In mathematics, the ring of semi-invariants is a subring of the coordinate ring of a quiver that containing those functions which are invariant under the action of a certain algebraic group, up to a character of that group. Specifically, given a representation of a quiver with vertices Q0, there is a natural action of the algebraic group Πi∈Q0 GL(d(i)) by simultaneous base change. Such an action induces an action on the ring of functions. The functions which are invariant up to a character of the group are called semi-invariants. They form a ring whose structure reflects representation-theoretical properties of the quiver.
Definitions Let Q = (Q0,Q1,s,t) be a quiver. Consider a dimension vector d, that is an element in N {\displaystyle \mathbb {N} } Q0. The set of d-dimensional representations is given by
Rep ( Q , d ) := { V ∈ Rep ( Q ) : V i = d ( i ) } {\displaystyle \operatorname {Rep} (Q,\mathbf {d} ):=\{V\in \operatorname {Rep} (Q):V_{i}=\mathbf {d} (i)\}}
Once fixed bases for each vector space Vi this can be identified with the vector space
⨁ α ∈ Q 1 Hom k ( k d ( s ( α ) ) , k d ( t ( α ) ) ) {\displaystyle \bigoplus _{\alpha \in Q_{1}}\operatorname {Hom} _{k}(k^{\mathbf {d} (s(\alpha ))},k^{\mathbf {d} (t(\alpha ))})}
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:
G L ( d ) × Rep ( Q , d ) ⟶ Rep ( Q , d ) ( ( g i ) , ( V i , V ( α ) ) ) ⟼ ( V i , g t ( α ) ⋅ V ( α ) ⋅ g s ( α ) − 1 ) {\displaystyle {\begin{array}{ccc}GL(\mathbf {d} )\times \operatorname {Rep} (Q,\mathbf {d} )&\longrightarrow &\operatorname {Rep} (Q,\mathbf {d} )\\{\Big (}(g_{i}),(V_{i},V(\alpha )){\Big )}&\longmapsto &(V_{i},g_{t(\alpha )}\cdot V(\alpha )\cdot g_{s(\alpha )}^{-1})\end{array}}}
By definition two modules M,N ∈ Rep(Q,d) are isomorphic if and only if their GL(d)-orbits coincide. We have an induced action on the coordinate ring k[Rep(Q,d)] by defining:
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