Q0, the set of representations of Q with dim Vi = d(i) for each i has a vector space structure.
It is naturally endowed with an action of the algebraic group Πi∈Q0 GL(d(i)) by simultaneous base change.
They form a ring whose structure reflects representation-theoretical properties of the quiver.
The set of d-dimensional representations is given by Once fixed bases for each vector space Vi this can be identified with the vector space Such affine variety is endowed with an action of the algebraic group GL(d) := Πi∈Q0 GL(d(i)) by simultaneous base change on each vertex: By definition two modules M,N ∈ Rep(Q,d) are isomorphic if and only if their GL(d)-orbits coincide.
Consider the 1-loop quiver Q: For d = (n) the representation space is End(kn) and the action of GL(n) is given by usual conjugation.
The invariant ring is where the cis are defined, for any A ∈ End(kn), as the coefficients of the characteristic polynomial In case Q has neither loops nor cycles, the variety k[Rep(Q,d)] has a unique closed orbit corresponding to the unique d-dimensional semi-simple representation, therefore any invariant function is constant.
The function defined, for any B ∈ M(n), as detu(B(α)) is a semi-invariant of weight (u,−u) in fact The ring of semi-invariants equals the polynomial ring generated by det, i.e. For quivers of finite representation-type, that is to say Dynkin quivers, the vector space k[Rep(Q,d)] admits an open dense orbit.
In the example above the action of GL(n,n) has an open orbit on M(n) consisting of invertible matrices.
ii) For each dimension vector d, the algebra SI(Q,d) is complete intersection.