Modulated Quivers, Semi-invariant Pictures and Picture Groups

This book presents an accessible approach to an emerging theory of picture groups. Intended for graduate students and researchers, it explains the connections between several branches of algebra and topology, and demonstrates how they interact. It begins with foundational material on modulated quivers and their representations, cluster categories, and semi-invariants. The text then develops virtual analogues of classical results, allowing dimension vectors with negative coordinates. Finally, it defines the notion of a picture group associated to a semi-invariant picture, also introducing picture spaces which are CW-complexes constructed from semi-invariant pictures. For quivers of type $A_n$ the key theorem draws on K-theory and states that the associated picture space is a $K(G(A_n) , 1)$ connected CW-complex for the corresponding group $G(A_n)$ associated with the same quiver.