A Slice Theorem for Quivers With an Involution

Abstract

We study the Luna slice theorem in the case of quivers with an involution or supermixed quivers as introduced by Zubkov in [6]. We construct an analogue to the notion of a local quiver setting described in [3]. We use this technique to determine dimension vectors of simple supermixed representations.

Generalized quivers, orthogonal and symplectic representations, and Hitchin–Kobayashi correspondences

“…As usual, we'll often loosen our tongue, and speak of the supermixed quiver Q, omitting the involution and sign map from the notation. The following result is proven in [5]: As is well known, and as the proof above will easily recall, quiver representations can also be characterized as modules of a certain algebra associated with Q, called the path algebra CQ. In fact, the path algebra is a universal example of the semisimple algebras we have used in proof, and indeed it does not come with an associated dimension vector.…”

“…(3) if i = σ(i), Then V i ⊕ V σ(i) comes with the standard orthogonal pairing; (4) if g i = SL or SO, then V i comes with a volume form; (5) if h α = M , S + , S − , L + , or L − , then with respect to the previous conditions, φ α ∈ M (n α ), S + (n α ), S − (n α ), L + (n α ), or L − (n α ), respectively.…”

Group actions on quiver varieties and applications

“…Connections with previous work. Contravariant involutions of a quiver were studied by Derksen and Weyman [9], Zubkov [35,36], Shmelkin [31], Bocklandt [5] and later by Young [34], where Young's motivation comes from physics and as an application he constructs orientifold Donaldson-Thomas invariants. In [34], the action of a contravariant involution is also modified using what is called a 'duality structure', which corresponds to our notion of modifying families.…”

“…Therefore, one can start by studying the actions by subgroups of covariant automorphisms and actions by contravariant involutions. Since contravariant automorphisms of order two are studied in [9,31,35,36,5,34], we restrict our attention to subgroups Σ ⊂ Aut + (Q) of covariant automorphisms until §3.5 (…”

Classification of linear mappings between indefinite inner product spaces

“…The canonical matrices of the system r / / ± r ± are given in the next section. The procedure developed in [20,22] is based on Roiter's quivers with involution [20], which were also studied by Derksen and Weyman [4] (they use the term "symmetric quivers"), Bocklandt [1], Shmelkin [29], and Zubkov [31].…”