# Shadows of Finite Groups

### Shadows of Finite Groups

**Series:**Mathematics Colloquium

**Location:**Math 501

**Presenter:**Olivier Dudas, Institut de Mathematiques de Jussieu-Paris

Many properties of $\mathrm{GL}_n(q)$ - the general linear group over a finite field with $q$ elements - behave generically with respect to $q$. Among them: the order of the group, its Sylow theory and to some extent its representation theory, all of which are governed by the symmetric group $\mathfrak{S}_n$ acting on the lattice $\mathbb{Z}^n$. This holds more generally for a finite reductive group $G(q)$ such as $\mathrm{SO}_n(q), \mathrm{Sp}_{2n}(q),\ldots, E_8(q)$ where $\mathfrak{S}_n$ is replaced by a reflection group over $\mathbb{Z}$.

During a conference in 1993 on the Greek island of Spetses, Brou\'e, Malle and Michel observed that the reflection group on $\mathbb{Z}$ may be replaced by a \emph{complex} reflection group so that the numerical data (such as the order of $G(q)$ and the dimensions of its representations) seem to still make sense, without being attached to a group. Since then, that mysterious object is called \emph{Spets}, referring to the eponymous island.

Refreshments in Math Commons Room at 3:30pm