THE EQUIVARIANT COHOMOLOGY RINGS OF REGULAR NILPOTENTHESSENBERG VARIETIES IN LIE TYPE A : RESEARCH ANNOUNCEMENTHIRAKU ABE, MEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

Dedicated to the memory of Samuel Gitler (1933-2014)
Abstract. Let n be a fixed positive integer and h : {1, 2, . . . , n} → {1, 2, . . . , n} a Hessenberg function.
The main result of this manuscript is to give a systematic method for producing an explicit presentation
by generators and relations of the equivariant and ordinary cohomology rings (with Q coefficients) of any
regular nilpotent Hessenberg variety Hess(h) in type A. Specifically, we give an explicit algorithm, depending
only on the Hessenberg function h, which produces the n defining relations {fh(j),j }nj=1 in the equivariant
cohomology ring. Our result generalizes known results: for the case h = (2, 3, 4, . . . , n, n), which corresponds
to the Peterson variety P etn , we recover the presentation of HS∗ (P etn ) given previously by Fukukawa,Harada, and Masuda. Moreover, in the case h = (n, n, . . . , n), for which the corresponding regular nilpotent
Hessenberg variety is the full flag variety F`ags(Cn ), we can explicitly relate the generators of our ideal withthose in the usual Borel presentation of the cohomology ring of F`ags(Cn ). The proof of our main theorem
includes an argument that the restriction homomorphism HT∗ (F`ags(Cn )) → HS∗ (Hess(h)) is surjective. In
this research announcement, we briefly recount the context and state our results; we also give a sketch of
our proofs and conclude with a brief discussion of open questions. A manuscript containing more details
and full proofs is forthcoming.

Contents
Introduction
1. Background on Hessenberg varieties
2. Statement of the main theorem
3. Sketch of the proof of the main theorem
4. Open questions
References

1
2
3
6
7
7

Introduction
This paper is a research announcement and is a contribution to the volume dedicated to the illustrious
career of Samuel Gitler. A manuscript containing full details is in preparation [1].
Hessenberg varieties (in type A) are subvarieties of the full flag variety F`ags(Cn ) of nested sequencesof subspaces in Cn . Their geometry and (equivariant) topology have been studied extensively since thelate 1980s [6, 7, 8]. This subject lies at the intersection of, and makes connections between, many researchareas such as: geometric representation theory [26, 14], combinatorics [12, 23], and algebraic geometry andtopology [5, 20]. Hessenberg varieties also arise in the study of the quantum cohomology of the flag variety[22, 25].
The (equivariant) cohomology rings of Hessenberg varieties has been actively studied in recent years.
For instance, Brion and Carrell showed an isomorphism between the equivariant cohomology ring of aregular nilpotent Hessenberg variety with the affine coordinate ring of a certain affine curve [5]. In thespecial case of Peterson varieties P etn (in type A), the second author and Tymoczko provided an explicit
set of generators for HS∗ (P etn ) and also proved a Schubert-calculus-type “Monk formula”, thus giving a
Date: December 2, 2014.
2000 Mathematics Subject Classification. Primary: 55N91, Secondary: 14N15.
Key words and phrases. equivariant cohomology, Hessenberg varieties, flag varieties.
1presentation of HS∗ (P etn ) via generators and relations [16]. Using this Monk formula, Bayegan and thesecond author derived a “Giambelli formula” [3] for HS∗ (P etn ) which then yields a simplification of theoriginal presentation given in [16]. Drellich has generalized the results in [16] and [3] to Peterson varieties inall Lie types [10]. In another direction, descriptions of the equivariant cohomology rings of Springer varieties
and regular nilpotent Hessenberg varieties in type A have been studied by Dewitt and the second author [9],the third author [18], the first and third authors [2], and Bayegan and the second author [4]. However, it has
been an open question to give a general and systematic description of the equivariant cohomology rings of
all regular nilpotent Hessenberg varieties [19, Introduction, page 2], to which our results provide an answer(in Lie type A).
Finally, we mention that, as a stepping stone to our main result, we can additionally prove a fact (cf.
Section 3) which seems to be well-known by experts but for which we did not find an explicit proof in theliterature: namely, that the natural restriction homomorphism HT∗ (F`ags(Cn )) → HS∗ (Hess(h)) is surjective
when Hess(h) is a regular nilpotent Hessenberg variety (of type A).
1. Background on Hessenberg varieties
In this section we briefly recall the terminology required to understand the statements of our main results;
in particular we recall the definition of a regular nilpotent Hessenberg variety, denoted Hess(h), along witha natural S 1 -action on it. In this manuscript we only discuss the Lie type A case (i.e. the GL(n, C) case).
We also record some observations regarding the S 1 -fixed points of Hess(h), which will be important in later
sections.
By the flag variety we mean the homogeneous space GL(n, C)/B which may also be identified withF`ags(Cn ) := {V• = ({0} ⊆ V1 ⊆ V2 ⊆ · · · Vn−1 ⊆ Vn = Cn ) | dimC (Vi ) = i}.
A Hessenberg function is a function h : {1, 2, . . . , n} → {1, 2, . . . , n} satisfying h(i) ≥ i for all 1 ≤ i ≤ n
and h(i + 1) ≥ h(i) for all 1 ≤ i < n. We frequently denote a Hessenberg function by listing its values insequence, h = (h(1), h(2), . . . , h(n) = n). Let N : Cn → Cn be a linear operator. The Hessenberg variety(associated to N and h) Hess(N, h) is defined as the following subvariety of F`ags(Cn ):(1.1)

Hess(N, h) := {V• ∈ F`ags(Cn ) | N Vi ⊆ Vh(i) for all i = 1, . . . , n} ⊆ F`ags(Cn ).

If N is nilpotent, we say Hess(N, h) is a nilpotent Hessenberg variety, and if N is a principal nilpotent
operator then Hess(N, h) is called a regular nilpotent Hessenberg variety. In this manuscript we restrict
to the regular nilpotent case, and as such we denote Hess(N, h) simply as Hess(h) where N is understood tobe the standard principal nilpotent operator, i.e. N has one Jordan block with eigenvalue 0.
Next recall that the following standard torus

(1.2)


g1





T = 













∗
|
g
∈
C
(i
=
1,
2,
.
.
.
n)

i







g2
..

.
gn

n

acts on the flag variety F lags(C ) by left multiplication. However, this T -action does not preserve thesubvariety Hess(h) in general. This problem can be rectified by considering instead the action of the following
circle subgroup S of T , which does preserve Hess(h) ([17, Lemma 5.1]):

(1.3)


g





S := 







g2
..

.
g









 | g ∈ C∗ .





n

(Indeed it can be checked that S −1 N S = gN which implies that S preserves Hess(h).) Recall that the
T -fixed points F lags(Cn )T of the flag variety F lags(Cn ) can be identified with the permutation group Sn
2

on n letters. More concretely, it is straightforward to see that the T -fixed points are the set{(hew(1) i ⊂ hew(1) , ew(2) i ⊂ · · · ⊂ hew(1) , ew(2) , . . . , ew(n) i = Cn ) | w ∈ Sn }where e1 , e2 , . . . , en denote the standard basis of Cn .
It is known that for a regular nilpotent Hessenberg variety Hess(h) we haveHess(h)S = Hess(h) ∩ (F lags(Cn ))T
so we may view Hess(h)S as a subset of Sn .
2. Statement of the main theorem
In this section we state the main result of this paper. We first recall some notation and terminology. Let Ei
denote the subbundle of the trivial vector bundle F lags(Cn ) × Cn over F lags(Cn ) whose fiber at a flag V• isjust Vi . We denote the T -equivariant first Chern class of the line bundle Ei /Ei−1 by τ̃i ∈ HT2 (F lags(Cn )). Let
Ci denote the one dimensional representation of T through the map T → C∗ given by diag(g1 , . . . , gn ) 7→ gi .
In addition we denote the first Chern class of the line bundle ET ×T Ci over BT by ti ∈ H 2 (BT ). It is wellknown that the t1 , . . . , tn generate H ∗ (BT ) as a ring and are algebraically independent, so we may identify
H ∗ (BT ) with the polynomial ring Q[t1 , . . . , tn ] as rings. Furthermore, it is known that HT∗ (F lags(Cn )) is
generated as a ring by the elements τ̃1 , . . . , τ̃n , t1 , . . . , tn . Indeed, by sending xi to τ̃i and the ti to ti weobtain the following isomorphism:HT∗ (F lags(Cn )) ∼= Q[x1 , . . . , xn , t1 , . . . , tn ]/(ei (x1 , . . . , xn ) − ei (t1 , . . . , tn ) | 1 ≤ i ≤ n).
Here the ei denote the degree-i elementary symmetric polynomials in the relevant variables. In particular,
since the odd cohomology of the flag variety F lags(Cn ) vanishes, we additionally obtain the following:(2.1)

H ∗ (F lags(Cn )) ∼= Q[x1 , . . . , xn ]/(ei (x1 , . . . , xn ) | 1 ≤ i ≤ n).

As mentioned in Section 1, in this manuscript we focus on a particular circle subgroup S of the usualmaximal torus T . For this subgroup S, we denote the first Chern class of the line bundle ES ×S C over BS
by t ∈ H 2 (BS), where by C we mean the standard one-dimensional representation of S through the map
S → C∗ given by diag(g, g 2 , . . . , g n ) 7→ g. Analogous to the identification H ∗ (BT ) ∼= Q[t1 , . . . , tn ], we may
also identify H ∗ (BS) with Q[t] as rings.
Consider the restricion homomorphism(2.2)

HT∗ (F`ags(Cn )) → HS∗ (Hess(h)).

Let τi denote the image of τ̃i under (2.2). We next analyze some algebraic relations satisfied by the τi . For
this purpose, we now introduce some polynomials fi,j = fi,j (x1 , . . . , xn , t) ∈ Q[x1 , . . . , xn , t].
First we define
(2.3)

pi :=

i
X

(xk − kt)

(1 ≤ i ≤ n).

k=1

For convenience we also set p0 := 0 by definition. Let (i, j) be a pair of natural numbers satisfying n ≥ i ≥j ≥ 1. These polynomials should be visualized as being associated to the (i, j)-th spot in an n × n matrix.
Note that by assumption on the indices, we only define the fi,j for entries in the lower-triangular part ofthe matrix, i.e. the part at or below the diagonal. The definition of the fi,j is inductive, beginning with thecase when i = j, i.e. the two indices are equal. In this case we make the following definition:(2.4)

fj,j := pj(1 ≤ j ≤ n).

Now we proceed inductively for the rest of the fi,j as follows: for (i, j) with n ≥ i > j ≥ 1 we define:(2.5)
fi,j := fi−1,j−1 + xj − xi − t fi−1,j .
3Again for convenience we define f∗,0 := 0 for any ∗. Informally, we may visualize each fi,j as being associatedto the lower-triangular (i, j)-th entry in an n × n matrix, as follows:


f1,1
0
··· ···0 f2,1 f2,2

0
···




.
.


.
(2.6)
 f3,1 f3,2 f3,3

 .

 ..

fn,1 fn,2 · · ·fn,nTo make the discussion more concrete, we present an explicit example.
Example 1. Suppose n = 4. Then the fi,j have the following form.
fi,i = pi (1 ≤ i ≤ 4)
f2,1 = (x1 − x2 − t)p1
f3,2 = (x1 − x2 − t)p1 + (x2 − x3 − t)p2
f4,3 = (x1 − x2 − t)p1 + (x2 − x3 − t)p2 + (x3 − x4 − t)p3
f3,1 = (x1 − x3 − t)(x1 − x2 − t)p1
f4,2 = (x1 − x3 − t)(x1 − x2 − t)p1 + (x2 − x4 − t){(x1 − x2 − t)p1 + (x2 − x3 − t)p2 }f4,1 = (x1 − x4 − t)(x1 − x3 − t)(x1 − x2 − t)p1
For general n, the polynomials fi,j for each (i, j)-th entry in the matrix (2.6) above can also be expressedin a closed formula in terms of certain polynomials ∆i,j for i ≥ j which are determined inductively, startingon the main diagonal. As for the fi,j , we think of ∆i,j for i ≥ j as being associated to the (i, j)-th box in
an n × n matrix. In what follows, for 0 < k ≤ n − 1, we refer to the lower-triangular matrix entries in the(i, j)-th spots where i − j = k as the k-th lower diagonal. (Equivalently, the k-th lower diagonal is the“usual” diagonal of the lower-left (n − k) × (n − k) submatrix.) The usual diagonal is the 0-th lower diagonalin this terminology. We now define the ∆i,j as follows.
(1) First place the linear polynomial xi − it in the i-th entry along the 0-th lower (i.e. main) diagonal,so ∆i,i := xi − it.
(2) Suppose that ∆i,j for the k − 1-st lower diagonal have already been defined. Let (i, j) be on the k-th
lower diagonal, so i − j = k. Define!
jX∆i,j :=∆i−j+`−1,` (xj − xi − t).
`=1

In words, this means the following. Suppose k = i − j > 0. Then ∆i,j is the product of (xj − xi − t) withthe sum of the entries in the boxes which are in the “diagonal immediately above the (i, j) box” (i.e. theboxes which are in the (k − 1)-st lower diagonal), but we omit any boxes to the right of the (i, j) box (i.e.
in columns j + 1 or higher). Finally, the polynomial fi,j is obtained by taking the sum of the entries in the(i, j)-th box and any boxes “to its left” in the same lower diagonal. More precisely,(2.7)

fi,j =jX∆i−j+k,k .

k=1

We are now ready to state our main result.
Theorem 2.1. Let n be a positive integer and h : {1, 2, . . . , n} → {1, 2, . . . , n} a Hessenberg function. Let
Hess(h) ⊂ F`ags(Cn ) denote the corresponding regular nilpotent Hessenberg variety equipped with the circle
S-action described above. Then the restriction map
HT∗ (F`ags(Cn )) → HS∗ (Hess(h))
is surjective. Moreover, there is an isomorphism of Q[t]-algebrasH ∗ (Hess(h)) ∼= Q[x1 , . . . , xn , t]/I(h)Ssending xi to τi and t to t and we identify H ∗ (BS) = Q[t]. Here the ideal I(h) is defined by(2.8)

I(h) := (fh(j),j | 1 ≤ j ≤ n).
4

We can also describe the ideal I(h) defined in (2.8) as follows. Any Hessenberg function h : {1, 2, . . . , n} →{1, 2, . . . , n} determines a subspace of the vector space M (n×n, C) of matrices as follows: an (i, j)-th entry is
required to be 0 if i > h(j). If we represent a Hessenberg function h by listing its values (h(1), h(2), · · · , h(n)),
then the Hessenberg subspace can be described in words as follows: the first column (starting from the left)
is allowed h(1) non-zero entries (starting from the top), the second column is allowed h(2) non-zero entries,
et cetera. For example, if h = (3, 3, 4, 5, 7, 7, 7) then the Hessenberg subspace is? ? ? ? ? ? ? 







? ? ? ? ? ? ?











? ? ? ? ? ? ?
0 0 ? ? ? ? ? ⊆ M (7 × 7, C).
0 0 0 ? ? ? ? 










0
0
0
0
?
?
?






0 0 0 0 ? ? ?
Then, using the association of the polynomials fi,j with the (i, j)-th entry of the matrix (2.6), the ideal
I(h) can be described as being “generated by the fi,j in the boxes at the bottom of each column in
the Hessenberg space”. For instance, in the h = (3, 3, 4, 5, 7, 7, 7) example above, the generators are{f3,1 , f3,2 , f4,3 , f5,4 , f7,5 , f7,6 , f7,7 }.
Our main result generalizes previous known results.
Remark 1. Consider the special case h = (2, 3, . . . , n, n). In this case the corresponding regular nilpotent
Hessenberg variety has been well-studied and it is called a Peterson variety P etn (of type A). Our
result above is a generalization of the result in [11] which gives a presentation of HS∗ (P etn ). Indeed, for1 ≤ j ≤ n − 1, we obtain from (2.5) and (2.3) that
fj+1,j = fj,j−1 + (xj − xj+1 − t)fj,j= fj,j−1 + (−pj−1 + 2pj − pj+1 − 2t)pj
and since fn,n = pn we haveHS∗ (P etn ) ∼= Q[x1 , . . . , xn , t]/ fj,j−1 + (−pj−1 + 2pj − pj+1 − 2t)pj , pn | 1 ≤ j ≤ n − 1= Q[x1 , . . . , xn , t]/ (−pj−1 + 2pj − pj+1 − 2t)pj , pn | 1 ≤ j ≤ n − 1∼= Q[p1 , . . . , pn−1 , t]/ (−pj−1 + 2pj − pj+1 − 2t)pj | 1 ≤ j ≤ n − 1which agrees with [11]. (Note that we take by convention p0 = pn = 0.)
The main theorem above also immediately yields a computation of the ordinary cohomology ring. Indeed,since the odd degree cohomology groups of Hess(h) vanish [29], by setting t = 0 we obtain the ordinary
cohomology. Let fˇi,j := fi,j (x, t = 0) denote the polynomials in the variables xi obtained by setting t = 0.
A computation then shows that
j
i
X
Y
ˇ
fi,j =xk(xk − x` ).
k=1(For the case i = j we take by convention

`=j+1

Qi

`=j+1 (xk − x` ) = 1.) We have the following.

Corollary 2.2. Let the notation be as above. There is a ring isomorphism∼ Q[x1 , . . . , xn ]/I(h)ˇH ∗ (Hess(h)) =ˇwhere I(h):= fˇh(j),j | 1 ≤ j ≤ n .
Remark 2. Consider the special case h = (n, n, . . . , n). In this case the condition in (1.1) is vacuousand the associated regular nilpotent Hessenberg variety is the full flag variety F`ags(Cn ). In this caseˇˇ n, . . . , n) with the power sums
we can explicitly relate the generators fˇh(j)=n,j of our ideal I(h)= I(n,Pnr
pr (x) = pr (x1 , . . . , xn ) :=k=1 xk , thus relating our presentation with the usual Borel presentation asin (2.1), see e.g. [13]. More explicitly, for r be an integer, 1 ≤ r ≤ n, define
qr (x) = qr (x1 , . . . , xn ) :=

n+1−r
X
k=1
5

xk

n
Y
`=n+2−r(xk − x` ).

ˇ n, . . . , n). The polynomials qr (x)Note that by definition qr (x) = fˇn,n+1−r so these are the generators of I(n,and the power sums pr (x) can then be shown to satisfy the relations(2.9)

qr (x) =r−1
X(−1)i ei (xn+2−r , . . . , xn )pr−i (x).

i=0Remark 3. In the usual Borel presentation of H ∗ (F`ags(Cn )), the ideal I of relations is taken to be generated by the elementary symmetric polynomials. The power sums pr generate this ideal I when we consider
the cohomology with Q coefficients, but this is not true with Z coefficients. Thus our main Theorem 2.1
does not hold with Z coefficients in the case when h = (n, n, . . . , n), suggesting that there is some subtlety
in the relationship between the choice of coefficients and the choice of generators of the ideal I(h).
3. Sketch of the proof of the main theorem
We now sketch the outline of the proof of the main result (Theorem 2.1) above. As a first step, we showthat the elements τi satisfy the relations fh(j),j = fh(j),j (τ1 , . . . , τn , t) = 0. The main technique of this partof the proof is (equivariant) localization, i.e. the injection(3.1)

HS∗ (Hess(h)) → HS∗ (Hess(h)S ).

Specifically, we show that the restriction fh(j),j (w) of each fh(j),j to an S-fixed point w ∈ Hess(h)S is equal
to 0; by the injectivity of (3.1) this then implies that fh(j),j = 0 as desired. This part of the argument israther long and requires a technical inductive argument based on a particular choice of total ordering on
Hess(h)S which refines a certain natural partial order on Hessenberg functions. Once we show fh(j),j = 0
for all j, we obtain a well-defined ring homomorphism which sends xi to τi and t to t:(3.2)

ϕh : Q[x1 , . . . , xn , t]/(fh(j),j | 1 ≤ j ≤ n) → HS∗ (Hess(h)).

We then show that the two sides of (3.2) have identical Hilbert series. This part of the argument is rather
straightforward, following the techniques used in e.g. [11].
The next key step in our proof of Theorem 2.1 relies on the following two key ideas: firstly, we use
our knowledge of the special case where the Hessenberg function h is h = (n, n, . . . , n), for which theassociated regular nilpotent Hessenberg variety is the full flag variety F`ags(Cn ), and secondly, we consider
localizations of the rings in question with respect to R := Q[t]\{0}. For the following, for h = (n, n, . . . , n)
we let H := Hess(h = (n, n, . . . , n)) = F`ags(Cn ) denote the full flag variety and let I denote the associatedideal I(n, n, . . . , n). In this case we know that the map ϕ := ϕ(n,n,...,n) is surjective since the Chern classes τiare known to generate the cohomology ring of F`ags(Cn ). Since the Hilbert series of both sides are identical,
we then know that ϕ is an isomorphism.
The following commutative diagram is crucial for the remainder of the argument.

R−1 ϕ
R−1 Q[x1 , . . . , xn , t]/I−−−−→
R−1 HS∗ (H)−−−−→R−1 HS∗ (HS )
∼
∼
=
=



surj

surj
y
y
y
−1

R ϕh
R−1 Q[x1 , . . . , xn , t]/I(h) −−−−→ R−1 HS∗ (Hess(h)) −−−−→ R−1 HS∗ (Hess(h)S )∼=The horizontal arrows in the right-hand square are isomorphisms by the localization theorem. Since ϕ is
an isomorphism, so is R−1 ϕ. The rightmost and leftmost vertical arrows are easily seen to be surjective,
implying that R−1 ϕh is also surjective. A comparison of Hilbert series shows that R−1 ϕh is an isomorphism.
Finally, to complete the proof we consider the commutative diagram
Q[x1 , . . . , xn , t]/I(h)

inj
y

ϕh

−−−−→

HS∗ (Hess(h))

inj
y

R−1 ϕhR−1 Q[x1 , . . . , xn , t]/I(h) −−−−→ R−1 HS∗ (Hess(h))∼=

for which it is straightforward to see that the vertical arrows are injections. From this it follows that ϕh is
an injection, and once again a comparison of Hilbert series shows that ϕh is in fact an isomorphism.
64. Open questions
We outline a sample of possible directions for future work.
• In [24], Mbirika and Tymoczko suggest a possible presentation of the cohomology rings of regular
nilpotent Hessenberg varieties. Using our presentation, we can show that the Mbirika-Tymoczko ringis not isomorphic to H ∗ (Hess(h)) in the special case of Peterson varieties for n − 1 ≥ 2, i.e. whenh(i) = i + 1, 1 ≤ i < n and n ≥ 3. (However, they do have the same Betti numbers.) In the casen = 4, we have also checked explicitly for the Hessenberg functions h = (2, 4, 4, 4), h = (3, 3, 4, 4),
and h = (3, 4, 4, 4) that the relevant rings are not isomorphic. It would be of interest to understandthe relationship between the two rings in some generality.
• In [15], the last three authors give a presentation of the (equivariant) cohomology rings of Peterson
varieties for general Lie type in a pleasant uniform way, using entries in the Cartan matrix. It would
be interesting to give a similar uniform description of the cohomology rings of regular nilpotent
Hessenberg varieties for all Lie types.
• In the case of the Peterson variety (in type A), a basis for the S-equivariant cohomology ring was
found by the second author and Tymoczko in [16]. In the general regular nilpotent case, and following
ideas of the second author and Tymoczko [17], it would be of interest to construct similar additive
bases for HS∗ (Hess(h)). Additive bases with suitable geometric or combinatorial properties could
lead to an interesting ‘Schubert calculus’ on regular nilpotent Hessenberg varieties.
• Fix a Hessenberg function h and let S : Cn → Cn be a regular semisimple linear operator, i.e.
a diagonalizable operator with distinct eigenvalues. There is a natural Weyl group action on thecohomology ring H ∗ (Hess(S, h)) of the regular semisimple Hessenberg variety corresponding to h(cf. for instance [30, p. 381] and also [28]). Let H ∗ (Hess(S, h))W denote the ring of W -invariants
where W denotes the Weyl group. It turns out that there exists a surjective ring homomorphism
H ∗ (Hess(N, h)) → H ∗ (Hess(S, h))W which is an isomorphism in the special case of the Peterson
variety. (Historically this line of thought goes back to Klyachko’s 1985 paper [21].) In an ongoingproject, we are investigating properties of this ring homomorphism for general Hessenberg functionsh.
References[1] H. Abe, M. Harada, T. Horiguchi and M. Masuda, The equivariant cohomology rings of regular nilpotent Hessenberg
varietiesin Lie type A, in preparation.
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Osaka City University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: hirakuabe@globe.ocn.ne.jp
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario
L8S4K1, Canada
E-mail address: Megumi.Harada@math.mcmaster.ca
URL: http://www.math.mcmaster.ca/∼haradam
Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: d13saR0z06@ex.media.osaka-cu.ac.jp
Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: masuda@sci.osaka-cu.ac.jp8