Semi-invariant of a quiver explained

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 ∈

N ^Q₀, 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 Π_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

N ^Q₀. The set of d-dimensional representations is given by\operatorname{Rep}(Q, d ): = \{V\in \operatorname{Rep}(Q) : V _{i} = d (i)\}

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

o plus _{\alpha\in Q _{1}} \operatorname{Hom} ^{d (s(\alpha))} _{k(k} , k ^{d(t(\alpha))} )

Such affine variety is endowed with an action of the algebraic group GL(d) := Π_{i∈ Q0} GL(d(i)) by simultaneous base change on each vertex:

\begin{array}{ccc} GL( d ) x \operatorname{Rep}(Q, d ) & \longrightarrow & \operatorname{Rep}(Q, d )\\ ( (g _{i),} (V _{i,} V(\alpha)) ) & \longmapsto & (V _{i,g} _{t(\alpha)} ⋅ V(\alpha) ⋅ g ^{-1} _{s(\alpha)} ) \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:

\begin{array}{ccc} GL( d ) x k[\operatorname{Rep}(Q, d )] & \longrightarrow & k[\operatorname{Rep}(Q, d )]\\ (g, f) & \longmapsto & g ⋅ f( - ): = f(g ^{-1} . - ) \end{array}

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

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

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

I(Q, 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

\det(A - t I ) = t ^{n -1} _{1 (A)t} + … + ( -1 ) ^{n} c _{n(A)}

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) := Π SL(d(i)) form a ring, SI(Q,d), with a richer structure called ring of semi-invariants. It decomposes as

SI(Q, d ) = o plus _{\sigma\in Z ^{Q _{0}}} SI(Q, d ) _{\sigma}

where

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

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

Example

Consider the quiver Q:

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

(g _{1,g} _{2) ⋅} {\det} ^{u} (B) = {\det} ^{-1} _{2} B g _{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.

SI (Q, d ) = k[\det]

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 = 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[''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 a 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:

SI(Q, d ) = (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})

See also

• Wild problem