Atlas: Dimensionvectors of irreducible representations of ${\Bbb Z}_{p}*{\Bbb Z}_{q}$ by Jan Adriaenssens and Raf Bocklandt

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Rings, Modules, and Representations
August 14-18, 2000
Ovidius University
Constanta, Romania

Organizers
Laszlo Marki, Fred van Oystaeyen, Klaus W. Roggenkamp, Mirela Stefanescu

Dimensionvectors of irreducible representations of Z_p*Z_q
by
Jan Adriaenssens and Raf Bocklandt
University of Antwerp
Coauthors: Geert Van de Weyer

To a representation of the free product of finite abelian groups Z_p*Z_q we can associate a quiver Q_pq, a corresponding quiver-representation and a character \theta. As vertex-vectorspaces it has the p+q eigenspaces of the actions of Z_p and Z_q on the n-dimensional vectorspace V; thus we have a dimensionvector \alpha = (a₁, ... , a_p, b₁, ... , b_q), whereas \sum_{i=1, ... , p}a_i = \sum_{j=1, ... , q}b_j = n.

Using results of Le Bruyn and Procesi on local quivers, one obtains that if a dimensionvector is to belong to an irreducible representation, it has to satisfy the conditions

n=1 or for alli=1, ... , p, j=1, ... , q:a_i + b_j <= n.

This restriction on the \alpha allows us to classify the moduli spaces M_\alpha^ss(Q_pq, \theta) which are projective smooth varieties.

Date received: July 28, 2000