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 ∈ $\mathbb$^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 $\mathbb$^Q₀. The set of d-dimensional representations is given by

: $\operatorname(Q,\mathbf):=\$

Once fixed bases for each vector space ''V''_''i'' this can be identified with the vector space

: $\bigoplus_ \operatorname_k(k^, k^)$

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:

: $\begin GL(\mathbf) \times \operatorname(Q,\mathbf) & \longrightarrow & \operatorname(Q,\mathbf)\\ \Big((g_i), (V_i, V(\alpha))\Big) & \longmapsto & (V_i,g_\cdot V(\alpha)\cdot g_^ ) \end$

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'' ep(Q,d)by defining:

:
\begin
GL(\mathbf) \times k operatorname(Q,\mathbf)& \longrightarrow & k operatorname(Q,\mathbf)\
(g, f) & \longmapsto & g\cdot f(-):=f(g^. -)
\end

Polynomial invariants

An element ''f'' ∈ ''k'' ep(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,\mathbf):=k operatorname(Q,\mathbf)

is in general a subalgebra of ''k'' ep(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

: I(Q,\mathbf)=k _1,\ldots,c_n/math>

where the ''c''_''i''s are defined, for any ''A'' ∈ End(''k''^''n''), as the coefficients of the characteristic polynomial

: $\det(A-t \mathbb)=t^n-c_1(A)t^+\cdots+(-1)^n c_n(A)$

Semi-invariants

In case Q has neither loops nor cycles the variety ''k'' ep(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

: $SI(Q,\mathbf)=\bigoplus_ SI(Q,\mathbf)_$

where

: $SI(Q,\mathbf)_:= \.$

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

Example

Consider the quiver Q:

:$1 \xrightarrow 2$

Fix d = (''n'',''n''). In this case ''k'' ep(''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

:$(g_1,g_2)\cdot ^u (B) = ^u(g_2^B g_1)= ^u(g_1) ^(g_2) ^u(B)$

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

: \mathsf(Q,\mathbf)=k det/math>

Characterization of representation type through semi-invariant theory

For quivers of finite representation-type, that is to say Dynkin quivers, the vector space ''k'' ep(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 $\mathbb$.

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'' ep(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 Z_i 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'' et

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'' ep(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:

: $SI(Q,\mathbf)=\frac$

References

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*{{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

Directed graphs
Invariant theory
Representation theory