills are bad but roughly it is said that the connection to the root systems is not due to Gabriel but due to Tits and both were teaching at that time in Bonn, where this might have happened. Gabriel himself called this result the "theorem of Yoshii" (Yoshii was a student of Nakayama).

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  • $\begingroup$ Thank you very much for sharing this information with me. $\endgroup$
    – José Hdz. Stgo.
    Commented Oct 14, 2021 at 18:02

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