Semi-Invariants, Root Systems, Associahedra and Nilpotent Groups
by Kiyoshi IGUSA (Brandeis University)
Abstract:
To every Dynkin diagram there is an associated nilpotent group. It also
has a root system consisting of positive and negative roots. The
relationship between the rational cohomology of the nilpotent group and
the root system is classical work of Bott, Kostant and Nomizu.
The theory of semi-invariants, developed recently by Schofield, Derksen and
Weyman, describes a triangulation of the n-1 dimensional simplex with
positive roots at the vertices. Tilting theory gives an algebraic
description of this triangulation. Buan, Marsh, Reineke, Reiten and
Todorov constructed the cluster category extending this to a
triangulation of the n-1 sphere. Our work (joint with Kent Orr, Jerzy
Weymann and Gordana Todorov) extends this theory to the n-sphere
creating a fundamental cell in the cell decomposition of the
corresponding nilmanifold. In the case of An the result is the
Stasheff associahedron. In general it is a subcomplex of the Coxeter
complex of the Dynkin diagram.
At the end of the talk I will show some diagrams to indicate what
happens in the case of the affine diagram Ãn.