#
Monday Lecture and Colloquium

**Monday, April 14, 2014**

Freie Universität Berlin

Institut für Informatik

Takustr. 9

14195 Berlin

room 005

** Lecture - 14:15**

### Peter McMullen -
University College London

### Realizations of symmetric sets

- - - **Handout** - - -

*Abstract:*

A * symmetric set * is a pair (**V, G**) consisting of a finite
set ** V ** and a subgroup **G** of its permutations that
acts transitively on ** V**. A * realization * is a mapping
(**V, G**) → (V, G), where G is a
representation of **G** in the orthogonal group **O**(𝔼)
in some euclidean space 𝔼, and the action of G on V ⊆
𝔼 is induced by that of **G** on ** V **.
Different realizations of (**V, G**)
can be combined in various geometric ways, analogous to scaling,
addition and multiplication, leading to a kind of algebra of
realizations. In particular, the * normalized * realizations
(where V lies in a unit sphere) naturally form a compact convex set. An
inner product on realizations yields orthogonality relations
reminiscent of those for representations of groups. However,
representations play only a minor part in what is a very geometric
theory.

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Letzte Aktualisierung:
15.04.2014