The graph cohomology ring of the GKM graph of a flag manifold of type
Suppose a compact torus acts on a closed smooth manifold . Under certain conditions, Guillemin and Zara associate to a labeled graph where the labels lie in . They also define the subring of , where is the set of vertices of and we call the “graph cohomology” ring of . It is known that the equivariant cohomology ring of can be described by using combinatorial data of the labeled graph. The main result of this paper is to determine the ring structure of equivariant cohomology ring of a flag manifold of type directly, using combinatorial techniques on the graph . This gives a new computation of the equivariant cohomology ring of a flag manifold of type . (See .)
Key words and phrases:flag manifold, GKM graph, equivariant cohomology
2010 Mathematics Subject Classification:Primary 14M15; Secondary 55N91
Suppose that a closed smooth manifold has an action of a compact torus . If the -action is “nice”, then it is known that we can describe its equivariant cohomology ring by using combinatorial way, namely by using GKM theory. By “nice” we mean that the -action on is GKM, namely the fixed point set of the -action is a finite set and the equivariant one-skeleton of is a union of points or 2-spheres. Then we construct a graph by replacing fixed points with vertices and 2-spheres with edges. This graph equipped with more information is defined by Guillemin and Zara  to be the GKM graph associated with . The object we study in this manuscript, the flag manifold of type with a standard maximal torus action, is GKM.
Interestingly, we can describe the equivariant cohomology ring of by using the data of the GKM graph. The equivariant cohomology ring of is defined to be the ordinary cohomology ring of Borel construction of , namely
where is the total space of the universal principal -bundle . Since acts on , we can consider the diagonal -action on , and its orbit space is called the Borel construction of . In this paper, we treat the equivariant cohomology ring with -coefficients, so we abbreviate as . Since a GKM space satisfies the condition that is torsion free as a module over , the restriction map
is injective. Moreover, is a finite set, hence is isolated, so we have
where . Therefore, we shall regard as a subring of through the map . Guillemin and Zara defined the subring, denoted by , of , by using the combinatorial data of the graph . Then, according to the result of Goresky-Kottwitz-MacPherson in , we have
If is a flag manifold then is isomorphic to without tensoring with (see , for example). Namely, by using data of the GKM graph, we can compute .
Our purpose is to determine the ring structure of for a flag manifold of different Lie types. In the paper , we computed the ring structure of for a flag manifold of classical type directly, namely without using the fact that , and our computation of confirms that is isomorphic to . The goal of this paper is to similarly determine the ring structure of for a flag manifold of type . The ring structure of is given by the following theorem.
Let be the labeled graph associated with the flag manifold of type . Then
where , and resp. is the elementary symmetric polynomial in resp. .
To prove Theorem 1.1 we will use the computation of the graph cohomology of type , because the labeled graph of type contains two labeled subgraphs isomorphic to the labeled graph of type . In fact, the labeled graph of type can be viewed as the total space of a “GKM fiber bundle” in the sense of Guillemin, Sabatini and Zara  where the type subgraph is the fiber, although we do not use this perspective in our computation.
This paper is organized as follows. In Section 2, we recall the labeled graph of a flag manifold. The labeled graph is a graph with weight attached to each edge. In Section 3, we give a different description of the Weyl group of type (the vertex set of the labeled graph) which allows us to describe the labeled graph more concretely. In Section 4, we compute the graph cohomology ring of the labeled graph.
2. The labeled graph
In this section, we recall the definition of the labeled graph for a flag manifold . For an -dimensional torus , let be a basis of , so that can be identified with the polynomial ring . We take an inner product on such that the basis is orthonormal. The following is a simplified version of the definition of GKM graph given in . To distinguish our graph from theirs, we call ours a labeled graph.
(See 2.2, ) Let be a flag manifold of classical type or exceptional type, namely a flag manifold is a homogeneous space where is a compact Lie group and is a maximal torus of . Suppose that be the root system of that type and be the Weyl group. (We regard as a subset of . ) The labeled graph has as a vertex set. Two vertices and in are connected by an edge if and only if there is an element in such that , where is the reflection determined by . The label of the edge , denoted by , is given by .
Guillemin and Zara  introduced the notion of a GKM graph which is a graph equipped with an axial function which satisfies a certain compatibility condition. They defined the graph cohomology for a GKM graph, but this definition does not use the compatibility condition of the axial function. One can also see that the graph cohomology is independent of the signs of the labels. Therefore, we omit the axial function in the data in definition 2.1, and will often disregard the signs of the labels on our labeled graph.
Example 2.3 ( type).
Let be the flag manifold of type , namely where is the maximal torus of . Then, the root system is and the Weyl group is the permutation group on three letters. We use the one-line notation for permutations. We denote by the labeled graph associated with . It is shown in Figure 1.
3. Labeled graph of type
In this section we concretely describe the labeled graph of the flag manifold associated to the compact Lie group of exceptional type . Then the root system of type is known to be
where . Let , and , then
so it is easy to see that has as a subset (but ’s play a role of ’s). We denote by the labeled graph associated with . The graph has the Weyl group of type as the vertex set. Let and be the simple roots, then has a presentation
where is the reflection defined by for . It is a dihedral group of order 12.
We shall give another description of as a set not as a group, which turns out to be convenient for our purpose, and rewrite the condition about edges and labels. Let
Let be the reflection group determined by , namely
because the reflections determined by the roots in are given by
@ It follows from (3.1) that the relations
hold, so we can identify with . We choose a group isomorphism between and as follows;
where is the transposition of and . We note that
where . (Note that is the rotation by angle . ) We record the preceding discussion in a lemma.
Let be the map from to defined as follows; for any in ,
Then is bijective, so that one can identify with as a set through the map .
By using the bijection , we can concretely describe the edge and the label of the graph . The following lemma tells us the way to find the label in the Definition 2.1 more concretely.
For any and in connected by an edge labeled by for some in , namely , one of the following occurs.
Case 1: both and are in . In this case there are distinct integers and in such that and .
Case 2: both and are in so that there are unique elements in such that for . In this case there are distinct integers and in such that and .
Case 3: one of and is in and the other is in . Without loss of generality, we may assume . Then there is an element in such that . In this case there are distinct integers and in such that and , where .
From the definition of , we have for . Since the graph cohomology is independent of the signs of the label, we do not need to be careful of signs of labels of the labeled graph when we consider the graph cohomology ring of the labeled graph. Therefore, in this proof, we sometime disregard signs in front of roots.
First, we prove the lemma for cases 1 and 2. In these cases is in , so for some distinct in .
Case 1. By assumption and are in . Remember that and . Since is a group isomorphism, we have
Therefore . In addition, since , we have , so in order to show , it is enough to show that
To prove this, it is enough to treat the case when or , because and are the generators of . Since and , we can check (3.3) this easily. In fact, for any two roots and , is given by where is the inner product on which we defined in section 2. Therefore we have
Case 2. By assumption for , where and are in . Remember that and . Since ,
Therefore , and we have
Here preserves up to sign because is the rotation by angle , so this completes the proof for case (2).
Case 3. By assumption is in and with . In this case is not in , namely for some . We have
This equation means because and are in . If , then is in and this is contradiction, so there is some such that
up to sign. The label of the edge which connects and is
up to sign.
Using the above lemma, we can redescribe the labeled graph associated with the root system , denoted by , as follows;
The vertex set is .
and are connected by an edge if and only if there are some integers and such that .
The label of the edge is if , and if where .
See Figure 2.
Note that in Figure 2, the parallel edges have the same label.
Let (resp. ) be a subset of defined to be , and (resp. ) be the labeled full subgraph of with (resp. ) as a vertex set. Clearly and are isomorphic as a labeled graph to the labeled graph associated with the root system of type . Guillemin, Sabatini and Zara introduced the notion of a GKM fiber bundle in  (in a GKM fiber bundle, the total space, fiber and base space are all labeled graphs). In fact, it can be seen that is the total space of a GKM fiber bundle, with fibers isomorphic to . In this sense it is natural to expect that the result of type plays a role when we determine the ring structure of below.
4. Graph cohomology ring of
In this section, we state our main result which describes the ring structure of the graph cohomology ring of the labeled graph . We review the definition of the graph cohomology ring of a labeled graph first.
Let be a labeled graph with a label taking values in and let be the vertex set of . We identify with where is the set of all maps from to . Then the graph cohomology of is defined to be the set of all which satisfies the so-called “GKM condition”, namely for any two vertices and connected by an edge , is divisible by . The ring structure on induces a ring structure on .
To become familiar with a graph cohomology ring, we shall remember the graph cohomology ring of .
We define elements of denoted by ’s and ’s as follows;
We regard an element in as a set of six polynomials in such that each polynomial corresponds to some vertex, because has six verticies. So the elements ’s are described as the following figures.
One can easily check that ’s are in . One can also check that the elements ’s and ’s generate as a ring, and
where and (resp. ) is the -elementary symmetric polynomial in (resp. ). (See .)
Now we consider elements in . We set for . For each , we define elements , of by
One can check that ’s, ’s are elements of . We set in , namely , where . In addition, we define an element of by
The restriction of to the subgraph (resp. ) is (resp. ) in . In other words in is a lift of the in . The element comes from some element in the graph cohomology ring of the labeled graph of the base space. In fact, the labeled graph of the base space, denoted by , is described as Figure 6, and Figure 6 describes an element in . The element in is the pullback of the element in Figure 6 by the projection .
Guillemin, Sabatini and Zara  construct module generators of the graph cohomology of the total space over the graph cohomology of the base space, by using module generators of the graph cohomology of fiber. (They consider the graph cohomology with coefficient.)
Let be the labeled graph associated with the root system of type . Then
where , and resp. is the elementary symmetric polynomial in resp. .
The rest of the paper is devoted to the proof of Theorem 4.4.
is generated by , and as a ring.
The idea of the proof of the lemma is same as that of Lemma 3.2 in .
For any homogeneous element of , there is a polynomial in ’s and ’s such that the restrictions of and to the subgraph coincide.
Proof of Claim. We set
Let be the degree of and
and assume that
For any , there is a unique vertex for each such that and are connected by an edge labeled by . (Namely .) Then is divisible by for , so there is a homogeneous polynomial of degree such that
On the other hand, since for any ,
whenever for or . We set
Let be the other vertex in . (Namely, .) Then and are connected by an edge labeled by and there is a unique vertex for each such that and are connected by an edge labeled by