On the Null Cone for Quiver Representations Under Reflection Functors

Abstract

Let d be a prehomogeneous dimension vector for a connected quiver Q with the property that c ^ℓ d has a negative entry for some ℓ ∈ ℕ where c is the Coxeter transformation corresponding to an admissible numbering of the vertices Q ₀ of Q. Denote by rep(Q, d) the variety of d-dimensional representations of Q and by Sl(d) the product of the special linear groups at all vertices Q ₀. We show how to find the irreducible components of the null cone of the algebraic quotient rep(Q, d)//Sl(d).

Key Words:

• Quivers
• Representations
• Semi-invariants

2010 Mathematics Subject Classification:

• 14L24
• 16G20

ACKNOWLEDGMENTS

The results presented in this article are part of my doctoral dissertation, written at the University of Bern under the supervision of Professor Ch. Riedtmann. My very special thanks go to her for many fruitful discussions, the guidance, and encouragement all along. I would like to thank the referee for useful remarks. I am grateful to the Swiss National Science Foundation for financial support.

Notes

Communicated by D. Zacharia.