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In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space Vi to each vertex and a linear map V: V] → V] to each arrow α, where s, t are, respectively, the starting and the ending vertices of α. Given an element d ∈ N {\displaystyle \mathbb {N} } , the set of representations of Q with dim Vi = d for each i has a vector space structure.
It is naturally endowed with an action of the algebraic group Πi∈Q0 GL] by simultaneous base change. Such action induces one on the ring of functions. The ones which are invariants up to a character of the group are called semi-invariants. They form a ring whose structure reflects representation-theoretical properties of the quiver.