Signed Quivers, Symmetric Quivers and Root Systems

Abstract

Abstract. We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of symmetric quivers, in particular we describe the indecomposable symmetric representations. Their dimensions constitute root systems corresponding to certain symmetrizable generalized Cartan matrices. Introduction.Let Q be a finite quiver, i.e., an oriented graph. Fix the notation as follows. We denote by Q 0 and Q 1 the sets of the vertices and the arrows of Q, respectively. For any arrow ϕ ∈ Q 1 denote by tϕ and hϕ its tail and its head, respectively. A representation V of Q over k consists in defining a vector space V (i) over k, for any i ∈ Q 0 , and a k-linear map V (ϕ) : V (tϕ) → V (hϕ), for any ϕ ∈ Q 1 . The dimension vector dim V is the collection of dim V (i), i ∈ Q 0 . For a fixed dimension α, we may fix V (i) = k αi . Then the set R(Q, α) of the representations of dimension α is converted into the vector spaceHom(k αtϕ , k α hϕ ).A homomorphism H of a representation U of Q to another representation, V is a collection of linear maps H(i), U (i) → V (i) ∈ Q 0 such that for any ϕ ∈ Q 1 holds V (ϕ)H(tϕ) = H(hϕ)U (ϕ). The endomorphisms, automorphisms, and isomorphisms are defined naturally. An easy but very fruitful observation is that the isomorphism classes of representations of Q are the orbits of a reductive group (2) GL(α) = i∈Q0 GL(α i ) acting naturally on R(Q, α), as follows (3) (g(V ))(ϕ) = g(hϕ)V (ϕ)(g(tϕ)) −1 .Futhermore, by the Krull-Schmidt theorem, each representation has a unique, modulo isomorphisms and permutations of summands, decomposition into indecomposable ones. The classification of representations modulo isomporphism is therefore reduced to that for the indecomposable ones. The latter problem is solved for the finite and tame quivers. Moreover, by Kac's Theorem [Kac2] the dimensions of the indecomposable representations are exactly the positive roots of the symmetric Kac-Moody algebra corresponding to the underlying graph of Q. From the point of view of Invariant Theory, the above notation introduces a nice set of reductive linear groups, (GL(α), R(Q, α)). This set is nice because the developed language of quiver representations allows (at least in some cases) to describe orbits, invariants, semi-invariants etc. Another important feature of this set is that the underlying quiver Q determines many natural properties of the groups, for example, if Q is finite, then (GL(α), R(Q, α)) contains finitely many orbits for any α.The above set of groups was extended in [DW] to that of generalized quivers. These can be described as actions of certain reductive groups in the spaces of either orthogonal or symplectic representations of a symmetric quiver S, i.e. a quiver with an involutive anti-automorphism. In this setting one can also classify the orbits in terms of the indecomposable representations; moreover, an important result of [DW, Proposition 2.7...

Invariants of quivers under the action of classical groups

“…• If g v ∈ {GL, O , Sp} for all v ∈ Q 0 , then Q-mixed representations are supermixed representations from [32,34], or equivalently, representations of signed quivers from [26].…”

“…The motivation for these generalizations of quivers from the point of view of the representational theory of algebraic groups was given in [4,26], where symmetric and signed quivers, respectively, of tame and finite type were classified.…”

Group actions on quiver varieties and applications

“…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 (…”

Free relations for matrix invariants in the modular case

“…The notion of supermixed representations of a quiver was introduced by Zubkov [18] and [19]. It is equivalent to the notion of representations of a signed quiver considered by Shmelkin in [14]. Orthogonal and symplectic representations of symmetric quivers studied by Derksen and Weyman in [2] as well as mixed representations of quivers are partial cases of supermixed representations.…”