Quivers, Invariants and Quotient Correspondence

TL;DR

This paper explores the geometric and algebraic structures of moduli spaces of fence-type quivers, establishing quotient presentations, extending to parabolic quivers, and connecting these to representation theory via reciprocity algebra.

Contribution

It introduces quotient presentations of quiver varieties, extends the framework to parabolic quivers, and links geometric structures to representation theory through reciprocity algebra.

Findings

Two quotient presentations of quiver varieties provided.

Extension to parabolic quivers and actions of parabolic subgroups.

Connection established between geometry of quivers and representation theory.

Abstract

This paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type. We first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized Gelfand-MacPherson correspondence. Next, we introduce parabolic quivers and extend the above from the actions of reductive groups to the actions of parabolic subgroups. Interestingly, the above geometry finds its natural counterparts in the representation theory as the branching rules and transfer principle in the context of the reciprocity algebra. The last half of the paper establishes this connection.