Semi invariant of a quiver

Definitions

Let Q = (Q₀,Q₁,s,t) be a quiver. Consider a dimension vector d, that is an element in ℕ^Q₀. The set of d-dimensional representations is given by

Once fixed bases for each vector space V_i this can be identified with the vector space

Such affine variety is endowed with an action of the algebraic group GL(d) := ∏_{i∈ 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.

We have an induces action on the coordinate ring k[Rep(Q,d)] by defining:

Polynomial invariants

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

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

Example

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

Semi-invariants

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

where

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

Example

Consider the quiver Q:

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

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

Characterization of representation type through semi-invariant theory

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 other words, it is a prehomogenous vector space. Sato and Kimura described the ring of semi-invariants in such case.

Sato–Kimura theorem

Let Q be a Dynkin quiver, d a dimension vector. Let ∑ 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 ∑ are linearly independent over ℚ.

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

Example

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.

Skowronski–Weyman theorem

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.

Example

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: