Semi-invariant of a quiver

In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space "V""i" to each vertex and a linear map "V"("α"): "V"("s"("α")) → "V"("t"("α")) to each arrow "α", where "s"("α"), "t"("α") are, respectively, the starting and the ending vertices of α. Given an element d ∈ ℕQ0, the set of representations of Q with dim "V""i" = d(i) for each "i" has a vector space structure.
It is naturally endowed with an action of the algebraic group ∏i∈Q0 GL(d("i")) 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.
Definitions.
Let Q = (Q0,Q1,"s","t") be a quiver. Consider a dimension vector d, that is an element in ℕQ0. 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 induced 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("k""n") and the action of GL("n") is given by usual conjugation. The invariant ring is
where the "c""i"s are defined, for any "A" ∈ End("k""n"), 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 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" = "Z"1 ∪ ... ∪ "Z""t" where each "Z""i" is closed and irreducible. We can assume that the "Z""i"s are arranged in increasing order with respect to the codimension so that the first "l" have codimension one and Zi is the zero-set of the irreducible polynomial "f"1, then SI(Q,d) = "k"["f"1, ..., "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 ("A"1, "A"2, "A"3, "A"4) 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: