Semi-invariant of a quiver

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

: $\backslash operatorname\{Rep\}(Q,\backslash mathbf\{d\}):=\backslash \{V\backslash in\; \backslash operatorname\{Rep\}(Q)\; :\; V\_i\; =\; \backslash mathbf\{d\}(i)\backslash \}$

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

: $\backslash bigoplus\_\{\backslash alpha\backslash in\; Q\_1\}\; \backslash operatorname\{Hom\}\_k(k^\{\backslash mathbf\{d\}(s(\backslash alpha))\},\; k^\{\backslash mathbf\{d\}(t(\backslash alpha))\})$

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:

: $\backslash begin\{array\}\{ccc\}$

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:

: $\backslash begin\{array\}\{ccc\}$

GL(\mathbf{d}) \times k[\operatorname{Rep}(Q,\mathbf{d})] & \longrightarrow & k[\operatorname{Rep}(Q,\mathbf{d})]\\

(g, f) & \longmapsto & g\cdot f(-):=f(g^{-1}. -)

\end{array}

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

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

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

Consider the 1-loop quiver Q:

: File:1-loop quiver.svg

For d = (n) the representation space is End(kⁿ) and the action of GL(n) is given by usual conjugation. The invariant ring is

: $I(Q,\backslash mathbf\{d\})=k[c\_1,\backslash ldots,c\_n]$

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

: $\backslash det(A-t\; \backslash mathbb\{I\})=t^n-c\_1(A)t^\{n-1\}+\backslash cdots+(-1)^n\; c\_n(A)$

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) := Π_i∈Q0 SL(d(i)) form a ring, SI(Q,d), with a richer structure called ring of semi-invariants. It decomposes as

: $SI(Q,\backslash mathbf\{d\})=\backslash bigoplus\_\{\backslash sigma\backslash in\; \backslash mathbb\{Z\}^\{Q\_0\}\}\; SI(Q,\backslash mathbf\{d\})\_\{\backslash sigma\}$

where

: $SI(Q,\backslash mathbf\{d\})\_\{\backslash sigma\}:=\; \backslash \{f\backslash in\; k[\backslash operatorname\{Rep\}(Q,\backslash mathbf\{d\})]\; :\; g\backslash cdot\; f\; =\; \backslash prod\_\{i\backslash in\; Q\_0\}\backslash det(g\_i)^\{\backslash sigma\_i\}\; f,\; \backslash forall\; g\backslash in\; GL(\backslash mathbf\{d\})\backslash \}.$

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

Consider the quiver Q:

:$1\; \backslash xrightarrow\{\backslash \; \backslash \; \backslash alpha\backslash \; \}\; 2$

Fix d = (n,n). In this case k[Rep(Q,(n,n))] is congruent to the set of n-by-n matrices: 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

:$(g\_1,g\_2)\backslash cdot\; \{\backslash det\}^u\; (B)\; =\; \{\backslash det\}^u(g\_2^\{-1\}B\; g\_1)=\; \{\backslash det\}^u(g\_1)\; \{\backslash det\}^\{-u\}(g\_2)\; \{\backslash det\}^u(B)$

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

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

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.

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 $\backslash mathbb\{Q\}$.

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 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.

Consider the Euclidean quiver Q:

: File:4-subspace quiver.svg

Pick the dimension vector d = (1,1,1,1,2). An element V ∈ k[Rep(Q,d)] can be identified with a quadruple (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:

: $SI(Q,\backslash mathbf\{d\})=\backslash frac\{k[D\_\{1,2\},D\_\{3,4\},D\_\{1,4\},D\_\{2,3\},D\_\{1,3\},D\_\{2,4\}]\}\{D\_\{1,2\}D\_\{3,4\}+D\_\{1,4\}D\_\{2,3\}-D\_\{1,3\}D\_\{2,4\}\}$

• Wild problem

• {{Citation | last1=Derksen | first1= H.| last2=Weyman | first2=J. | title=Semi-invariants of quivers and saturation for Littlewood–Richardson coefficients. | url=http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00331-3/home.html|mr=1758750 | year=2000 | journal=J. Amer. Math. Soc. | issue=13 | pages=467–479 | volume=3| doi= 10.1090/S0894-0347-00-00331-3| doi-access=free }}
• {{Citation | last1=Sato | first1= M.| last2=Kimura | first2=T. | title=A classification of irreducible prehomogeneous vector spaces and their relative invariants. | url=http://projecteuclid.org/euclid.nmj/1118796150|mr=0430336 | year=1977 | journal=Nagoya Math. J. | pages=1–155 | volume=65| doi= 10.1017/S0027763000017633| doi-access=free }}
• {{Citation | last1=Skowronski | first1= A.| last2=Weyman | first2=J. | title=The algebras of semi-invariants of quivers. |mr=1800533 | year=2000 | journal=Transform. Groups | issue=4 | pages=361–402 | volume=5 | doi=10.1007/bf01234798| s2cid= 120708005}}

Category:Directed graphs

Category:Invariant theory

Category:Representation theory