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...

We suggest a geometrical approach to the semi-invariants of quivers based on Luna's slice theorem and the Luna-Richardson theorem. The locally semi-simple representations are defined in this spirit but turn out to be connected with stable representations in the sense of GIT, Schofield's perpendicular categories, and Ringel's regular representations. As an application of this method we obtain an independent short proof of a theorem of Skowronski and Weyman about semi-invariants of the tame quivers.

Let U(G) be a maximal unipotent subgroup of one of the classical groups Sp(V). Let W be a direct sum of copies of V and its dual V

We present some theorems and algorithms for calculating perpendicular categories and locally semi-simple decompositions. We implemented a computer program TETIVA based on these algorithms and we offer this program for everybody's use.

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