THE EQUIVARIANT COHOMOLOGY RINGS OF PETERSONVARIETIES IN ALL LIE TYPESMEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

Abstract. Let G be a complex semisimple linear algebraic group and let P et
be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology
ring HS∗ (P et) of the Peterson variety as a quotient of a polynomial ring by an
ideal J generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group S ∼= C∗ is a certain
subgroup of a maximal torus T of G. Our description of the ideal J uses the
Cartan matrix and is uniform across Lie types. In our arguments we use theMonk formula and Giambelli formula for the equivariant cohomology rings ofPeterson varieties for all Lie types, as obtained in the work of Drellich. Our
result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was
only for Lie type A.

Contents
1. Introduction
2. Background on the Peterson variety
3. Quadratic relations satisfied by the Peterson Schubert classes psi
4. The main theorem
References

1
2
5
8
10

1. Introduction
The main result of this paper is an explicit and efficient presentation of the equivariant cohomology ring of Peterson varieties in all Lie types in terms of generatorsand relations, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. We briefly recall the setting of our results. The Peterson
variety has been much studied due to its relation, for example, to the quantum
cohomology of the flag variety [11, 13]. Thus it is natural to study their topology,
e.g. the structure of their (equivariant) cohomology rings. In Lie type A the Peterson variety P et can be easily described in a concrete manner. Specifically, it isdefined to be the following subvariety of the full flag variety F`ags(Cn ):(1.1)

P et := {V• | N Vi ⊆ Vi+1 for all i = 1, . . . , n − 1}

Date: September 1, 2014.
2000 Mathematics Subject Classification. Primary: 55N91, Secondary: 14N15.
Key words and phrases. equivariant cohomology, Peterson varieties, flag varieties, Monk formula,
Giambelli formula.
1

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MEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

where V• denotes a nested sequence 0 ⊆ V1 ⊆ V2 ⊆ · · · ⊆ Vn−1 ⊆ Vn = Cn ofsubspaces of Cn and dimC Vi = i for all i and N : Cn → Cn denotes a principal nilpotent operator. For a general complex semisimple linear algebraic groupG, there is also a concrete Lie-theoretic description of the Peterson variety as a
subvariety of its flag variety G/B, which we recall in Section 2 below.
In addition, there is a natural subgroup S ∼= C∗ (which is a subgroup of the
maximal torus T of G) which acts on P et (for details see Section 2). The inclusionof P et into G/B induces a natural ring homomorphism(1.2)

HT∗ (G/B) → HS∗ (P et).

The purpose of this manuscript is to give an efficient presentation of the equivariant
cohomology ring HS∗ (P et) in all Lie types, using the recent work of Drellich [4] which
gives a Monk formula and a Giambelli formula for Peterson varieties in all Lie types.
In particular, we are able to obtain a uniform description of the relevant ideal J,
valid for all Lie types, using the Cartan matrix associated to the Lie algebra g of
G. In particular, our analysis shows that the ideal J is generated by quadratics.
Our proof uses Hilbert series and regular sequences, in a similar spirit to previous
work of Fukukawa and the first and third authors [7], which was in turn motivatedby the work of [5, 6] which computes the graph cohomology of the GKM graphs of
the flag varieties of classical type and of G2 .
Brion and Carrell give a different description of the equivariant cohomology ringsof Peterson (and other nilpotent Hessenberg) varieties as the coordinate rings ofcertain affine curves [3, Theorem 3] using techniques from algebraic geometry. Ina related direction, by extending the ideas of the present manusciprt, we give in[1] a uniform description (with explicit generators and relations) of the equivariant
cohomology rings of arbitrary regular nilpotent Hessenberg varieties – a class ofvarieties which includes the Peterson variety and the full flag variety – in Lie typeA.
This paper is organized as follows. We briefly recall the necessary backgroundin Section 2. We derive the relevant quadratic relations in Section 3. In particular,a key computation is contained in Lemma 3.3. The main theorem is proven in
Section 4.
Acknowledgements. The first author is supported in part by an NSERC Discovery Grant (Individual), an Ontario Early Researcher Award, a Canada Research
Chair (Tier 2) Award, and a Japan Society for the Promotion of Science Invitation
Fellowship for Research in Japan (Fellowship ID L-13517). The first author additionally thanks the Osaka City University Advanced Mathematics Institute for its
hospitality, which made this collaboration possible. The third author is partially
supported by a JSPS Grant-in-Aid for Scientific Research 25400095.
2. Background on the Peterson variety
In this section we record some facts about Peterson varieties which we requirein this manuscript.
Let G be a complex semisimple linear algebraic group of rank n. We fix B a
Borel subgroup and T a maximal torus of G such that T ⊆ B ⊆ G. These choices
then determine the following data:• a set of simple roots ∆ = {α1 , . . . , αn },• the associated Weyl group W ,EQUIVARIANT COHOMOLOGY RINGS OF PETERSON VARIETIES

3• the associated Lie algebras t ⊆ b ⊆ g, and• root spaces gα ⊆ g for each root α.
PDefinition. Let Eα be a basis element of the root space gα and let N0 = α∈∆ Eα ,a regular nilpotent operator. In this setting we may define the Peterson variety(associated to g) as
M
P et := {gB ∈ G/B | Ad(g −1 )(N0 ) ∈ b ⊕
gα }.
α∈−∆

As is well-known, the maximal torus T acts on G/B by left multiplication.
This action does not in general preserve the Peterson variety. However, using thehomomorphism φ : T → (C∗ )n defined by t 7→ (α1 (t), . . . , αn (t)) and defining S to
be the connected component of the identity in
φ−1 ({(c, c, . . . , c) | c ∈ C∗ })it can be seen that the restriction of the T -action on G/B to the subgroup S does
preserve P et ([8, Lemma 5.1]).
Next recall that the T -fixed points of G/B are in bijective correspondence with
the Weyl group W of G. Moreover, since the S-fixed points P etS of the Peterson
variety satisfy the relation
P etS = P et ∩ (G/B)T
we may view P etS as a subset of the Weyl group W . Indeed, the fixed point set
P etS may be described concretely as follows. For a subset K of the set ∆ simple
roots, let WK denote the parabolic subgroup generated by K and let wK denotethe longest element of WK . Then it is known [8, Proposition 5.8] thatP etS = {wK | K ⊆ ∆}.
Here and below we always use complex coefficients C for our cohomology ringsand hence omit it from our notation. Let αi : T → C∗ be a homomorphism which
thus determines a complex 1-dimensional representation of T . Let ET ×T C → BT
be the corresponding complex line bundle and by slight abuse of notation we let
αi ∈ H 2 (BT ) also denote the corresponding first Chern class. With this notationin place we haveH ∗ (BT ) = C[α1 , . . . , αn ].
Consider the 1-dimensional representation of the diagonal subgroup {(c, c, . . . , c) :c ∈ C∗ } ⊆ (C∗ )n obtained via the projection (c, c, . . . , c) → c. Composing withthe restriction to S of the above homomorphism φ, we obtain a 1-dimensional
representation of S and an associated line bundle ES ×S C → BS with first Chernclass denoted t ∈ H 2 (BS). With this notation in place we haveH ∗ (BS) = C[t].
Next we recall that the inclusion homomorphism S ,→ T induces a homomorphism
π : H ∗ (BT ) → H ∗ (BS) and from the definition of φ we obtain(2.1)π(αi ) = t(i = 1, 2, . . . , n).

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MEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

We now consider the following commutative diagramM
HT∗ (G/B) −−−−→HT∗ (w)
w∈(G/B)T =W

(2.2)



ρy




πy

M

HS∗ (P et) −−−−→

HS∗ (w)

w∈P etS ⊆W

where all the maps are induced from inclusions of subgroups or inclusions of subspaces. As is well-known, the odd cohomology H odd (G/B) of G/B vanishes. The
same holds for the Peterson variety, i.e. H odd (P et) = 0 [12]. Thus we obtainthat both horizontal maps in (2.2) are injective, and we may identify HT∗ (G/B)(respectively HS∗ (P et)) with its image under these maps. For w ∈ (G/B)T ∼=W(respectively w ∈ P etS ⊆ W ) and f ∈ HT∗ (G/B) (respectively f ∈ HS∗ (P et)) wewill denote by f (w) the restriction of f to the w-th factor HT∗ (w) = H ∗ (BT ) =C[α1 , . . . , αn ] (resp. HS∗ (w) = H ∗ (BS) = C[t]) in the direct products on the righthand sides of (2.2).
For v ∈ W , we let σv denote the corresponding equivariant Schubert class inHT∗ (G/B), and let pv denote its image ρ(σv ) in HS∗ (P et). We call pv a Peterson
Schubert class (associated to v). Let si be the simple reflection corresponding toa simple root αi . The vertices of the Dynkin diagram corresponding to the set ofsimple roots ∆ = {α1 , . . . , αn } is in 1-1 correspondence with ∆. Here and below,
we assume to be fixed an ordering of the simple roots as given in [4, Figure 1](which in turn agrees with the standard ordering in [9, p.58]). With respect tothis ordering, given any subset K = {αa1 , αa2 , . . . , αak } of the simple roots witha1 < a2 < · · · < ak , we define an element vK of W by the formulavK := sa1 sa2 · · · sak .
The Peterson Schubert classes pvK corresponding to the Weyl group elements vK
defined above satisfy the following property.
Proposition 2.1 (Theorem 3.5 in [4]). The Peterson Schubert classes {pvK | K ⊆∆} form a C[t]-module basis for HS∗ (P et).
It follows from Proposition 2.1 that the ρ in (2.2) is surjective. It is well-known
that the equivariant Schubert classes σsi generate HT∗ (G/B) as a C[α1 , . . . , αn ]algebra. From the surjectivity of the homomorphism ρ we immediately obtain thefollowing.
Proposition 2.2. The Peterson Schubert classes psi (i = 1, 2, . . . , n) generateHS∗ (P et) as a C[t]-algebra.
Since the odd cohomology H odd (P et) of the Peterson variety vanishes, we knowthat as a C[t]-module the equivariant cohomology HS∗ (P et) is isomorphic to C[t] ⊗H ∗ (P et). It is known [3, Theorem 3] that
F (H ∗ (P et), s) = (1 + s2 )n(2.3)(1 + s2 )n
1 − s2where the left hand sides denotes the Hilbert series of the graded rings H ∗ (P et)
and HS∗ (P et) with respect to the variable s (of degree 1).
F (HS∗ (P et), s) =

EQUIVARIANT COHOMOLOGY RINGS OF PETERSON VARIETIES5

Fix an integer i with 1 ≤ i ≤ n and a subset K ⊆ ∆. From Proposition 2.1 itfollows that the product psi ·pvK can be written uniquely as a C[t]-linear combinationof the pvJ (for J ⊆ ∆). The so-called Monk’s formula gives a concrete computationof the coefficients in this linear combination.
Theorem 2.3 (Monk’s formula for Peterson varieties for all Lie types, Theorem
4.2 in [4]). The Peterson Schubert classes satisfy the following relation:X
psi · pvK = psi (wK ) · pvK +cJi,K · pvJ
J⊃K|J|=|K|+1

where the coefficient cJi,K are non-negative rational numbers. More specifically, wehave
pv (wJ ).
cJi,K = (psi (wJ ) − psi (wK )) · K
pvJ (wJ )Next we recall the so-called Giambelli’s formula. From Proposition 2.2 it follows that each module generator pvK can be expressed as a polynomial (with C[t]coefficients) in the (ring) generators psi . The Giambelli formula gives a concrete
expression for this polynomial as follows.
Theorem 2.4 (Giambelli’s formula for Peterson varieties for all Lie types, Theorem
5.5 in [4]). Suppose K is a subset of the simple roots ∆. Assume that the Dynkin
diagram corresponding to the subset K is connected. ThenY
|K|!
· pv K =psi|R(vK )|
αi ∈Kwhere |R(vK )| denotes the number of distinct reduced-word expressions for vK .
Remark (cf. Theorem 5.3 in [4]). The connectedness assumption in the above
theorem is not serious, in the following sense. Suppose J, K ⊆ ∆ are two subsets of∆ such that their corresponding Dynkin diagrams are connected. Suppose, however,that J ∪ K has corresponding Dynkin diagram that is not connected. Then pvJ∪K
is simply the product of pvJ and pvK , i.e.
pvJ∪K = pvJ · pvK .
3. Quadratic relations satisfied by the Peterson Schubert classes psi
In this section, we derive certain quadratic relations satisfied by the cohomologydegree-2 Peterson Schubert classes psi (i = 1, 2, . . . , n) by using Monk’s formula(Theorem 2.3), Giambelli’s formula (Theorem 2.4), and Billey’s formula recalled
below. We will then show in the next section that these relations are sufficient todetermine the equivariant cohomology ring HS∗ (P et) of the Peterson variety.
Theorem 3.1 (Billey’s formula, Theorem 4 in [2]). Let w ∈ W and fix a reduced
word decomposition w = sb1 sb2 · · · sbm of w. Set r(i, w) := sb1 sb2 · · · sbi−1 (αbi ). For
an equivariant Schubert class σv for v ∈ W we have the following:
σv (w) =`Y

X

reduced words i=1
v=sbj sbj ···sbj
12`

r(ji , w).

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MEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

We begin with some elementary computations involving Peterson Schubert classes.
First, from Monk’s formula (Theorem 2.3) applied to the case K = {αi } andvK = si we obtain
X j
(3.1)
p2si = psi (si ) · psi +ci · pv{αi ,αj }
j6=i

where
(3.2)

cji = (psi (w{αi ,αj } ) − psi (si )) ·psi (w{αi ,αj } ).
pv{αi ,αj } (w{αi ,αj } )More specifically, since Theorem 3.1 implies that σsi (si ) = αi , from (2.1) we conclude(3.3)

psi (si ) = t.

We record the following.
Lemma 3.2. In (3.1), if si and sj commute, then cji = 0.
Proof. Since si and sj commute, we have w{αi ,αj } = si sj . Moreover from Theorem 3.1 we can computeσsi (w{αi ,αj } ) = σsi (si sj ) = αi .
From (2.1) we get psi (w{αi ,αj } ) = t. Then the equations (3.2), (3.3) imply cji = 0
as desired.
2
In the case when si and sj do not commute, the Dynkin diagram corresponding
to the subset K = {αi , αj } is connected, so Giambelli’s formula (Theorem 2.4)
yields1(3.4)
pv{αi ,αj } = psi psj .
2
In this case, the coefficient appearing in (3.1) can be expressed in terms of the
Cartan matrix.
Lemma 3.3. In (3.1), if si and sj do not commute, thencji = −hαi , αj iwhere hαi , αj i denotes the Cartan integer.
Proof. From (3.2), (3.3), (3.4) we can compute
(3.5)

cji =

2(psi (w{αi ,αj } ) − t)psj (w{αi ,αj } )so it suffices to compute psi (w{αi ,αj } ) and psj (w{αi ,αj } ). In what follows we use
the notation
aij := hαi , αj i (i 6= j), a := aij aji .
With this notation in place, note that by definition of the Cartan integers we havethat the action of the simple reflections sj on the simple roots αj may be expressedas(αi − aij αj(i 6= j),(3.6)
sj (αi ) =−αi(i = j).
In order to prove the lemma, we consider each of the possible cases.

EQUIVARIANT COHOMOLOGY RINGS OF PETERSON VARIETIESdi7(i) In the case when the Dynkin diagram corresponding to {αi , αj } is of the formdthe order of si sj is 3 so we havejw{αi ,αj } = si sj si = sj si sj .

Using Theorem 3.1 and (3.6) in this case we can compute that
σsi (w{αi ,αj } ) = σsi (si sj si ) = αi + si sj (αi ) = aαi − aij αj ,σsj (w{αi ,αj } ) = σsj (sj si sj ) = αj + sj si (αj ) = aαj − aji αi .
Then (2.1) implies
psi (w{αi ,αj } ) = (a − aij )t,psj (w{αi ,αj } ) = (a − aji )t.

Finally (3.5) yields
cji =

(3.7)

2(a − aij − 1)
(a − aji )and substituting a = aij aji , aij = −1 we obtain cji = −aij as desired.
d the order of s s is 4 so we have(ii) In the case di jijw{αi ,αj } = si sj si sj = sj si sj si .
Using the above together with Theorem 3.1 and (3.6) we may computeσsi (w{αi ,αj } ) = σsi (si sj si sj ) = αi + si sj (αi ) = aαi − aij αj ,σsj (w{αi ,αj } ) = σsj (sj si sj si ) = αj + sj si (αj ) = aαj − aji αiwhich is the same as case (i) above. Thus (3.7) also holds in this case and sincea = aij aji = 2 we obtain cji = −aij as required.
d(iii) Finally, in the case dthe element si sj has order 6 and thusijw{αi ,αj } = si sj si sj si sj = sj si sj si sj si .
In this case we have a = 3 so Theorem 3.1 and (3.6) yield that
σsi (w{αi ,αj } ) = σsi (si sj si sj si sj ) = αi + si sj (αi ) + (si sj )2 (αi ) = 4αi − 2aij αj ,σsj (w{αi ,αj } ) = σsj (sj si sj si sj si ) = αj + sj si (αj ) + (sj si )2 (αj ) = 4αj − 2aji αi .
Then from (2.1) we computepsi (w{αi ,αj } ) = (4 − 2aij )t,psj (w{αi ,αj } ) = (4 − 2aji )t.

Equation (3.5) then impliescji =2(3 − 2aij )4 − 2ajiand finally using that aij aji = 3 we get that cji = −aij as desired.
This completes the proof of the lemma.

2

From the above considerations we obtain the following proposition.
Proposition 3.4. In the equivariant cohomology ring HS∗ (P et) of the Peterson
variety, the following quadratic relations are satisfied:nX
j=1

hαi , αj ipsi psj − 2tpsi = 0 (1 ≤ i ≤ n).

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MEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

Proof. If si and sj commute then hαi , αj i = 0, so by Lemma 3.2 the conclusionof Lemma 3.3 holds in this case. From this and (3.4) we see that (3.1) can be
expressed as
1X
hαi , αj ipsi psj .
p2si = t · psi −2
j6=i

Since hαi , αi i = 2 for any i, the above equation can be re-written to be of the form
given in the statement of the proposition.
2
4. The main theorem
Let (hαi , αj i)1≤i,j≤n be the Cartan matrix associated to a rank n semisimple Liealgebra g. Using the coefficients in the Cartan matrix, we define an ideal J in thepolynomial ring C[x1 , . . . , xn , t] as follows:nX
J :=  hαi , αj ixi xj − 2txi | 1 ≤ i ≤ n .
j=1

From Proposition 2.2 and Proposition 3.4 it then follows that the map sending xi
to psi defines a surjective C[t]-algebra homomorphism(4.1)

ϕ : C[x1 , . . . , xn , t]/J  HS∗ (P et).

Here H ∗ (BS) = C[t] and P et denotes the Peterson variety associated to the
Lie algebra g. Since H odd (P et) = 0, as a H ∗ (BS)-module we have HS∗ (P et) ∼=H ∗ (BS) ⊗ H ∗ (P et). Defining the ideal Jˇ as


n
X
(4.2)
Jˇ =  hαi , αj ixi xj | 1 ≤ i ≤ n
j=1

we then also have a surjective ring homomorphism(4.3)
ϕ̌ : C[x1 , . . . , xn ]/Jˇ  H ∗ (P et).
The following is the main theorem of this paper.
Theorem 4.1. The maps ϕ and ϕ̌ of (4.1) and (4.3) are both isomorphisms.
In order to prove Theorem 4.1 we use the theory of regular sequences. For
reference we briefly recall the definition and a key property of regular sequences(cf. [7]).
Definition. Let R be a graded commutative algebra over C and let R+ denote thepositive-degree elements in R. Then a homogeneous sequence θ1 , . . . , θr ∈ R+ isa regular sequence if θk is a non-zero-divisor in the quotient ring R/(θ1 , . . . , θk−1 )for every 1 ≤ k ≤ r. This is equivalent to saying that θ1 , . . . , θr is algebraicallyindependent over C and R is a free C[θ1 , . . . , θr ]-module.
It is a well-known fact (see for instance [14, p.35]) that a homogeneous sequenceθ1 , . . . , θr ∈ R+ is a regular sequence if and only if
r
Y
(4.4)
F (R/(θ1 , . . . , θr ), s) = F (R, s)(1 − sdeg θk )k=1

where F (R/(θ1 , . . . , θr ), s) and F (R, s) denote the Hilbert series of the graded rings
R/(θ1 , . . . , θr ) and R, respectively.

EQUIVARIANT COHOMOLOGY RINGS OF PETERSON VARIETIES

9

The following proposition gives a convenient characterization of regular sequences.
Proposition 4.2. [7, Proposition 5.1] A sequence of positive-degree homogeneous
elements θ1 , . . . , θr in the polynomial ring C[z1 , . . . , zr ] is a regular sequence if andonly if the solution set in Cr of the equations θ1 = 0, . . . , θr = 0 consists only ofthe origin {0}.
We can now prove our main theorem.
Proof of Theorem 4.1. We first claim that if ϕ̌ is an isomorphism then it follows
that ϕ is an isomorphism. To see this, suppose that ϕ̌ is an isomorphism. Thenthe sequencenXθi : =
hαi , αj ixi xj − 2txi for 1 ≤ i ≤ n,j=1

θn+1 : = t
in C[x1 , . . . , xn , t] is regular, where deg(xi ) = deg(t) = 2. Indeed,F (C[x1 , . . . , xn , t]/(θ1 , . . . , θn , θn+1 ), s)
ˇ s)=F (C[x1 , . . . , xn ]/J,=(1 + s2 )n1· (1 − s4 )n (1 − s2 )=(1 − s2 )n+1
=F (C[x1 , . . . , xn , t], s)n+1Y(1 − sdeg θi )i=1

so this follows from (4.4). Note that a subsequence θ1 , . . . , θn of a regular sequenceθ1 , . . . , θn+1 is again a regular sequence, so from (4.4) and (2.3) we obtain
F (C[x1 , . . . , xn , t]/J, s) = F (C[x1 , . . . , xn , t]/(θ1 , . . . , θn ), s)=nY
1(1 − sdeg θi )(1 − s2 )n+1 i=1(1 + s2 )n
1 − s2= F (HS∗ (P et), s)=from which it follows that ϕ is an isomorphism.
Thus it suffices to check that ϕ̌ is an isomorphism. We already know that ϕ̌ issurjective and from equation (2.3) we know that F (H ∗ (P et), s) = (1 + s2 )n . Thusin order to show that ϕ̌ is injective it suffices to show thatˇ s) = (1 + s2 )n .
(4.5)
F (C[x1 , . . . , xn ]/J,
Pn
Note that by (4.4), the equality (4.5) is equivalent to the statement that j=1 hαi , αj ixi xj(1 ≤ i ≤ n) is a regular sequence. Furthermore, by Proposition 4.2 , in order toprove (4.5) it in turn suffices to show that the zero set of the collection of quadratic
equationsnX(4.6)
hαi , αj ixi xj = 0 (1 ≤ i ≤ n),
j=1

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MEGUMI HARADA, TATSUYA HORIGUCHI, AND MIKIYA MASUDA

given by the generators of the ideal Jˇ of (4.2) is {0}, i.e., the equations (4.6) have
only the trivial solution.
Suppose in order to derive a contradiction that (4.6) has a non-trivial solution(b1 , . . . , bn ). In particular, setting I = {i | bi 6= 0}, we have I 6= ∅ and so sincebi 6= 0 for i ∈ I we obtain from (4.6) thatX

hαi , αj ibj = 0(i ∈ I).

j∈I

Since (hαi , αj i)i,j∈I is a |I|×|I| square matrix which is again the Cartan matrix of asemisimple Lie algebra, it must be positive definite [10, section 2.4] and in particular
non-singular. Thus the bi must be 0 for i ∈ I, contradicting the assumption on I.
Thus (4.6) has only the trivial solution, as desired.
2
Remark. Theorem 4.1 is a generalization to all Lie types of the computation given
in [7]. Indeed, the generators of the ideal given in [7] are the same as those given
above, up to a scalar factor of 1/2.
Remark. In fact, Theorem 4.1 holds also with Q coefficients. Indeed, since bothϕ and ϕ̌ can be defined over Z, if the maps become isomorphisms upon tensoringwith C then they are also isomorphisms upon tensoring with Q.

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[13] K. Rietsch, Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc., 16(2):363–392 (electronic), 2003.
[14] R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition 1996, Birkhäuser,
Boston.

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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,
JapanE-mail address: d13saR0z06@ex.media.osaka-cu.ac.jp
Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585,
JapanE-mail address: masuda@sci.osaka-cu.ac.jp