On the complement of the dense orbit for a quiver of type
Let be the directed quiver of type with vertices. For each dimension vector there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the irreducible components in the complement of the dense orbit. Then we compare this result with already existing ones by Knight and Zelevinsky, and by Ringel. Moreover, we compare with the fan associated to the quiver and derive a new formula for the number of orbits using nilpotent classes. In the complement of the dense orbit we determine the irreducible components and their codimension. Finally, we consider several particular examples.
- 1 Introduction
- 2 Description of the Orbits
- 3 Proof of the main theorem
- 4 Further descriptions
- 5 Examples
- 6 Parabolic group actions
The principal aim of this note is to describe the complement of the generic orbit in the representation space of a directed quiver of type with vertices and arrows . For a dimension vector and a representation with we define
Let be defined as the complement in
of the generic orbit
In the complement we define closed (not neccesarily irreducible) varieties
We claim in our main result that all irreducible components of are among the and at most of the occure as irreducible components in .
For the formulation of the main result we need to define a set of pairs
and a subset
consisting of alle elements in satisfying one of the
following properties (where we define )
(ii) and we define to be the minimal index with . Then for all .
(iii) and we define to be the maximal index with . Then for all .
If then we show that is irreducible (Theorem 1.1) and there exists a unique representation whose orbit is dense in . For any irreducible component of there exists a representation whose orbit is dense in this component. Such a representation is called almost generic.
Then we prove the following theorem.
Assume for each .
is the decomposition of into pairwise different irreducible components.
(ii) Each component is the closure of an orbit corresponding to an almost generic representation .
(iii) For any the codimension of is .
(iv) The irreducible components in of codimension are in bijection with the pairs with and for all .
In fact we prove the following stronger results. First of all is irreducible precisely when is in (Prop 3.1, Cor. 3.4 ). Then obviously decomposes into the union of all possible (Lemma 3.3). Next we show that any for is already contained in a union of some other (Prop 3.2). Moreover, we interprete our result in terms of multisegments and nilpotent classes in Section 4.
Note that the techniques are similar to the ones in [BH], our case corresponds to therein, however, the index sets are different, no case follows from the other. With some technical modifications on the index sets one can also handle the case for the remaining values of in a similar way (Section 6).
The paper is organized as follows. In Section 2 we only collect the details we need for the proof of the main result in Section 3. Then we proceed in Section 4 with some further descriptions related to tilting modules and trees, the structure of the fan associated to tilting modules and other combinatorial descriptions. The associated simplicial complex of the fan coincides with the simplicial complex considered by Riedtman and Schofield ([RS]). Then, in Section 5 we consider several examples that are of interest: convex and concave dimension vectors, pure and generic dimension vectors, and symmetric ones. In the last section we compare with the results in [BH] and mention some generalizations without proofs.
We always work over an infinite field , the results here do not depend on the ground field. For finite fields, one needs to modify the definition of a dense orbit slightly: an orbit is dense, if it is dense over the algebraic closure. For a partition we denote by the corresponding nilpotent class defined by
All varieties are considered over the algebraic closure and might be reducible. Also the action of the group should be understood over the algebraic closure. We will always identify isomorphism classes of representations of (with directed orientation) with so-called multisegments defined below. With we denote the number of integers in the interval .
Acknowledgment: This work started during a stay of both authors in Oberwolfach. We are indebted to the Institut for the perfect working conditions. The second author was supported by the DFG priority program SPP 1388 representation theory.
2. Description of the Orbits
In this section we recall some of the various descriptions of the isomorphism classes of representations of with the directed orientation that we need in the proof. Moreover, we recall some well-known facts from the classification of tilting modules and compute the extension groups. We proceed with these descriptions in Section 4. Further related results can be found in [KZ] and in the classical papers [AF] and [AFK].
A multisegment consists of a union of intervalls with , written as (since a multisegments represents an isomorphism class of representations we write ’direct sum’ instead of ’union’). The dimension vector of such a multisegemt is defined as
There are natural bijections between the multisegments of dimensions vector , the isomorphism classes of representations of , and the orbits of the –action on . Moreover, for any dimension vector there exists a unique multisegment corresponding to the dense orbit. This multisegment can be constructed recursively as follows: Define to be the minimum of the entries in . Then we consider and consider the longest interval in with minimal . Then
is nonnegative for some maximal and we proceed with instead of in the same way. Eventually, we obtain a multisegment with at most different direct summands. A second way to obtain this multisegment is described in in [H1], Section 8 (this is a similar, but not the same, construction as in [BHRR]) as follows: consider the unique diagram with vertices in the th column and connect each vertex in the th column and the th row with the vertex in the th column and the th row (if it exists). Roughly one connects all neighboured vertices in the same row. The connected components of this diagram are the direct summands and this diagram represents the multisegment (see Section 5 for examples).
2.2. Extensions and homomorphisms
The category of finite dimensional representations of is a hereditary category and the Euler characteristic , respectively the Hom- and Ext-spaces are
All this follows from direct calculations using a projective or an injective resolution
a) A multisegment has no selfextension precisely when for each
pair of direct summands and of one of the
following conditions hold
(i) , (ii) , (iii) , or (iv) .
b) The multisegment has no selfextension and any other multisegment of dimension vector satisfies .
c) A multisegement satisfies precisely when it contains two segments and with , , and as a direct summand and the complement of equals , where and the complement of equals , where .
d) A multisegmet is almost generic precisely when the direct sum of the pairwise non-isomorphic direct summands satisfies and one of the direct summands with non-trivial Ext–group occurs with multiplicity one in .
a) and the first claim of b) is a direct consequence from the formula
for the extension groups above. The uniqueness in b) follows either
directly from the construction, or since is irreducible (it
can contain at most one dense orbit). To prove c) one uses that
is additive, thus there is at most one non–vanishing
extension group. Finally, to prove d) we note that for
we have by a simple computation of the Euler characteristic
Thus, the stabilizer of
the orbit of and differ by one and then the dimension of the
orbits also differ by one. The closure of the orbit of obviously
contains the orbit of . Now assume is a multisegmet as in the
claim and it is neither generic nor almost generic. Take two direct
summands and of with . We define a new multisegment of the same dimension vector by
deleting and and replacing it by (if we replace it just by ). Then is in the closure of the orbit of and is
almost generic precisely when is already generic. This in turn is
equivalent to the second condition in d), proving one direction of the
Now assume satisfies . Then the closure of the orbit of is of codimension one in the space of all representations of dimensions vector . Thus it is some irreducible component in the complement of the dense orbit. Now we add the remaining segments to so that we obtain . The multisegment is then also a direct summand of , since on can get from by extending only two segments in .
Assume contains two indecomposable direct summands and , both occuring with multiplicity at least two and , then we can again (using extensions) construct an orbit that is not generic and contains in ist closure. Consequently, such an is not almost generic.
The proof also follows directly from Zwara’s result [Z] that the partial order of the Ext-degeneration and the partial order for the geometric degeneration coincide. In the proof above we only used the trivial direction.
2.3. Rank conditions
To any representation of one can associate the ranks of the compositions of the corresponding matrices. Consider . Then we define the rank triangle
Moreover, it is convenient to define the extended rank triangle with and to define whenever or . Obviously, we must have and (using generic matrices) the set
is open and dense in . In fact, the set consists of all representations isomorphic to , since by construction.
We fix a dimension vector and consider any triangle of non-negative integers satisfying . Then
defines an open (possibly empty) subvariety in a closed, non-empty algebraic subvariety (not necessarily irreducible) of . The rank triangles are partially ordered by iff has only non-negative entries. It turns out that some of the are irreducible (we determine which ones) and the rank conditions are very useful for determining the components in the orbit closures. Moreover, one can reconstruct the multisegment from the rank condition , where the orbit of is dense in with minimal: A direct sum is a direct summand of (with maximal possible ) if and only if . Consequently, is empty, if some is negative. Otherwise is dense in .
Conversely, given a multisegment we can easily determine its rank vector as follows
In the particular case of a segment , we obtain just the characteristic function of a triangle as the rank triangle
a) If then . In particular,
contains each and (consisting of the zero matrix) is
contained in each .
b) The variety is irreducible precisely when it is the closure of one –orbit.
c) is non-empty precisely when is a sum of functions of the form and this is equivalent to for all pairs .
Proof. Assertion a) is obvious, since implies for any .
To prove b) we decompose in a disjoint union of –orbits. This is possible, since is –invariant. Thus we obtain a set of multisegments with
Consequently, is the union of a finite number of orbit closures for a finite number of multisegments . We can assume this set is minimal. Thus is irreducible precisely when for some maximal in .
For c), note that is nonempty, precisley when there exists a multisegment with . This is also equivalent to is the sum of rank functions of segments. To prove the last characterization we note that for we obtain . Conversely, if then we define .
3. Proof of the main theorem
We start this section by showing that some of the are irreducible and compute their dimension. Then we show that all for not in are already contained in some union of other ones. This allows a reduction to the case for . Finally we show that is already contained in the union of all .
3.1. Irreducible varieties
Assume , then is irreducible of codimension in .
Proof. We consider the projection of a representation of to the quiver with vertices and its subvarieties in and in defined by . Then is a direct product of with some affine space and is a product of with some affine space. Thus is irreducible precisely when is irreducible. Consequently, it is sufficient to prove the claim for in .
We now assume and for any .
Now we consider a multisegment consisting of and for . A computation of the ranks yields and for all . Thus the equation and for defines an orbit and is the closure of this orbit containing . Consequently it is irreducible, and it coincides with .
Finally, we need to compute the codimension of the orbit closure . For this we compute the dimension of the stabilizer of and of constructed above. To make the computation easier, we delete the common direct summands that contribute with the same dimension to the stabilizer and assume without loss of generality . Then we need to compute
Back to , we decompose it into with maximal and . Then is and
Consequently, the codimension of the orbit of equals and this equals the codimension of . Finally, note that under the reduction from arbitrary to the codimension does not change.
3.2. The reduction process
a) Assume then there exists some with and for all . In particular, . In this case we have an inclusion
b) If with then there exists an with and . In this case we obtain .
c) If with then there exists an with and . In this case we obtain .
Without loss of generality we may assume in the proof.
a) Consider the maps , , and . We consider two cases.
Assume , then or since factors through with .
Assume , then
b) Consider the maps and . Since there exist some with and . Then from follows .
c) This case is opposite to case b).
Proof. The dense orbit is defined by the condition . Thus, the complement satisfies for at least one pair with . Since we finish the proof of the first equality.
To prove the second one we use the proposition above. From Proposition 3.2 a) we obtain and from part b) and c) .
The variety is irreducible precisely when .
Thanks to Proposition 3.1, we only need to prove that
is not irreducible for not in . Take
not in , thus there
exists an with and is
by Proposition 3.2 a) we have . Assume first that We claim that and this is a proper decomposition, none contains the
other. To see the equality, consider any element in . Then or . From each of the inequalities follows . On the other hand, there exists a representation with
and and vice versa, proving also the last claim.
In the second case We construct two different subvarieties that contain and none contains the other. To simplify the arguments, we assume without loss of generality and, using the first case, is the minimal entry of between and . The first variety is just the orbit closure , the second one is defined by and . Using multisegments (or rank conditions) one can show that we obtain at least two irreducible components in this way ( is irreducible, the other variety need not to be). Anyway, we obtain at least two irreducible components.
4. Further descriptions
In this section we proceed with the various descriptions of the irreducible components and the tilting modules started in Section 2. In particular, we use trees and fans to describe the irreducible components and we relate our description to the nilpotent class representations defined in [H2].
4.1. Trees and tilting modules
Let be a –regular tree with one root and leaves, where the leaves are enumerated by . We denote the set of those trees with . With we denote all trees that have precisely on vertex with four neighbours, all other vertices have three neighbours and admit one root and leaves. There is a natural map from to the set of unordered pairs of of trees in by “resolving” the vertex with four neighbours and replacing it by two –regular vertices (see Figure 1).
Figure 1. a tree in and the two associated –regular trees in .
We always draw a tree in the plane and fix the numbering of the leaves from left to right. Two trees are considered to be equal, if the abstract graphs are isomorphic and the numbering of the leaves is preserved under the isomorphism. Then each vertex defines the set of leaves (in fact an interval) above the vertex . This way, each vertex defines a segment . To any tree in or we can associate multisegments as follows. Assume and denote by the vertices in , then we define
to be the union of the multisegments above of . If and and are the two associated –regular trees with unique vertex and (these are the only vertices defining a segment that is not obtained from the other tree), we define
The module for has pairwise nonisomorphic direct summands and the module has pairwise nonisomorphic direct summands.
a) If is a –regular tree, then for . In
particular, for any tree in .
b) If is in then . In particular, is almost generic.
c) If is generic, then there exists a unique so that and have the same indecomposable direct summands. Thus is a direct summand of several copies of .
d) If defines the open dense subset in an irreducible component of , then there exists some tree so that is a direct summand of copies of .
e) For each multisegment with there exists some with and as direct summand of .
Proof. Using Prop. 2.1 a) one sees immediately that the segments in satisfy the vanishing condition for the extension groups. Thus, the only non-vanishing extension group in are in the complement of , that consists of two segments. This proves a) and b) (see also the proof in [H1]).
Part c) also follows from the arguments in loc. cit.:
Each multisegment with non-vanishing extension group can be completed
to one with non-isomorphic direct summands. Finally, any
multisegment with precisely indecomposable summands, all pairwise
non-isomorphic and vanishing extension group is isomorphic to
for some in and the segments determine uniquely.
Using the description of an almost generic multisegment in Prop. 2.1 d) we find two segments and in with non-vanishing extension group. Moreover, we can assume that has non-isomorphic direct summands (otherwise we add further ones to ). Deleting all summands of the form defines a tree , deleting the other direct summand defines a different tree by c). By construction, both trees come from a common in so that and contain the same indecomposable direct summands up to isomorphism. This proves d). The two summands in e) are just , respectively .
4.2. Nilpotent class representations
There is an obvious formula for the number of orbits in . We just count the number of multisegments defined by a non-strict triangle
This function is also called Kostants partition function for type . It is for large not efficiently computable, thus an easier formula is desirable. For we define numbers for any two partitions of and of
The number of multisegments coincides with the sum, taken over all sequences of partitions with a partition of , and are both trivial, of the product of the numbers
Proof. We only mention the idea of the proof, the details can be found in [H2], Section 4.2. First we consider the preprojective algebra of and the cyclic quiver with two vertices together with the natural projection maps
If we denote an element in by , then it satisfies , where and . The projections are defined by
In particular, each element defines a sequence of partitions (defined by the partition of the nilpotent class of ). By definition, and are always the trivial ones corresponding to the zero matrix. It is known (see [KZ] or [P]) that is equidimensional and the irreducible components are in bijection with the –orbits on . Thus is just the number of irreducible components in . Now we determine the irreducible components in a different way using the projection map above. First note, that is the number of irreducible components of
If one fixes the sequence of partitions , then one can compute the number of irreducible components in as the sum of the products taken over all such sequences of partitions with and trivial.
The advantage of the formula above is twofold. First, it is
independent of the orientation of the quiver. We can, for any
orientation of the quiver of type define such a sequence of
partitions. Secondly, the formula in Prop. 4.2 is much more
efficient than just a simple counting.
Note that for the generic representation for a a quiver of type with an arbitrary orientation the corresponding sequence of partitions is just the trivial one (all are zero).
4.3. The fan and the volume
The sets of trees and define a graph that is the dual graph of the simplicial complex of tilting modules defined in [RS]. This simplicial complex has a natural realisation as a fan in the positive quadrant of the real Grothendieck group , where . This fan is described in [H1]. From the fan, one can again determine the irreducible compenents in a simple way.
We start to define the graph . The vertices are just the trees in . The set of edges is . The end points of the edge consists of the two resolutions and