Colloquium ICJ
# !!!! ANNULÉ !!!! Perpetuants -- A lost treasure

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by

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Europe/Paris

Salle Fokko (UCBL-Braconnier)
### Salle Fokko

#### UCBL-Braconnier

21 av Claude Bernard, 69100 VILLEURBANNE

Description

Perpetuant is one of the several names invented by J. J. Sylvester. It appears in one of the first issues of the American Journal of Mathematics which he had founded a few years before. It is a name which will hardly appear in a mathematical paper of the last 70 years.

We were surprised to find an entry in Wikipedia where it is mentioned that the following beautiful result was conjectured by MacMahon in 1884 and proved by \name{Emil Stroh} in 1890.

THEOREM.

The dimension of the space of perpetuants of degree $k>2$ and weight $g$ is the coefficient of $x^g$ in

${\frac {x^{2^{k-1}-1}}{(1-x^{2})(1-x^{3})\cdots (1-x^{k})}}$

For $k=1$ there is just one perpetuant, of weight 0, and for $k=2$ the number is given by the coefficient of $x^g$ in $x^2/(1-x^2)$.

We will start with a short history of Classical Invariant Theory of binary forms, showing why the formula above comes as a surprise.

We will explain the notion since it still has some mathematical interest, and also \name{Stroh}'s proof which is quite remarkable and in a way very modern. With our method we are even able to exhibit a {\em basis of perpetuants} which is definitely a new result.

(joint work with Claudio Procesi)

Organized by

Christophe Garban