Semi-invariant of a quiver

An element f ∈ k[Rep(Q,d)] is called an invariant (with respect to GL(d)) if g⋅f = f for any g ∈ GL(d). The set of invariants

${\displaystyle I(Q,\mathbf {d} ):=k[\operatorname {Rep} (Q,\mathbf {d} )]^{GL(\mathbf {d} )}}$

is in general a subalgebra of k[Rep(Q,d)].

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

${\displaystyle I(Q,\mathbf {d} )=k[c_{1},\ldots ,c_{n}]}$

where the c_is are defined, for any A ∈ End(kⁿ), as the coefficients of the characteristic polynomial

${\displaystyle \det(A-t\mathbb {I} )=t^{n}-c_{1}(A)t^{n-1}+\cdots +(-1)^{n}c_{n}(A)}$

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.

Elements which are invariants with respect to the subgroup SL(d) := ∏_{{i ∈ Q0}} SL(d(i)) form a ring, SI(Q,d), with a richer structure called ring of semi-invariants. It decomposes as

${\displaystyle SI(Q,\mathbf {d} )=\bigoplus _{\sigma \in \mathbb {Z} ^{Q_{0}}}SI(Q,\mathbf {d} )_{\sigma }}$

where

${\displaystyle SI(Q,\mathbf {d} )_{\sigma }:=\{f\in k[\operatorname {Rep} (Q,\mathbf {d} )]:g\cdot f=\prod _{i\in Q_{0}}\det(g_{i})^{\sigma _{i}}f,\forall g\in GL(\mathbf {d} )\}.}$

A function belonging to SI(Q,d)_σ is called semi-invariant of weight σ.

Consider the quiver Q:

${\displaystyle 1{\xrightarrow {\ \ \alpha \ }}2}$

Fix d = (n,n). In this case k[Rep(Q,(n,n))] is congruent to the set of square matrices of size n: M(n). The function defined, for any B ∈ M(n), as det^u(B(α)) is a semi-invariant of weight (u,−u) in fact

${\displaystyle (g_{1},g_{2})\cdot {\det }^{u}(B)={\det }^{u}(g_{2}^{-1}Bg_{1})={\det }^{u}(g_{1}){\det }^{-u}(g_{2}){\det }^{u}(B)}$

The ring of semi-invariants equals the polynomial ring generated by det, i.e.

${\displaystyle {\mathsf {SI}}(Q,\mathbf {d} )=k[\det ]}$

Let Q be a Dynkin quiver, d a dimension vector. Let Σ be the set of weights σ such that there exists f_σ ∈ SI(Q,d)_σ non-zero and irreducible. Then the following properties hold true.

i) For every weight σ we have dim_k SI(Q,d)_σ ≤ 1.

ii) All weights in Σ are linearly independent over ℚ.

iii) SI(Q,d) is the polynomial ring generated by the f_σ's, σ ∈ Σ.

Furthermore, we have an interpretation for the generators of this polynomial algebra. Let O be the open orbit, then k[Rep(Q,d)] \ O = Z₁ ∪ ... ∪ Z_t where each Z_i is closed and irreducible. We can assume that the Z_is are arranged in increasing order with respect to the codimension so that the first l have codimension one and Z_i is the zero-set of the irreducible polynomial f₁, then SI(Q,d) = k[f₁, ..., f_l].

In the example above the action of GL(n,n) has an open orbit on M(n) consisting of invertible matrices. Then we immediately recover SI(Q,(n,n)) = k[det].

Skowronski–Weyman provided a geometric characterization of the class of tame quivers (i.e. Dynkin and Euclidean quivers) in terms of semi-invariants.

Let Q be a finite connected quiver. The following are equivalent:

i) Q is either a Dynkin quiver or an Euclidean quiver.

ii) For each dimension vector d, the algebra SI(Q,d) is complete intersection.

iii) For each dimension vector d, the algebra SI(Q,d) is either a polynomial algebra or a hypersurface.

Consider the Euclidean quiver Q:

Pick the dimension vector d = (1,1,1,1,2). An element V ∈ k[Rep(Q,d)] can be identified with a 4-ple (A₁, A₂, A₃, A₄) of matrices in M(1,2). Call D_i,j the function defined on each V as det(A_i,A_j). Such functions generate the ring of semi-invariants:

${\displaystyle SI(Q,\mathbf {d} )={\frac {k[D_{1,2},D_{3,4},D_{1,4},D_{2,3},D_{1,3},D_{2,4}]}{D_{1,2}D_{3,4}+D_{1,4}D_{2,3}-D_{1,3}D_{2,4}}}}$