ON REIDEMEISTER TORSION OF FLAG MANIFOLDS OF COMPACT SEMISIMPLE LIE GROUPS

ON REIDEMEISTER TORSION OF FLAG MANIFOLDS OF

COMPACT SEMISIMPLE LIE GROUPS

HABIB BASBAYDAR & CENAP OZEL & YASAR SOZEN & EROL YILMAZ

Abstract. In this paper we calculate Reidemeister torsion of flag manifoldK/T of compact semi-simple Lie group K = SU(n + 1) using Reidemeister

torsion formula and Schubert calculus, where T is maximal torus of K. We findthat this number is 1. Also we explicitly calculate ring structure of integral

cohomology algebra of flag manifold K/T of compact semi-simple Lie group

K = SU(n + 1).

1. Introduction

Reidemeister torsion is a topological invariant and was introduced by Reidemeis-ter in 1935. Up to PL equivalence, he classified the lens spaces S3/Γ, where Γ is afinite cyclic group of fixed point free orthogonal transformations [33]. In [12], Franzextended the Reidemeister torsion and classified the higher dimensional lens spacesS2n+1/Γ, where Γ is a cyclic group acting freely and isometrically on the sphereS2n+1.

In 1964, the results of Reidemeister and Franz were extended by de Rham tospaces of constant curvature +1 [11]. Kirby and Siebenmann proved the topologicalinvariance of the Reidemeister torsion for manifolds in 1969 [22]. Chapman provedfor arbitrary simplicial complexes [7, 8]. Hence, the classification of lens spaces ofReidemeister and Franz was actually topological (i.e. up to homeomorphism).

Using the Reidemeister torsion, Milnor disproved Hauptvermutung in 1961. Heconstructed two homeomorphic but combinatorially distinct finite simplicial com-plexes. He identified in 1962 the Reidemeister torsion with Alexander polynomialwhich plays an important role in knot theory and links [29,31].

In paper [34], Y. Sozen explained the claim mentioned in [41, p. 187] aboutthe relation between a symplectic chain complex with ω−compatible bases and theReidemeister torsion of it (Theorem 2.7). Moreover, he applied Theorem 2.7 to thechain-complex

0→ C2(Σg; Ad%)∂2⊗id→ C1(Σg; Ad%)

∂1⊗id→ C0(Σg; Ad%)→ 0,

where Σg is a compact Riemann surface of genus g > 1, where ∂ is the usualboundary operator, and where % : π1(Σg) → PSL2(R) is a discrete and faithfulrepresentation of the fundamental group π1(Σg) of Σg [34].

In the article [34] , oriented closed connected 2m−manifolds (m ≥ 1) are consid-ered and he proved the following formula for computing the Reidemeister torsionof them. Namely,

1991 Mathematics Subject Classification. 57Q10;16Z05;14M15;14N15;22E67.Key words and phrases. Reidemeister torsion of manifolds, Flag manifolds, Weyl groups, Schu-

bert calculus, Groebner-Shirshov bases; Graded inverse lexicographic order.

1

2 HABIB BASBAYDAR & CENAP OZEL & YASAR SOZEN & EROL YILMAZ

Theorem 1.1. Let M be an oriented closed connected 2m−manifold (m ≥ 1). Forp = 0, . . . , 2m, let hp be a basis of Hp(M). Then, the Reidemeister torsion of Msatisfies the following formula:∣∣T(M, {hp}2m0 )

∣∣ =

m−1∏p=0

|detHp,2m−p(M)|(−1)p√|detHm,m(M)|

(−1)m

,

where detHp,2m−p(M) is the determinant of the matrix of the intersection pairing(·, ·)p,2m−p : Hp(M)×H2m−p(M) → R in bases hp,h2m−p.

It is well known that Riemann surfaces and Grasmannians have many appli-cations in a wide range of mathematics such as topology, differential geometry,algebraic geometry, symplectic geometry, and theoretical physics (see, e.g, [2]-[6], [13, 15], [34]- [41], and the references therein). He also applies Theorem 1.1to Riemann surfaces and Grasmannians.

In this work we calculate Reidemeister torsion of compact flag manifold K/Tfor G = Al, where K is compact simply connected semi-simple Lie group and T ismaximal torus.

The content of the paper is as follows. In Section 2 we provide the basic defini-tions and facts about the Reidemeister torsion of a general chain complex. More-over, we explain symplectic chain complex. In Section 3 we give all details of cupproduct formula in the cohomology ring of flag manifolds which is called Schubertcalculus. In the last section we calculate the Reidemesiter torsion of flag manifoldSU(n+ 1)/T for n ≥ 3.

The results of this paper were obtained during M.Sc studies of Habib Basbaydarat Abant Izzet Baysal University and are also contained in his thesis [1].

2. CALCULATION OF THE REIDEMEISTER TORSION OFCOMPACT MANIFOLDS.

In this section we will give the formula for calculation of Reidemeister torsionof compact manifolds giving the required definitions and the basic facts about theReidemeister torsion. The general reference for this section is [37]. The detailedproofs and more information can be found in [32,34,41], and the references therein.

2.1. Reidemeister torsion of a chain complex. We shall reserve F to denote

the field of real R or complex C numbers. Let (C∗, ∂∗) = (Cn∂n→ Cn−1 → · · · →

C1∂1→ C0 → 0) be a chain complex of finite dimensional vector spaces over F.

Let Hp(C∗) = Zp(C∗)/Bp(C∗) denote the p−th homology of C∗, where Bp(C∗) =Im{∂p+1 : Cp+1 → Cp}, and Zp(C∗) = ker{∂p : Cp → Cp−1}.

Clearly, we have the following short-exact sequences: 0 → Zp(C∗) → Cp →Bp−1(C∗)→ 0 and 0→ Bp(C∗)→ Zp(C∗)→ Hp(C∗)→ 0. Assume that bp, hp arebases of Bp(C∗), Hp(C∗), respectively. Assume also that `p : Hp(C∗)→ Zp(C∗), sp :Bp−1(C∗) → Cp are sections of Zp(C∗) → Hp(C∗), Cp → Bp−1(C∗), respectively.Then, we obtain a new basis of Cp, namely bp ⊕ `p(hp)⊕ sp(bp−1).

Definition 2.1. Let C∗ : Cn∂n→ Cn−1 → · · · → C1

∂1→ C0 → 0 be a chain complexof finite dimensional vector spaces over F. For p = 0, . . . , n, let cp, bp, hp bebases of Cp, Bp(C∗), Hp(C∗), respectively, and let `p : Hp(C∗) → Zp(C∗), sp :Bp−1(C∗) → Cp be sections of Zp(C∗) → Hp(C∗), Cp → Bp−1(C∗), respectively.

REIDEMEISTER TORSION OF FLAG MANIFOLDS 3

The Reidemeister torsion of C∗ with respect to bases {cp}np=0, {hp}np=0 is thealternating product

T(C∗, {cp}n0 , {hp}n0 ) =

n∏p=0

[bp ⊕ `p(hp)⊕ sp(bp−1), cp](−1)(p+1)

,

where [ep, fp] denotes the determinant of the change-base-matrix from basis fp toep of Cp.

Remark 2.2. Milnor proved that the Reidemeister torsion does not depend on basesbp, sections sp, `p [30]. Let c′p,h

′p be other bases respectively for Cp, Hp(C∗). Then,

by an easy computation we have the following change-base-formula:

T(C∗, {c′p}n0 , {h′p}n0 ) =

n∏p=0

([c′p, cp]

[h′p,hp]

)(−1)p

T(C∗, {cp}n0 , {hp}n0 ). (2.1)

By the independence of the Reidemeister torsion from bp and sections sp, `p,formula (2.1) is easily obtained. Note that if, for example, [c′p, cp] = 1, [h′p,hp] =−1, then the torsions are the same for odd n, and torsions have opposite sign foreven n.

It follows from Snake Lemma that for short-exact sequence (2.2) of chain com-plexes

0→ A∗ı→ B∗

π→ D∗ → 0, (2.2)

there is also the long-exact sequence of vector spaces C∗ of length 3n+ 2. Namely,

C∗ : · · · → Hp(A∗)ıp→ Hp(B∗)

πp→ Hp(D∗)δp→ Hp−1(A∗)→ · · · , (2.3)

where C3p = Hp(D∗), C3p+1 = Hp(A∗), and C3p+2 = Hp(B∗).Clearly, the bases hDp , hAp , and hBp serve as bases for C3p, C3p+1, and C3p+2,

respectively.The following result of Milnor states that the alternating product of the torsions

of the chain complexes in (2.2) is equal to the torsion of (2.3). More precisely,

Theorem 2.3. ( [30]) Let cAp , cBp , and cDp be bases respectively for Ap, Bp, and

Dp. Let hAp , hBp , and hDp be bases of Hp(A∗), Hp(B∗), and Hp(D∗), respectively. If,

moreover, cAp , cBp , cDp are compatible in the sense that [cBp , cAp ⊕ cDp ] = ±1, where

π(cDp

)= cDp , then

T(B∗, {cBp }n0 , {hBp }n0 ) = T(A∗, {cAp }n0 , {hAp }n0 )T(D∗, {cDp }np=0, {hDp }n0 )

× T(C∗, {c3p}3n+20 , {0}3n+2

0 ),

where [ep, fp] is the determinant of the change-base-matrix from basis fp to ep ofBp.

For future reference, let us give the following sum-lemma:

Lemma 2.4. Let A∗, D∗ be two chain complexes. Let cAp , cDp , hAp , and hDp be basesof Ap, Dp, Hp(A∗), and Hp(D∗), respectively. Then,

T(A∗⊕D∗, {cAp ⊕cDp }n0 , {hAp ⊕hDp }n0 ) = T(A∗, {cAp }n0 , {hAp }n0 )T(D∗, {cDp }n0 , {hDp }n0 ).


Independently, it is explained in [2, 34] that a general chain complex can (un-naturally) be splitted as a direct sum of an acyclic and ∂−zero chain complexes.Moreover, it is proved independently in [2, Proposition 1.5] and [34, Theorem 2.0.4]that the Reidemeister torsion T(C∗) of a general complex C∗ can be interpreted as

an element of ⊗np=0(det(Hp(C∗)))(−1)p+1

. For detailed proof and further informa-tion, we may refer the readers to [2, 34].

Definition 2.5. A symplectic chain complex of length q is (C∗, ∂∗, {ω∗,q−∗}),where C∗ : 0 → Cq

∂q→ Cq−1 → · · · → Cq/2 → · · · → C1∂1→ C0 → 0 is

a chain complex with q ≡ 2(mod 4), and for p = 0, . . . , q/2, ωp,q−p : Cp ×Cq−p → R is a ∂−compatible anti-symmetric non-degenerate bilinear form. To bemore precise, ωp,q−p(∂p+1a, b) = (−1)p+1ωp+1,q−(p+1)(a, ∂q−pb) and ωp,q−p(a, b) =

(−1)p(q−p)ωq−p,p(b, a).

Note that by q ≡ 2(mod 4), we easily have ωp,q−p(a, b) = (−1)pωq−p,p(b, a).It follows from the ∂−compatibility of the non-degenerate anti-symmetric bilinearmaps ωp,q−p : Cp × Cq−p → R that one can easily extend these to homologies [34].

Definition 2.6. Let C∗ be a symplectic chain complex. We say that bases cp ofCp and cq−p of Cq−p are ω−compatible if the matrix of ωp,q−p in bases cp, cq−p

equals to the k × k identity matrix Ik×k when p 6= q/2 and

[0l×l Il×l−Il×l 0l×l

]when

p = q/2, where k = dimCp = dimCq−p and 2l = dimCq/2.

Similarly, considering [ωp,q−p] : Hp(C∗)×Hq−p(C∗)→ R, one can also define the[ω]-compatibility of bases hp of Hp(C∗) and hq−p of Hq−p(C∗).

The existence of ω−compatible bases enabled us to prove in [34] that a symplecticchain complex C∗ can be splitted ω−orthogonally as a direct sum of an exact and∂−zero symplectic complexes. Moreover, Y. Sozen proved Theorem 2.7, which isone of the main results of [34]. Namely,

Theorem 2.7. ( [34]) Let C∗ be a symplectic chain complex. For p = 0, . . . , q, letcp, hp be any bases of Cp, Hp(C∗), respectively. Then, for the Reidemeister torsionof C∗ with respect to {cp}q0, {hp}

q0, the following formula

T(C∗, {cp}q0, {hp}q0) =

(q/2)−1∏p=0

(det[ωp,q−p])(−1)p

√det[ω

q/2,q/2](−1)q/2

holds, where det[ωp,q−p] is the determinant of the matrix of the non-degeneratepairing [ωp,q−p] : Hp(C∗)×Hq−p(C∗)→ R in bases hp, hq−p.

The proof and unexplained subjects can be found in [34]. For further applicationsof Theorem 2.7, we refer the reader to [35,36].

2.2. The Reidemeister Torsion of a Manifold. Let M be an m−manifold witha cell decomposition K. If cp = {cp1, . . . , cpnp} is the geometric basis for the p−cells

Cp(K;Z), p = 0, . . . ,m, then one can associate to M the following chain complex

0→ Cm(K)∂m→ Cm−1(K)→ · · · → C1(K)

∂1→ C0(K)→ 0,

where Z is the set of integers and ∂p is the usual boundary operator.


Definition 2.8. Let M be an m−manifold with a cell decomposition K. Forp = 0, . . . ,m, let cp and bp be bases of Cp(K;Z) and Hp(M ;Z), respectively.T(C∗(K), {cp}m0 , {hp}m0 ) is called the Reidemeister torsion of M.

Using similar arguments introduced in [34, Lemma 2.0.5], one can prove:

Lemma 2.9. The Reidemeister torsion of M is independent of cell decomposition.

Hence, the Reidemeister torsion T(C∗(K), {cp}m0 , {hp}m0 ) of M is well-defined.Thus, we let T(M, {hp}m0 ) denote the Reidemeister torsion of M in the bases hp ofHp(M), p = 0, . . . ,m.

By [2, Proposition 1.5] and [34, Theorem 2.0.4], one concludes that the Rei-demeister torsion of M is an element of the dual of 1−dimensional vector space⊗np=0(det(Hp(M))(−1)p .

2.3. The Main Result. In this section, we give Theorem 1.1. To alleviate thenotation, let us introduce the following which is used throughout the paper. Let Ybe an oriented closed connected manifold of dimension d. For p = 0, . . . , d, let hYpand hYd−p be bases of Hp(Y ) and Hd−p(Y ), respectively. We denote the matrix of

the intersection pairing (·, ·)p,d−p : Hp(Y )×Hd−p(Y )→ R in the bases hYp and hYd−pby Hp,d−p(Y ). As convention, we let Hp,d−p(Y ) = 1 when Hp(Y ) = Hd−p(Y ) = 0.

Now we shall give the symplectic chain complex associated to a compact evendimensional manifolds. So we have calculation formulas for Reidemeister torsion ofcompact manifolds.

Theorem 2.10. Let M be an oriented closed connected 2m−manifold with m oddand χ(M) ≡ 0 (mod 4). For p = 0, . . . , 2m, let hp be a basis of Hp(M). Then,

∣∣T(M, {hp}2m0 )∣∣ =

m−1∏p=0

|detHp,2m−p(M)|(−1)p√

detHm,m(M)(−1)m

.

Theorem 2.11. If M is an oriented closed connected 2m−manifold with m evenand χ(M) even, and if hp is a basis of Hp(M), p = 0, . . . , 2m, then

∣∣T(M, {hp}2m0 )∣∣ =

m−1∏p=0


(−1)m

.

Theorem 2.12. Let M be an oriented closed oriented 2m−manifold with m evenand for p = 0, . . . , 2m, let hp be a basis of Hp(M). Then, for the Reidemeistertorsion of M, the formula

∣∣T(M, {hp}2m0 )∣∣ =

m−1∏p=0


(−1)m

is valid.

The Euler characteristic χ(M) of an oriented closed connected 2m−manifold Mwith m odd is even. In Theorem 2.10, we obtain a formula for Reidemeister torsionof such M with χ(M) ≡ 0 (mod 4). In this section, we consider the case whenχ(M) ≡ 2 (mod 4).


Theorem 2.13. Let M be an oriented closed connected 2m−manifold with m oddand χ(M) ≡ 2 (mod 4). For p = 0, . . . , 2m, let hp be a basis of Hp(M). Then,∣∣T(M, {hMp }2m0 )

∣∣ =

m−1∏p=0


(−1)m

.

In the following section, we discuss the Reidemeister torsion of oriented closedconnected odd dimensional manifolds.

Theorem 2.14. Let M be an oriented closed connected m−manifold with m odd.Let hp be a basis for Hp(M), p = 0, . . . ,m. Then, |T(M, {hp}m0 )| = 1.

3. Differential operators, Lie algebra cohomology and generalizedSchubert cocycles.

First, we will give some facts about Kac-Moody Lie algebras and associatedgroups which will be used in this section. The general reference is [18] of V. Kac.

Definition 3.1. LetA = {aij}n×n be a complex matrix of rank l. A realization ofAis a triple (h, π, πV ), where h is a complex vector space of dimension n+ corankA,π = {αi}1≤i≤n ⊆ h∗ and πV = {hi}1≤i≤n ⊆ h are free indexed sets satisfyingαj(hi) = aij .

Definition 3.2. Two realizations (h, π, πV ) and (h1, πi, πV1 ) are called isomorphic

if there exists a vector space isomorphism φ : h → h1 such that φ(πV ) = πV1 andφ∗(π1) = π.

From [18], we have

Theorem 3.3. There exists a unique up to isomorphism realization for every n×nmatrix.

Definition 3.4. A generalized Cartan matrix A = {aij}n×n is a matrix of integerssatisfying aii = 2 for all i and aij 6 0 if i 6= j, aij = 0 implies aji = 0.

Definition 3.5. Given a realization (h, π, πV ) of a n×n generalized Cartan matrixA, the Kac-Moody algebra g = g(A) is the Lie algebra over C, generated by h andthe elements ei and fi for 1 ≤ i ≤ l such that this basis elements satisfy thefollowing relations:

[h,h] = 0, [h, ei] = αi(h)ei, [h, fi] = −αi(h)fi

for h ∈ h and all 1 ≤ i ≤ l; [ei, fi] = δijhj for all 1 ≤ i, j ≤ n;

(ad ei)1−aij (ej) = 0 = (ad fi)

1−aij (fj)

for all 1 ≤ i 6= j ≤ n. The elements hi, ei, fi are called Chevalley generators andthe subalgebra h of g(A) is called the Cartan subalgebra.

The Kac-Moody algebra g = g(A) has a root space decomposition.

Theorem 3.6. For 0 6= α, the root space gα = {x ∈ g : [h, x] = α(h)x, ∀h ∈ h} isfinite dimensional and there is a root space decomposition

g = h⊕⊕α∈∆

gα = h⊕⊕α∈∆+

gα ⊕⊕α∈∆−

gα

where ∆+ (resp. ∆−) is the positive (resp. negative) root system.


We define fundamental reflections ri ∈ AutC(h), 1 ≤ i ≤ n, by ri(h) = h −αi(h)hi. They generate the Weyl group W , which is a Coxeter group on {ri}1≤i≤n.

We define the following Lie algebras.

n =⊕α∈∆+

gα, n− =⊕α∈∆−

gα.

Then, g = h ⊕ n ⊕ n−, where b = h ⊕ n is called the Borel algebra. We have aunique complex linear involution ω of g defined by ω(fi) = ei for all 1 ≤ i ≤ l andω(h) = −h for h ∈ h. The involution ω leaves the real points of g stable. ω iscalled the Chevalley involution. Also, we have a unique conjugate linear involutionω0 which agrees with ω on the real points of g. We can define a nondegenerate g-invariant, symmetric complex bilinear form σ on h∗ such that σ(αi, αj) = 〈hαi , hαj 〉where 〈 , 〉 is the standard complex inner product on g. This form is called theKilling form. This gives a Hermitian form { , } on g defined by {x, y} = −〈x, ω0(y)〉for x, y ∈ g.

Now, we will mention the highest weight module category of a Kac-Moody alge-bra g. The fundamental reference is [18] of V. Kac. Let V be a g-module, λ ∈ h∗.We define

Vλ = {x ∈ V : h · x = λ(h)x for ∀h ∈ h}.Then, Vλ is a subspace of V . If Vλ 6= 0, λ is called a weight of the g-module V ,Vλ the weight space corresponding to λ, and dimVλ the multiplicity of λ. If λ isa weight of V , then any non-zero vector of Vλ is called a weight vector of λ. Wedenote by

P (V ) = {λ ∈ h∗ : Vλ 6= 0},the set of weights of the g-module V .

Lemma 3.7. For any α ∈ ∆ ∪ {0} and λ ∈ h∗, we have

gα · Vλ ⊆ Vλ+α.

We setD(λ) = {λ− α : α ∈ Q+},

where Q+ =⊕i

Z+αi. For any subset F ⊆ h∗, we define

D(F ) =⋃λ∈F

D(λ).

We can define a partial ordering > on h∗ by

λ > µ ⇐⇒ λ− µ ∈ Q+ ⇐⇒ µ ∈ D(λ).

We will give the definition of category O of g-modules.

Definition 3.8. The objects of O are g-modules V which satisfy the conditions

(1) V is h-diagonalizable, i.e.,

V =⊕λ∈h∗

Vλ,

(2) dimVλ <∞ for all λ ∈ h∗,(3) there exists a finite set F ⊆ h∗ such that P (V ) ⊆ D(F ), and whose mor-

phisms are g-module homomorphisms.

By the property 3 of the definition of O, we have


Proposition 3.9. Every non-zero g-module in O has at least one maximal weight.

Definition 3.10. A g-module V is called a highest weight module, if V has a uniquemaximal weight Λ and V is generated by some weight vector vΛ ∈ VΛ.

Theorem 3.11. Let V be a highest weight module with maximal weight Λ. Then,

V = U(g) · vΛ = U(n−) · vΛ

for any weight vector vΛ of Λ; V ∈ O, dimVΛ = 1 and P (V ) ⊆ D(Λ); gα · VΛ = 0for any α ∈ ∆+; V has a unique maximal submodule, hence a unique quotientsimple module; the homomorphic image of V is also a highest weight module withmaximal weight Λ.

Since b = h⊕n+, we can regard VΛ as a b-module with n+ acting on it trivially.We define

M(Λ) = U(g)⊗U(b) VΛ

where VΛ is 1-dimensional weight space corresponding to the weight Λ. The g-module M(Λ) is called Verma module corresponding to the weight Λ.

Theorem 3.12. The Verma module M(Λ) is a highest weight module with highestweight Λ. Any highest weight module with highest weight Λ is a homomorphicimage of the Verma module M(Λ). M(Λ) has a unique maximal submodule withsimple quotient L(Λ). Any irreducible highest weight module with highest weight Λis isomorphic to L(Λ).

Now, we will give the definition of lowest weight module. Let L(Λ) be an ir-reducible highest weight module with the highest weight Λ. Let L(Λ)∗ be theg-module contragredient to L(Λ). Then

L(Λ)∗ =∏λ∈h∗

L(Λ)∗λ.

The subspace

L∗(Λ) =⊕λ∈h∗

L(Λ)∗λ

is a submodule of the g-module L(Λ)∗. The module L∗(Λ) is irreducible and forv ∈ L(Λ)∗λ, we have

n− · v = 0 and h · v = −Λ(h)v for h ∈ h.

The module L∗(Λ) is called an irreducible module with lowest weight −Λ.

Theorem 3.13. There is a bijection between h∗ and irreducible lowest weight mod-ules given by Λ→ L∗(−Λ).

We denote by πΛ the action of g on L(Λ). We give a new action π∗Λ on the spaceL(Λ) by

π∗Λ(g)v = πΛ(ω(g))v,

where ω is the Chevalley involution of g. (L(Λ), π∗Λ) is an irreducible g-modulewith lowest weight −Λ. By the uniqueness of irreducible lowest weight moduleswith the lowest weight −Λ, this module can be identified with L∗(Λ).

Definition 3.14. A g-module L is called quasi-simple if it is a highest weightmodule with highest weight vector x0 such that there exists n ∈ Z+ with fi

n(x0) = 0for all 1 ≤ i ≤ i.


From [14], we have

Proposition 3.15. The quasi-simple g-modules are indexed by the positive integralweights.

We will denote by L(λ) the quasi-simple module with highest weight λ. We willdenote the derived algebra [g,g] by g′. From [42], we have

Theorem 3.16. g′ is the subalgebra of g generated by the Chevalley generators eianf fi for 1 ≤ i ≤ l and we have the decomposition

g′ = h′ ⊕ n⊕ n−,

where h′ = g′ ∩ h.

Definition 3.17. A g′-module (V, π) is called integrable if π(e) is locally nilpotentwhenever e ∈ gα for any real root.

Let G∗ be the free product of the additive groups {gα}α∈∆re with canonicalinclusions iα : gα → G∗. For any integrable g′-module (V, π), we define a homo-morphism π∗ : G∗ → AutC(V ) by π∗(iα(e)) = exp(π(e)) for e ∈ gα. Let N∗ be theintersection of all kerπ∗ and let q : G∗ → G∗/N∗ be the canonical homomorphism.We put G = G∗/N∗. The next result comes from [21].

Proposition 3.18. G is an algebraic group in the sense of Safarevic.

We call G the group associated to the Kac-Moody Lie algebra g. G may be ofthree different types: finite, affine and wild. The finite type Kac-Moody groupsare simply-connected semi-simple finite dimensional algebraic groups. The affinetype Kac-Moody groups are the circle group extension of the group of polynomialmaps from S1 to a group of finite type, or a twisted analogue. There is no concreterealization of the wild type groups. Now, we will introduce some subgroups of theKac-Moody group G. For e ∈ gα, we put exp(e) = q(iα(e)) so that Uα = exp gαis an additive one parameter subgroup of G. We denote by U (resp. U−) thesubgroup of G generated by the Uα (resp. U−α) for α ∈ ∆+. For 1 ≤ i ≤ l, there

exists a unique homomorphism ϕi : SL2(C) → G, satisfying ϕ

(1 z0 1

)= exp(zei)

and ϕ

(1 0z 1

)= exp(zfi) for all z ∈ C. We define

Hi =

{ϕ

(z 00 z−1

): z ∈ C∗

};

Gi = ϕ(SL2(C)). Let Ni be the normalizer of Hi in Gi, H the subgroup of Ggenerated by all Hi and N the subgroup of G generated by all Ni. There is anisomorphism W → N/H. We put B = HU . B is called standard Borel subgroupof G. Also, we can define the negative Borel subgroup B− as B− = HU−. G hasBruhat and Birkhoff decompositions. Details can be found in [20]. The conjugatelinear involution ω0 of g gives to an involution ω0 on G. Let K denote the setof fixed points of this involution. K is called the standard real form of G. Also,this involution preserves the subgroups Gi, Hi and H; we denote by Ki, Ti and Trespectively the corresponding fixed point subgroups. Then, Ki = ϕ(SU2) and

Ti =

{ϕ

(u 00 u−1

): |u| = 1

}


is a maximal torus of Ki and T =∏Ti is a maximal torus in K.

Now, we will give some facts about the topology of K. Let D (resp. D◦) be theunit disk (resp. its interior) in C and let S1 be the unit circle. Given u ∈ D, let

z(u) =

(u (1− |u|2)

1/2

−(1− |u|2)1/2

u

)∈ SU2,

and zi(u) = ϕi(z(u)). We also set

Yi = {zi(u) : u ∈ D◦} ⊆ Ki.

Let w = ri1 · · · rin be a reduced expression of w ∈W . We put Yw = Yi1 · · ·Yin . Wehave a fibration π : K → K/T . The topological space K/T is called the flag varietyof the K and G. Already we have given the topology of finite dimensional flagvariety in the Section 1 of this chapter. Now, we will give the topological structurein the infinite dimensional case. We define Cw = π(Yw). From [19], we have

Proposition 3.19. The decomposition

K/T =∐w∈W

Cw

defines a CW structure on K/T .

The closure of Cw is given by

Cw =∐w′6w

Cw′

The closures Cw are called Schubert varieties and they are finite dimensional com-plex spaces. The infinite type flag variety K/T is the inductive limit of these spacesand by Iwasawa decomposition in [20], we have a homeomorphism K/T → G/B.From [19], we have

Proposition 3.20. The flag variety K/T is an infinite dimensional complex pro-jective variety.

Proposition 3.21. The elements Cw are a basis form of free Z-module H∗(K/T,Z).

Now we will give the construction of the dual Schubert cocycles on the flagvariety by using the relative Lie-algebra cohomology tools. This construction wasdone by B. Kostant in [23] for finite type and extended by S. Kumar in [27] for theKac-Moody case.

First, we will introduce some notations for this section. Reference for the no-tations is [16]. By Λ(V ), we denote the exterior algebra on a g-module V . For aLie-algebra pair (g,h) and a left g-module M , let Λ(g,h,M t) denote the standardchain complex with coefficients in the right module M t, where M t is the right g-module, whose underlying space is M and on which g acts by the rule m ·g = −g ·mfor all g ∈ g and m ∈M , and let C(g,h,M) denote the standard cochain complexwith coefficients in M .

Let Lλ be the quasi-simple g-module with highest weight λ. Then, there is aninvariant positive definite Hermitian form {, } on Λ(n−) ⊗ Lλ due to V. Kac andD. Peterson [20]. Let ∂ : Λ(n−) ⊗ Lλ → Λ(n−) ⊗ Lλ be the differential of degree−1 of the chain complex Λ(n−, Lλ

t). We denote the adjoint of ∂ respect to {, } by∂∗. Λ(n−) ⊗ Lλ is a h-module and ∂ is a h-module map, as is ∂∗. We define the


Laplacian by ∇ = ∂∂∗ + ∂∗∂. We know from [14] that Λ(n−)⊗ Lλ decomposes asa direct sum of finite dimensional irreducible h-modules. From [27], we have

Theorem 3.22. Let g be the Kac-Moody Lie algebra and let Lλ be the quasi-simple g-module with the highest weight λ. Then the action of ∇ on Λ(n−, Lλ

t) isas follows. Let Sβ be an irreducible h-submodule of Λ(n−, Lλ

t) with highest weightβ, then ∇ reduces to a scalar operator on Sβ and the scalar is

1

2[σ(λ+ ρ, λ+ ρ)− σ(β + ρ, β + ρ)]

where ρ is the sum of all simple roots.

From [14], we have

Theorem 3.23. With the notations as in Theorem 3.22, the 2jth homology spaceH2j(n

−, Lλt) is finite dimensional and it is isomorphic as an h-module to the direct

sum ⊕`(w)=j

M(w(λ+ρ)−ρ)

of non-isomorphic irreducible h-modules.

C(g,h) denotes the standard cochain complex with differential d associated tothe Lie algebra pair (g,h) with trivial coefficients. That is, C(g,h) is defined to be∑s≥0

Homh(Λs(g/h),C) such that h acts trivially on C. We define

C =∑s≥0

Cs

where Cs = HomC(Λs(g/h),C). We put the topology of pointwise convergence on

Cs, i.e., fn → f in Cs if and only if fn(x)→ f(x) in C with usual topology, for allx ∈ Λ(g,h). From [9], we have

Theorem 3.24. Cs is a complete, Hausdorff, topological vector space with respectto the pointwise topology.

In [27], a continous map ∂ : Cs → Cs−1, and a cochain map of b on C are defined.

We define ∂, b to be the restrictions of ∂ and b to the subspace C(g,h). We definethe following operators on C(g,h): S = d∂ + ∂d and L = b∂ + ∂b. From [27], wehave

Proposition 3.25. kerS ⊕ imS = C.

Theorem 3.26. d and ∂ on C(g,h) are disjoint.

Proposition 3.27 (Hodge type decomposition). Let V be any vector spaceand d, ∂ : V → V be two disjoint operators such that d2 = ∂2 = 0. Further, assumethat kerS ⊕ imS = V where S = d∂ + ∂d. Then, kerS → ker d/ im d and kerS →ker ∂/ im ∂ are both isomorphisms.

By the Hodge type decomposition and Proposition 3.25, we have

Theorem 3.28. The canonical maps ψd,S : kerS → H(C, d) andψ∂,S : kerS →H(C, ∂) are both isomorphisms.


Now, we describe a basis for kerL. We fix w ∈ W of length s. We defineΦw = w∆− ∩ ∆+. Φw consists of real roots {γ1, · · · , γs}. We pick yγi ∈ g−γi ofunit norm with respect to the form {, } and let xγi = −ω0(yγi). Let M(wρ−ρ) be theirreducible h-submodule with highest weight (wρ− ρ). By Proposition 2.5 in [14],the corresponding highest weight vector is yγ1 ∧ · · · ∧ yγs . There exists a unique

element hw ∈ [M(wρ−ρ)⊗Λs(n)] such that hw = (2i)s(yγ1∧· · ·∧yγs∧xγ1∧· · ·∧xγs)mod Pw ⊗ Λs(n), where Pw is the orthogonal complement of yγ1 ∧ · · · ∧ yγs inM(wρ−ρ). Using the nondegenerate bilinear form 〈〉 on g, we have embedding

e :⊕k≥0

Λs(n⊕ n−)→⊕k≥0

[Λs(n⊕ n−)]∗.

Then hw = e(hw) ∈ kerL. These elements {hw}w∈W is a C-basis of kerL. Then,we can define sw = ψ∂,S

−1([hw]) ∈ H(C, ∂). From [27], [24], we have

Theorem 3.29. Let g be the Kac-Moody Lie algebra and let G be the group asso-ciated to the Kac-Moody algebra g and B be standard Borel subgroup of G. Then∫

Cw′

sw =

0 if w 6= w′,

(4π)2`(w)

∏ν∈w−1∆∩∆+

σ(ρ, ν)−1

if w = w′.

This gives the expression for the d, ∂ harmonic forms sw0 =sw

dwwhich are dual

to the Schubert cells where dw =

∫Cw

sw.

Theorem 3.30. (see [28])

H(

∫) : H∗(g,h)→ H∗(G/B,C)

is a graded algebra isomorphism.

Let εw denote the image of s0w by the integral map in last theorem. These

cohomology classes are dual to the closure of the Schubert cells, hence we have

Theorem 3.31. The elements εw, w ∈W , form a basis of the Z-module H∗(G/B,Z).

For a finite type flag variety G/B, εw denotes Pw−1 in the notation of Section1. Now, we give a cup product formula in the cohomology of any type flag varietyG/B. From [24],

Theorem 3.32. Let χi be the fundamental weight of G for 1 ≤ i ≤ l. For anysimple reflection ri and any element w ∈W , and a coroot γ∨,

εri · εw =∑

wγ−→w′

χi(γ∨)εw

′.

As an analogy of cohomology theory of the finite type flag space G/B, thecohomology of affine type flag space G/B and some operators will be introduced inthis section. The fundamental reference is [19] of V. Kac.

Let Q∨ =⊕i

Zhi, where hi is coroot, be the coroot lattice and let

P = {λ ∈ h′∗

: λ(hi) ∈ Z}


be the weight lattice dual to Q∨. Let S(P ) =⊕j≥0

Sj(P ) be the integral symmetric

algebra over the lattice P , and S(P )+ =⊕j>0

Sj(P ) the augmentation ideal. Given

a commutative ring F with unit, we denote S(P )F = S(P )F ⊗Z F. We define thecharacteristic homomorphism ψ : S(P )→ H∗(G/B,Z) as follows: given λ ∈ P , wehave the corresponding character of B and the associated line bundle Lλ on G/B.We put ψ(λ) ∈ H2(G/B,Z) equal to the Chern class of Lλ and we extend thismultiplicativity to the whole S(P ). We denote by ψF the extension of ψ by linearityto S(P )F. In order to describe the properties of ψF, we define BGG-operator ∆i

for 1 ≤ i ≤ l on S(P ) by

∆i(f) =f − ri(f)

αiand we extend this by linearity to S(P )F. We define

IF = {f ∈ S(P )+F : ∆i1 · · ·∆in(f) ∈ S(P )+

F ∀ sequence (i1, · · · .in)}.

Theorem 3.33. We have kerψF = IF and H∗(G/B,F) is a free module over imψF.

We will introduce certain operators on cohomology of the flag space G/B whichare basic tools in the study of this theory. These operators are extension of actionof the BGG-operators ∆i from the image of ψ to the whole cohomology operators.We know that the Weyl group W acts by right multiplication on K/T and thisaction induces an action of W on homology and cohomology of flag space. Onthe other hand, we have a fibration pi : K/T → K/KiT with fibre Ki/Ti. Sincethe odd degree cohomologies of Ki/Ti and K/KiT are trivial, then the Leray-Serre spectral sequence of the fibration degenerates after the second term. So,H∗(K/T,Z) is generated by im pi

∗, which is ri invariant and the element ψ(χi)where χi is fundamental weight. We define a Z-linear operator Ai on H∗(K/T,Z)lowering the degree by 2 such that ri leaves the image of Ai invariant and

x− ri(x) = Ai(x) ∪ ψ(αi)

for x ∈ H∗(K/T,Z). Similarly, we can define homology operators Ai onH∗(K/T,Z)raising the degree by 2 such that ri(Ai(v)) = −Ai(v) and

v + ri(v) = Ai(v) ∩ ψ(αi)

for v ∈ H∗(K/T,Z). The properties of the actions of the operators Ai (resp. Ai)on the cup product (resp. cap product) in the cohomology (resp. homology) can befound in [19]. Now, we will give the geometric interpretation of Ai. Given w ∈W ,we choose a reduced expression w = ri1 · · · ris and and define a map τw : D → K/Tgiven by τw(u1, · · · , us) = zi1 · · · zisT where D is the unit disk in the complex spaceCs and zi has been defined in the previous section. By Proposition 3.19, the relativehomology map τw∗ gives us an element sw ∈ H2`(w)(K/T,Z). By Proposition 3.21,these elements are a basis of H∗(K/T,Z); let εw be the dual basis of H∗(K/T,Z).

Proposition 3.34.

ri(εw) =

εw if `(wri) > `(w),

εw −∑

wriγ−→w′

〈αi, γ〉εw′

otherwise.

Similarly, we can give the reflection action on Schubert cycles sw.


Proposition 3.35.

Ai(εw) =

{εwri if `(wri) < `(w),

0 otherwise.

Ai(sw) =

{swri if `(wri) > `(w),

0 otherwise.

Proposition 3.36.

c1(Lλ) ∩ sw =∑

w′γ−→w

〈λ, γ〉sw′

The set of all functions from the Weyl group W to C will be denoted by C{W}.C{W} is an algebra under pointwise addition and multiplication. Now, we will givethe relation between C{W} and EndhH

∗(n−,C). From [24], we have

Theorem 3.37. Let (A, d) be a differential graded algebra over C and let δ be thederivation in End(A, d) induced by d such that

δξ = dξ − (−1)iξd for ξ ∈ Endi(A).

Then ι : H(End(A, d), δ)→ EndH(A, d) is an isomorphism graded algebras.

Theorem 3.38. The standard cochain complexes C(g,h) and C(n−) with the topol-ogy of pointwise convergence are both differential graded algebras over C.

Also, we can put the topology of pointwise convergence on End C(g,h) andEnd C(n−). Then, the derivation map δ : End C(n−) → End C(n−) is continuousunder this topology and it commutes with the action of h on End C(n−). We denoteby δ0, the restriction of δ to Endh C(n−). From [24], we have

Proposition 3.39. There exists a unique injective continuous map η : C(g,h) →Endh C(n−).

Lemma 3.40. We have η(kerS) ⊆ ker δ0.

The map η induces a map η : kerS → H(Endh C(n−), δ0) Also, ι induces a mapι0 : H(Endh C(n−), δ0) → EndhH

∗(n−,C). By Theorem 3.23, as an h-module,H2j(n−,C) is isomorphic to the direct sum⊕

`(w)=j

M(wρ−ρ)

of non-isomorphic irreducible h-submodules. By a property of the Hom functor,we have

EndhH∗(n−,C) ∼=

∏i≥0

EndhHi(n−,C) ∼=

∏i≥0

∏`(w)=i

EndhM(wρ−ρ).

Since M(wρ−ρ) is irreducible, EndhM(wρ−ρ) is 1-dimensional with a canonical gen-

erator 1w which is the identity map of M(wρ−ρ). This identifies EndhH∗(n−,C)

with∏w∈W C1w. The space

∏w∈W C1w is the vector space C{W} of all functions

from W to C. Let η be the composite map

kerSη−→ H(Endh C(n−), δ0)

ι0−→ EndhH∗(n−,C) ∼= C{W}.


Now, we will give filtrations of C(g,h) and C{W}. We define a decrasing filtra-

tion G = (Gp)p∈Z− by Gp =∑

0≥k+q≥p

Cq,k(g,h) where Cq,k(g,h) = Homh(Λq(n) ⊗

Λk(n−)). This gives rise to a filtration F = (Fp)p∈Z− of Endh C(n) by definingFp = η(Gp). By ι0, we have filtration H = (Hp)p∈Z+ of C{W}. GrC{W} willdenote the associated graded algebra with respect to the filtration of C{W}. That

is, GrC{W} =∑p≥0

Grp, where Grp = Hp/Hp+1. From [24], we have

Theorem 3.41. Let g be a symmetrizable Kac-Moody Lie algebra. Let h be theCartan subalgebra. Then, H∗(g,h)→ GrC{W} is a graded algebra isomorphism.

By Theorem 3.30, we can give the following corollary.

Corollary 3.42. H∗(G/B,C)→ GrC{W} is a graded algebra isomorphism.

Let g be an arbitrary Kac-Moody algebra associated to a generalized Cartanmatrix A, with its Cartan subalgebra h and Weyl group W. Let Q = Q(h∗) bethe field of the rational functions on h. The Weyl group W acts as a group ofautomorphisms on the field Q. Let QW be the smash product of Q with thegroup algebra C[W ], i.e., QW is a right Q-module with a basis {δw}w∈W and themultiplication is given by

(δvqv) · (δwqw) = δvw(w−1qv)qw

for v, w ∈ W and qv, qw ∈ Q. The module QW admits an involutary anti-automorphism t, defined by (δwq)

t = δw−1(wq) for w ∈W and q ∈ Q. We define

xi = −(δri + δe)1

αi=

1

αi(δri − δe) ∈ QW

where ri ∈W is a simple reflection and αi is the simple root.

Proposition 3.43. Let w ∈ W and let w = ri1 · · · rin be a reduced expression.Then the element xi1 · · ·xin ∈ QW does not depend upon the choice of reducedexpression of w.

The element xi1 · · ·xin ∈ QW will be denoted by xw and (xw−1)t

denoted by xw.

Proposition 3.44.

xv · xw =

{xvw if `(vw) = `(v) + `(w),

0 otherwise.

We know that QW is a right Q-module. Also, Q has a left QW -module structuredefined by (δwq)q

′ = w(qq′) for w ∈W and q, q′. We define subring R ⊆ QW givenby

R = {x ∈ QW : x · S ⊆ S}where S = S(h∗) is the polynomial algebra on h. Let SW be the smash product ofS with the group algebra C[W ]. Obviously SW ⊆ R since S has left SW -modulestructure.

Theorem 3.45. R is a free right S-module with basis {xw}w∈W In particular, any

x ∈ R can be uniquely written as x =∑w∈W

xwpw some pw ∈ S.


R will be referred as a nil-Hecke ring. Now, we will give the coproduct structureon QW . Let QW ⊗QQW be the tensor product, considering both the copies of QWas right Q-modules. We define the diagonal map ∆ : QW → QW ⊗Q QW by

∆(δwq) = δwq ⊗ δw = δw ⊗ δwqfor w ∈W and q ∈ Q. ∆ is right Q-linear.

Theorem 3.46. For any w ∈W , we have

∆(xw) =∑u,v6w

xu ⊗ xvpwu,v

for some homogeneous polynomials pwu,v ∈ S of degree `(u) + `(v) − `(w). In par-ticular, pwu,v = 0 unless `(u) + `(v) ≥ `(w).

Now, we will introduce some dual objects. Let Ξ = HomQ(QW , Q). Since anyξ ∈ Ξ is determined by its restriction to the Q-basis {δw}w∈W , we can regard Ξas the Q-module of all the functions W → Q with pointwise addition and scalarmultiplication defined by the structure (qξ)w = q · ξ(w) for q ∈ Q, ξ ∈ Ξ andw ∈ W . Ξ has a commutative Q-algebra structure with the product as pointwisemultiplication of functions on W . Also, Ξ has a left QW module structure definedby (x · ξ)y = ξ(xt · y) for x, y ∈ QW and ξ ∈ Ξ. We have the Weyl group action aswell as the Hecke-type operators Aw on Ξ defined by wξ = δw · ξ and Awξ = xw · ξfor w ∈W and ξ ∈ Ξ. We define the important subring Λ ⊆ Ξ as follows:

Λ = {ξ ∈ Ξ : ξ(Rt) ⊆ S and ξ(xw) = 0 for all but a finite number of w ∈W}

Proposition 3.47. λ is a S-subalgebra of Ξ. {ξw}w∈W is a S-basis of Λ where ξw

is dual to xw for w ∈W .

Proposition 3.48. Aiξw = ξriw if riw < w, 0 otherwise.

Proposition 3.49. ξri(w) = χi − w−1χi where χi is the fundamental weight dualto the coroot hi corresponding to simple root αi.

Now, we will give the important formula equivalent to the cup product formulain the cohomology of G/B where G is a Kac-Moody group.

Proposition 3.50.

ξu · ξv =∑u,v6w

pwu,vξw,

where pwu,v is a homogeneous polynomial of degree `(u) + `(v)− `(w).

Proposition 3.51.

riξw =

ξw if riw > w,

−(w−1αi)ξriw + ξw −

∑riw

γ−→w′

αi(γ∨)ξw

′otherwise.

Theorem 3.52. Let u, v ∈W . We write w−1 = ri1 · · · rin as a reduced expression.

pwu,v =∑

j1<···<jmrj1 ···rjm=v−1

Ai1 ◦ · · · ◦ Aij1 ◦ · · · ◦ Aijm ◦ · · · ◦Ain(ξu)(e)

where m = `(v) and the notation Ai means that the operator Ai is replaced by theWeyl group action ri.


Let C0 = S/S+ be the S-module where S+ is the augmentation ideal of S. Itis 1-dimensional as C-vector space. Since Λ is a S-module, we can define C0 ⊗S Λ.It is an algebra and the action of R on Λ gives an action of R on C0 ⊗S Λ. Theelements σw = 1⊗ ξw ∈ C0 ⊗S Λ is a C-basis form of C0 ⊗S Λ.

Proposition 3.53. C0 ⊗S Λ is a graded algebra associated with the filtration oflength of the element of the Weyl group W .

Proposition 3.54. The complex linear map f : C0 ⊗S Λ→ GrC{W} is a gradedalgebra homomorphism.

Theorem 3.55. Let K be the standard real form of the group G associated to asymmetrizable Kac-Moody Lie algebra g and let T denote the maximal torus of K.Then the map

θ : H∗(K/T,C)→ C0 ⊗S Λ

defined by θ(εw) = σw for any w ∈W is a graded algebra isomorphism. Moreover,the action of w ∈ W and Aw on H∗(K/T,C) corresponds respectively to that δwand xw ∈ R on C0 ⊗S Λ.

Corollary 3.56. The operators Ai on H∗(K/T,C) generate the nil-Hecke algebra.

Corollary 3.57. We can use Proposition 3.50 and Theorem 3.52 to determine thecup product εuεv in terms of the Schubert basis {εw}w∈W of H∗(K/T,Z).

4. THE REIDEMEISTER TORSION OF COMPACT FLAGMANIFOLDS K/T FOR K = Al

This section includes our calculations about Reidemeister torsion of flag mani-folds using Theorem 1.1 and Proposition 3.50 because χ(SUn+1/T ) = |W | = n! isalways an even number.

We know that the Weyl group W of G acts on the Lie algebra of the maximaltorus T . lt is a finite group of isometries of the Lie algebra t of the maximal torusT . lt preserves the coweight lattice T v. For each simple root α, the Weyl group Wcontains an element rα of order two represented by e((π/2)(eα+e−α)) in N(T ). Sincethe roots α can be considered as the linear functionals on the Lie algebra t of themaximal torus T , the action of rα on t is given by

rα(ξ) = ξ − α(ξ)hα for ξ ∈ t,

where hα is the coroot in t corresponding to simple root α.Also, we can give theaction of rα on the roots by

rα(β) = β − α(hβ)α for α, β ∈ t∗,

where t∗ is the dual vector space of t. The element rα is the reflection in thehyperplane Hα of t whose equation is α(ξ) = 0. These reflections rα generate theWeyl group W .

Set α1, α2, . . . , αn be roots of Weyl Group of SUn+1. Since the Cartan Matrixof Weyl Group of SUn+1 is


Mij =

2 i = j−1 |i− j| = 10 otherwise

,

rαi(αj) =

−αi, i = jαi + αj , |i− j| = 1αj , otherwise.

Proposition 4.1. The Weyl group W of SUn+1 is isomorphic to Coxeter GroupAn given by generators s1, s2, . . . , sn and relations

(i): s2i = 1 i = 1, 2, . . . , n

(ii): sisi+1si = si+1sisi+1 i = 1, 2, . . . , n− 1(iii): sisj = sjsi 1 ≤ i < j − 1 < n.

Proof. (i):

rαi ◦ rαi(β) = rαi(β− < αi, β > αi)

= β− < αi, β > αi− < β− < αi, β > αi, αi > αi

= β− < αi, β > αi− < β,αi > αi+ < αi, β >< αi, αi > αi

= β− < αi, β > αi− < αi, β > αi + 2 < αi, β > αi

= β.

(ii):

rαi ◦ rαi+1 ◦ rαi(β) = rαi ◦ rαi+1(β− < αi, β > αi)

= rαi(β− < αi, β > αi− < αi+1, β− < αi, β > αi > αi+1)

= rαi(β− < αi, β > αi− < αi+1, β > αi+1

+ < αi+1, < αi, β > αi > αi+1)

= rαi(β− < αi, β > αi− < αi+1, β > αi+1

+ < αi, β >< αi+1, αi > αi+1)

= rαi(β− < αi, β > αi− < αi+1, β > αi+1− < αi, β > αi+1)

= β− < αi, β > αi− < αi+1, β > αi+1− < αi, β > αi+1

− < αi, β− < αi, β > αi− < αi+1, β > αi+1

− < αi, β > αi+1 > αi


− < αi, β > αi+ < αi, β >< αi, αi > αi

+ < αi+1, β >< αi+1, αi > αi+ < αi, β >< αi+1, αi > αi


− < αi, β > αi + 2 < αi, β > αi− < αi+1, β > αi

− < αi, β > αi

= β− < αi, β > αi− < αi+1, β > αi− < αi+1, β > αi+1

− < αi, β > αi+1

= β − (< αi, β > + < αi+1, β >)(αi + αi+1).


rαi+1◦ rαi ◦ rαi+1

(β) = rαi+1◦ rαi(β− < αi+1, β > αi+1)

= rαi+1(β− < αi+1, β > αi+1− < αi, β− < αi+1, β > αi+1 > αi)

= rαi+1(β− < αi+1, β > αi+1− < αi, β > αi

+ < αi+1, β >< αi, αi+1 > αi)

= rαi+1(β− < αi+1, β > αi+1− < αi, β > αi− < αi+1, β > αi)

= β− < αi+1, β > αi+1− < αi, β > αi− < αi+1, β > αi

− < αi+1, β− < αi+1, β > αi+1− < αi, β > αi

− < αi+1, β > αi > αi+1

= β− < αi+1, β > αi+1− < αi+1, β > αi− < αi, β > αi

− < αi+1, β > αi+1+ < αi, β >< αi+1, αi > αi+1

+ < αi+1, β >< αi+1, αi > αi+1

+ < αi+1, β >< αi+1, αi+1 > αi+1


− < αi+1, β > αi+1 + 2 < αi+1, β > αi+1− < αi, β > αi+1

− < αi+1, β > αi+1


− < αi, β > αi+1

= β − (< αi+1, β > + < αi, β >)(αi+1 + αi).

Hence rαi+1◦ rαi ◦ rαi+1

(β) = rαi+1◦ rαi ◦ rαi+1

(β).(iii):

rαi ◦ rαj (β) = rαi ◦ (β− < αj , β > αj)

= β− < αj , β > αj− < αi, β− < αj , β > αj > αi

= β− < αj , β > αj− < αi, β > αi+ < αj , β >< αi, αj > αi

= β− < αj , β > αj− < αi, β > αi.

rαj ◦ rαi(β) = rαj ◦ (β− < αi, β > αi)

= β− < αi, β > αi− < αj , β− < αi, β > αi > αj

= β− < αi, β > αi− < αj , β > αj+ < αi, β >< αj , αi > αj

= β− < αi, β > αi− < αj , β > αj .

Hence rαi ◦ rαj (β) = rαj ◦ rαi(β).�

After this point si will represent rαi .Let us define the word

sij =

sisi+1 · · · sj i < jsi i = j1 i > j.


Theorem 4.2. (Theorem 3.1 of [4])The reduced Grobner-Shirshov basis of theCoxeter group An consist of relation

sijsi = si+jsij 1 ≤ i < j ≤ ntogether with defining relations of An.

The following lemma is equivalent of Lemma 3.2 in [4]. The only differenceis that order of generators s1 > s2 > . . . sn in our setting.

Lemma 4.3. Using elimination of leading words of relations, the reduced elementsof An are in the formsn+1,jn+1sn,jnsn−1,jn−1 · · · si,ji · · · s1,j1 1 ≤ i ≤ ji + 1 ≤ n+ 1.Notice that jn+1 + 1 = n+ 1 =⇒ jn+1 = n and sn+1,n = 1.

Algorithm 4.1. (Finding Inverse) Let w = sn,jnsn−1,jn−1· · · s1,j1 . The inverse of

w can be found using following algorithm.Invw = {};Conw = Reverse(w);For k = 1 to k = n

Find maximum sequence in Conw;list = {sk, sk+1, sk+2, . . . , skj};Invw = list ∪ Invw;

End For.

Example 4.4. Let s46s35s25s13. The inverse of its is S3s2s1s5s4s3s2s5s4s3s6s5s4.Invw = s14

S3s2s5s4s3s5s4s6s5

Invw = s25s14

S3s5s4s5s6

Invw = s35s25s14

s5s6

Invw = s56s35s25s14.

Lemma 4.5. Let w = (sn,jn)(sn−1,jn−1) · · · (si+1, ji+1

)(si,ji) · · · (s1,J1) andsiw = (sn,jn)(sn−1, jn−1

) · · · (si+1, ji+1)(si,ji) · · · (s1,j1

) where

siw =

ji+1 = ji + 1, ji = ji+1 if ji < ji+1

ji+1 = ji, ji = ji+1 − 1 if ji ≥ ji+1

jk = jk if k 6= i, i+ 1

Here if i = n, then we assume jn+1 = n.

Corollary 4.6. Let w = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1

)(si,ji) · · · (s1,J1) and

si−1(siw) = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1

)(si,ji) · · · (s1,J1)


where

si−1(siw) =

ji+1 = ji + 1, ji = ji−1 + 1, ji−1 = ji+1 if ji < ji+1, ji−1 < ji+1

ji+1 = ji + 1, ji = ji−1, ji−1 = ji+1 − 1 if ji < ji+1, ji−1 ≥ ji+1

ji+1 = ji, ji = ji−1 + 1, ji−1 = ji+1 − 1 if ji ≥ ji+1, ji−1 < ji+1 − 1

ji+1 = ji, ji = ji−1, ji−1 = ji+1 − 2 if ji ≥ ji+1, ji−1 ≥ ji+1 − 1

jk = jk if k 6= i− 1, i, i+ 1.

Proof. Let w = siw = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1

)(si,ji) · · · (s1,j1). Therefore

si−1(w) =

ji = ji−1 + 1, ji−1 = ji if ji−1 < jiji = ji−1, ji−1 = ji − 1 if ji−1 ≥ jijk = jk if k 6= i− 1, i.

(i): ji < ji+1 ⇒ ji+1 = ji + 1, ji = ji+1 So ji−1 < ji ⇒ ji−1 < ji+1,

ji+1 = ji+1 = ji + 1, ji = ji−1 + 1 = ji−1 + 1, ji−1 = ji = ji+1.

(ii): ji < ji+1 ⇒ ji+1 = ji + 1, ji = ji+1 So ji−1 ≥ ji ⇒ ji−1 ≥ ji+1,

ji+1 = ji+1 = ji + 1, ji = ji−1 = ji−1, ji−1 = ji − 1 = ji+1 − 1.

(iii): ji ≥ ji+1 ⇒ ji+1 = ji , ji = ji+1 − 1 So ji−1 < ji ⇒ ji−1 < ji+1,

ji+1 = ji+1 = ji + 1, ji = ji−1 = ji−1, ji−1 = ji − 1 = ji+1 − 1.

(iv): ji ≥ ji+1 ⇒ ji+1 = ji, ji = ji+1 − 1 So ji−1 ≥ ji ⇒ ji−1 ≥ ji+1 − 1,

ji+1 = ji+1 = ji , ji = ji−1 = ji−1, ji−1 = ji − 1 = ji+1 − 2.

�

Corollary 4.7. Let w = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1)(si,ji) · · · (s1,J1) and

si+1(siw) = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1

)(si,ji) · · · (s1,j1).

Then


si+1(siw) =

ji+2 = ji + 2, ji+1 = ji+2, ji = ji+1 if ji < ji+1, ji+1 < ji+2

ji+2 = ji + 1, ji+1 = ji+2 − 1, ji = ji+1 if ji < ji+1, ji + 1 ≥ ji+2

ji+2 = ji + 1, ji+1 = ji+2, ji = ji+1 − 1 if ji ≥ ji+1, ji < ji+2

ji+2 = ji, ji+1 = ji+2 − 1, ji = ji+1 − 1 if ji ≥ ji+1, ji ≥ ji+2.

jk = jk if k 6= i, i+ 1, i+ 2.

Proof. Let w = siw = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1

)(si,ji) · · · (s1,j1). Therefore

si+1(w) =

ji+2 = ji+1 + 1, ji+1 = ji+2 if ji+1 < ji+2

ji+2 = ji+1, ji+1 = ji+2 − 1 if ji+1 ≥ ji+2

jk = jk if k 6= i+ 1, i+ 2.

(i): ji < ji+1 ⇒ ji+1 = ji + 1, ji = ji+1 So ji+1 < ji+2 ⇒ ji + 1 < ji+2,

ji+2 = ji+1 + 1 = ji + 2 , ji+1 = ji+2 = ji+2, ji = ji = ji+1.

(ii): ji < ji+1 ⇒ ji+1 = ji + 1, ji = ji+1 So ji+1 ≥ ji+2 ⇒ ji + 1 ≥ ji+2

, ji+2 = ji+1 = ji + 1, ji+1 = ji+2 − 1 = ji+2 − 1, ji = ji = ji+1.

(iii): ji ≥ ji+1 ⇒ ji+1 = ji, ji = ji+1 − 1 So ji+1 < ji+2 ⇒ ji < ji+2 ,

ji+2 = ji+1 + 1 = ji + 1, ji+1 = ji+2 = ji+2, ji = ji − 1 = ji+1 − 1.

(iv): ji ≥ ji+1 ⇒ ji+1 = ji, ji = ji+1 − 1 So ji+1 ≥ ji+2 ⇒ ji ≥ ji+2,

ji+2 = ji+1 = ji, ji+1 = ji+2 − 1 = ji+2 − 1, ji = ji = ji+1 − 1.

�

Using Lemma 4.3 and definitions of Ai and ri operators, we can obtain thefollowings.

Lemma 4.8. Let w = (sn,jn)(sn−1,jn−1) · · · (si+1, ji+1

)(si,ji) · · · (s1,J1). Then

Ai(εw) =

{εw1 if ji ≥ ji+1

0 if ji < ji+1

where w1 = (sn,jn)(sn−1,jn−1) · · · (si+1,ji+1

)(si,ji) · · · (s1,j1) with ji+1 = ji,

ji = ji+1 − 1 and jk = jk if k 6= i, i+ 1.

Lemma 4.9. ri(εsj ) =

{εsi−1 − εsi − εsi+1 if i = jεsj if i 6= j.

The integral cohomology of SUn+1/T is generated by Schubert classes indexed


W = {snjnsn−1,jn−1. . . s1j1 : ji = 0 or i ≤ ji ≤ n}.

Let xi = εri ∈ H2(SUn+1/T,R). We would like to define an order between genera-tors of the integral cohomology of SUn+1/T . Since each element εsnjnsn−1,jn−1

...siji ...s1j1

can be represented by an n-tuple (jn−n+1, jn−1−(n−1)+1, . . . , ji−i+1, . . . , j1−1 + 1),

we can define an order between n-tuples.

Definition 4.10. (Graded Inverse Lexicographic Order)Let α = (α1, α2, . . . , αn) and β = (β1, β2, . . . , βn) ∈ Zn≥0. We say α > β

if |α| = α1 + α2 + . . . αn > |β| = β1 + β2 + . . . βn or |α| = |β| and in thevector difference α − β ∈ Zn, the right-most nonzero entry is positive. We willwrite εsnjnsn−1,jn−1

...siji ...s1j1 > εsnknsn−1,jk−1...siki ...s1j1 if (jn − n + 1, jn−1 − (n −

1) + 1, . . . , ji − i− 1, . . . , j1 − 1 + 1) > (kn − n+ 1, kn−1 − (n− 1) + 1, . . . , ki − i−1, . . . , k1 − 1 + 1).

Example 4.11. εs35s23s14 > εs35s24s13 since (3, 2, 4) > (3, 3, 3) in graded inverselexicographic order.

We will try to find a quotient ring Z[x1, x2, . . . , xn]/I which is isomorphic toH∗(SUn+1/T,R). We also define an order between monomials as follows.

Definition 4.12. We say xα11 xα2

2 . . . xαnn > xβ1

1 xβ2

2 . . . xβnn if |α| = α1+α2+. . . αn >|β| = β1+β2+. . . βn or |α| = |β| and in the vector difference α−β ∈ Zn the left-mostnon-zero entry is negative.

Example 4.13. x41x

22x

33 < x3

1x32x

33, since (4, 2, 3)− (3, 3, 3) = (1,−1, 0).

Lemma 4.14. xα11 xα2

2 . . . xαnn = εsnαnsn−1,αn−1...siαi ...s1α1 + lower terms.

Proof. To prove this, we use induction on degree of the monomials. By definitionxi = εsi . Let us compute xixj = εsiεsj . Here we may assume that i ≤ j. Ifj − i > 1, the inverse of sisj is sisj . Hence

P sjsisisj = rjAi(εsi) = rj(1) = 1

in the cup product. If j = i+ 1, the inverse of si+1si is sisi+1. In this case

Psi,si+1= Airi+1(εsi) = Ai(εsi) = ε{} = 1.

If i = j, then we have to consider the word si,i+1. Its inverse si+1si and

P si,i+1sisi = ri+1A

i(εsi) = ri+1(1) = 1.

Now we have to show that P skslsisj = 0 if εsksl > εsjsi . By definition of cup productthe coefficient of εsksl is not zero only if si → sksl and sj → sksl. However, this ispossible only if sksl = sjsi or sksl = si,i+1 when j = i+1. Clearly εsisi+1 < εsi+1si .Hence εsiεsi+1 = εsi+1si + lower terms and εsiεsj = εsjεsi if j − i > 1. In the casei = j, we have to look elements sisk and sksi. The inverse of sksi is equal to sksiitself if k − i > 1, hence

P sksisisj = Akri(εsi) = Ak(εsi−1 − εsi + εsi+1) = 0

since k−i > 1. Clearly εsisk < εsisi+1 if k < i. Hence εsiεsi = εsisi+1 +lower terms.Assumexα1

1 xα22 . . . xαnn = εsnαnsn−1,αn−1

...siαi ...s1α1 + lower terms.


We have to show xα11 xα2

2 . . . xαi+1i . . . xαnn = εsnαnsn−1,αn−1

...siαi+1...s1α1 +lower termsby Bruhat ordering.snαnsn−1,αn−1 . . . siαi+1 . . . s1α1 → w′ only if w′ = snαnsn−1,αn−1 . . . siαi . . . s1α1

where there exists an index j for which αj = αj + 1 and αk = αk if k 6= j.By given ordering

w′ = snαnsn−1,αn−1. . . siαi . . . s1α1

> snαnsn−1,αn−1. . . siαi+1 . . . s1α1

.

If j > i, then, by Algorithm 4.1, in w′−1 we will not have a subsequencesj−1, sj−2 . . . si after the elements sj . Hence in the cup product before applying Aj

we will not have the term εsj . It means Pw′

si,w = 0.

If j = i, then, again by Algorithm 4.1, in w′−1 we will not have a subsequencesj−1, sj−2 . . . si after the elements sj . Hence in the cup product before applying Aj

we will not have the term εsj . It means Pw′

si,w = 1 if and only if j > i. �

Example 4.15. Let l = 3;x1x2x3 = εs3s2s1 + lower terms.x2

1x2x3 = εs3s2s12 + lower terms. Then we haveεs3s23s1 > εs3s2s12 > εs23s12 > εs3s13 > εs2s13 . Sincethe inverse of s3s23s1 is s3s13 and the inverse of s3s2s1 is s13,A3r1r2r3(εs1) = A3r1(εs1) = A3(−εs1 + εs2) = 0.Similarly since the inverse of s3s2s12 is s2s13,A2r1r2r3(εs1) = A2r1(εs1) = A2(−εs1 + εs2) = 1.

Before finding the quotient ring Z[x1, . . . , xn]/I, we give some information aboutring k[x1, . . . , xn]/I.

Fix a monomial ordering on k[x1, . . . , xn]. Let f ∈ k[x1, . . . , xn]. The leadingmonomial of f , denoted by LM(f), is the highest degree monomial of f . Thecoefficient of LM(f) is called leading coefficient of and denoted by LC(f). Theleading term of f , LT (f) = LC(f)LM(f).

Let I ⊆ k[x1, . . . , xn] be an ideal. Define LT (I) = {LT (f) : f ∈ I}. Let< LT (I) > be an ideal generated by LT (I).

Proposition 4.16. (Proposition 1 and 4 of 5.3 in [10])

(i): Every f ∈ k[x1, . . . , xn] is congruent modulo I to a unique polynomialr which is a k-linear combination of the monomials in the complement of< LT (I) >.

(ii): The elements of {xα : xα 6∈< LT (I) >} are linearly independent moduloI.

(iii): k[x1, . . . , xn]/I is isomorphic as a k− vector space to

S = Span{xα : xα 6∈< LT (I) >}.

Theorem 4.17. (Theorem 6 of 5.3 in [10]) Let I ⊆ k[x1, . . . , xn] be an ideal

(i): The k-vector space k[x1, . . . , xn]/I is finite dimensional.(ii): For each i, 1 ≤ i ≤ n, there is a polynomial fi ∈ I such that LM(fi) =xmii for some positive integer mi.

Theorem 4.18. H∗(SUn+1/T,R) isomorphic to Z[x1, x2, . . . , xn]/ < f1, f2, . . . , fn >where LT (fi) = xn−i+2

i with respect to monomial order given by Definition 4.12.


Proof. Let I be the ideal such that H∗(Sn+1/T,R) ∼= Z[α1, α2, . . . , αn]/I. Sincewe found one to one correspondence between length l elements of H∗(SUn+1/T,R)and monomials xα1

1 xα22 · · ·xαnn , where α1 +α2 + · · ·αn = l and for each i, 1 ≤ i ≤ n,

αi ≤ n− i+ 1, there should be a polynomial fi ∈ I such that LT (fi) = xn−i+2i . �

Example 4.19. Let n = 3. Then we haveαi ≤ n− i+ 1, i = 1, 2, 3;α1 ≤ 3, α2 ≤ 2, α3 ≤ 1.For l = 1; x1, x2, x3; andfor l = 2; x2

1, x1x2, x1x3, x2x3, x22. So we must have a polynomial f3 with

LM(f3) = x23.

For l = 3; x31, x

21x2, x

21x3, x1x2x3, x1x

22, x

22x3, so

we must have a polynomial f2 with LM(f2) = x32.

For l = 4; x31x2, x

31x3, x

21x2x3, x

21x

22, x1x

22x3, so

we must have a polynomial f1 with LM(f1) = x41.

Since the unique highest element has length of n(n+1)2 , we now give the result

about the multiplication of elements of length k and of length n(n+1)2 − k.

Theorem 4.20. Let A = εsnjnsn−1,jn−1···s1j1 be an element of length k and B =

εsnpnsn−1,pn−1···s1p1 be an element of length n(n+1)

2 − k. The corresponding polyno-mials in Z[x1, x2, . . . , xn]/ < f1, f2, . . . , fn > has leading monomials

xj1−1+11 xj2−2+1

2 · · ·xji−i+1i · · ·xjn−n+1

1 and xp1−1+11 xp1−2+1

2 · · ·xpn−n+11 , respectively.

Then

A ·B =

{εsn,nsn−1,n,...,sin,...,s1n , if ji + pi + 1 = n+ i;0, if ji + pi + 1 6= n+ i.

Proof. The unique highest degree monomial in Z[x1, x2, . . . , xn]/ < f1, f2, . . . , fn >is xn1x

n−12 · · ·xn−i+1

i · · ·xn. The multiplication of leading monomials of correspond-ing monomials of A and B produce the monomial

xj1+p11 xj2+p2−2

2 · · ·xji+pi−2i+2i · · ·xjn+pn−2n+2

n .

If ji + pi − 2i + 2 = n − i + 1 → ji + pi + 1 = n + i for each i, i ≤ 1 ≤ n, thenthe multiplication gives the xn1x

n−12 . . . xn. Since this monomial correspondence the

element εsn,nsn−1,n···sin···s1n , A ·B = εsn,nsn−1,n···s1n If ji + pi + 1 6= n+ i, then theleading monomial so the monomials of lower degree must reduce to zero modulo< f1, f2, . . . , fn > in k[x1, x2, . . . , xn] when we apply the division algorithm. HenceA ·B = 0.

�

Now we can give the whole computation of the quotient ring Z[x1, x2, x3]/ <f1, f2, f3 >.

Example 4.21. Let x1 = εs1 , x2 = εs2 , x3 = εs3 .For l = 2, we have


x2x3 = εs3s2 + εs2s3

x22 = εs2s3 + εs2s1

x1x3 = εs3s1

x1x2 = εs2s1 + εs1s2

x21 = εs1s2 ,

andx2x3

x22

x1x3

x1x2

x21

= M

εs3s2

εs2s3

εs3s1

εs2s1

εs1s2

and

εs3s2

εs2s3

εs3s1

εs2s1

εs1s2

= M−1

x2x3

x22

x1x3

x1x2

x21

where

M =

1 1 0 0 00 1 0 1 00 0 1 0 00 0 0 1 10 0 0 0 1

M−1 =

1 −1 0 1 −10 1 0 −1 10 0 1 0 00 0 0 1 −10 0 0 0 1

. Then we have

εs3s2 = x2x3 − x22 + x1x2 − x2

1

εs2s3 = x22 − x1x2 + x2

2

εs3s1 = x1x3

εs2s1 = x1x2 − x21

εs1s2 = x21.

Here we must have a relation involving x23 and we have it as

x23 = εs3s2 = x2x3 − x2

2 + x1x2 − x21.


For l = 3;

x22x3 = εs3s2s3 + εs3s2s1 + εs2s3s1

x1x2x3 = εs3s2s1 + εs2s3s1 + εs3s1s2 + εs1s2s3

x1x22 = εs2s3s1 + εs2s1s2 + εs1s2s3

x21x3 = εs3s1s2 + εs1s2s3

x21x2 = εs2s1s2 + εs1s2s3

x31 = εs1s2s3 ,

andx2

2x3

x1x2x3

x1x22

x21x3

x21x2

x31

= M

εs3s2s3

εs3s2s1

εs2s3s1

εs3s1s2

εs2s1s2

εs1s2s3

and

εs3s2s3

εs3s2s1

εs2s3s1

εs3s1s2

εs2s1s2

εs1s2s3

= M−1

x2

2x3

x1x2x3

x1x22

x21x3

x21x2

x31

where

M =

1 1 1 0 0 00 1 1 1 0 10 0 1 0 1 10 0 0 1 0 10 0 0 0 1 10 0 0 0 0 1

M−1 =

1 −1 0 1 0 00 1 −1 −1 1 00 0 1 0 −1 00 0 0 1 0 −10 0 0 0 1 −10 0 0 0 0 1

.

Then we have

εs3s2s3 = x22x3 − x1x2x3 + x2

1x3

εs3s2s1 = x1x2x3 − x1x22 − x2

1x3 + x21x2

εs2s3s1 = x1x22 − x2

1x2

εs3s1s2 = x21x3 − x3

1

εs2s1s2 = x21x2 − x3

1

εs1s2s3 = x31.

Here we must have a relation involving x32 and we now we have it as

x32 = 2εs2s3s1 = 2(x1x

22 − x2

1x2).

For l = 4; we have


x1x22x3 = εs3s2s3s1 + εs3s2s1s2 + 2εs2s3s1s2 + 2εs3s1s2s3

x21x2x3 = εs3s2s1s2 + εs2s3s1s2 + εs3s1s2s3 + εs2s1s2s3

x21x

22 = εs2s3s1s2 + εs2s1s2s3

x31x3 = εs3s1s2s3

x31x2 = εs2s1s2s3

andx1x

22x3

x21x2x3

x21x

22

x31x3

x31x2

= M

εs3s2s3s1

εs3s2s1s2

εs2s3s1s2

εs3s1s2s3

εs2s1s2s3

and

εs3s2s3s1

εs3s2s1s2

εs2s3s1s2

εs3s1s2s3

εs2s1s2s3

= M−1

x1x

22x3

x21x2x3

x21x

22

x31x3

x31x2

where

M =

1 1 2 2 00 1 1 1 10 0 1 0 10 0 0 1 00 0 0 0 1

M−1 =

1 −1 −1 −1 20 1 −1 −1 00 0 1 0 −10 0 0 1 00 0 0 0 1

.

Then

εs3s2s3s1 = x1x22x3 − x2

1x2x3 − x21x

22 − x3

1x3 + 2x31x2

εs3s2s1s2 = x21x2x3 − x2

1x22 − x3

1x3

εs2s3s1s2 = x21x

22 − x3

1x2

εs3s1s2s3 = x31x3

εs2s1s2s3 = x31x2.

We must have a relation involving x41 which is

x1x31 = εs1 .εs1s2s3 = 0.

For l = 5;

x21x

22x3 = εs3s2s3s1s2 + εs3s2s1s2s3 + εs2s3s1s2s3

x31x2x3 = εs3s2s1s2s3 + εs2s3s1s2s3

x31x

22 = εs2s3s1s2s3

and x21x

22x3

x31x2x3

x31x

22

= M

εs3s2s3s1s2

εs3s2s1s2s3

εs2s3s1s2s3

and

εs3s2s3s1s2

εs3s2s1s2s3

εs2s3s1s2s3

= M−1

x21x

22x3

x31x2x3

x31x

22

where


M =

1 1 10 1 10 0 1

M−1 =

1 −1 00 1 −10 0 1

. So

εs3s2s3s1s2 = x21x

22x3 − x3

1x2x3

εs3s2s1s2s3 = x31x2x3 − x3

1x22

εs2s3s1s2s3 = x31x

22.

Hence we don’t have any relation.

For l = 6;x3

1x22x3 = εs3s2s3s1s2s3 and

εs3s2s3s1s2s3 = x31x

22x3.

Now let us multiple elements with lengths of k and 6− k.First M0 = 1 and |det(M0)| = 1.

Degree 1 ∗Degree 5

Elements Leading Monomial in Polynomial Ringεs1 x1

εs2 x2

εs3 x3

εs3s23s12 x21x

22x3

εs3s2s13 x31x2x3

εs23s13 x31x

22

εs3 * εs3s23s12 x21x

22x

23 0

εs3 * εs3s2s13 x31x2x

23 0

εs3 * εs23s13 x31x

22x3 1


32x3 0


22x3 1

εs2 * εs23s13 x31x

32 0


22x3 1

εs1 * εs3s2s13 x41x2x3 0

εs1 * εs23s13 x41x

22 0

Now we will calculate Reidemeister torsion of SU4/T using above multiplication.From multiplication of the second cohomology then we have

M2 =

0 0 10 1 01 0 0

and |det(M2)| = 1.



Elements Leading Monomial in Polynomial Ringεs3s2 x2x3

εs23 x22

εs3s1 x1x3

εs2s1 x1x2

εs12 x21

εs3s23s1 x1x22x3

εs3s2s12 x21x2x3

εs23s12 x21x

22

εs3s13 x31x3

εs2s13 x31x2

εs3s2 * εs3s23s1 x1x32x

23 0

εs3s2 * εs3s2s12 x21x

22x

23 0

εs3s2 * εs23s12 x21x

32x3 0

εs3s2 * εs3s13 x31x2x

23 0


22x3 1

εs23 * εs3s23s1 x1x42x3 0


32x3 0

εs23 * εs23s12 x21x

42 0

εs23 * εs3s13 x31x

22x3 1

εs23 * εs2s13 x31x

32 0


22x

23 0


23 0


22x3 1


23 0

εs3s1 * εs2s13 x41x2x3 0


32x3 0


22x3 1


32 0

εs2s1 * εs3s13 x41x2x3 0


22 0


22x3 1

εs12 * εs3s2s12 x41x2x3 0

εs12 * εs23s12 x41x

22 0

εs12 * εs3s13 x51x3 0

εs12 * εs2s13 x51x2 0


To calculate Reidemeister torsion of SU4/T we need multiplication of fourthcohomology bases elements and then we have

M4 =

0 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 01 0 0 0 0

and |det(M4)| = 1.


Elements Leading Monomial in Polynomial Ringεs3s23 x2

2x3

εs3s2s1 x22x3

εs23s1 x1x22

εs3s12 x21x3

εs2s12 x21x2

εs13 x31

εs3s23 x22x3

εs3s2s1 x1x2x3

εs23s1 x1x22

εs3s12 x21x3

εs2s12 x21x2

εs13 x31


23 0


23 0

εs3s23 * εs23s1 x1x42x3 0


22x

23 0


32x3 0

εs3s23 * εs13 x31x

22x3 1

εs3s2s1 * εs3s23 x1x32x

23 0

εs3s2s1 * εs3s2s1 x21x

22x

23 0

εs3s2s1 * εs23s1 x21x

32x3 0

εs3s2s1 * εs3s12 x31x2x

23 0

εs3s2s1 * εs2s12 x31x

22x3 1

εs3s2s1 * εs13 x41x2x3 0

εs23s1 * εs3s23 x1x42x3 0


32x3 0


42 0


22x3 1


32 0

εs23s1 * εs13 x41x

22 0



22x

23 0


23 0


22x3 1


23 0

εs3s12 * εs2s12 x41x2x3 0

εs3s12* εs13 x51x3 0


32x3 0


22x3 1


32 0

εs2s12 * εs3s12 x41x2x3 0


22 0

εs2s12 * εs13 x51x2 0

εs13 * εs3s23 x31x

22x3 1

εs13 * εs3s2s1 x41x2x3 0

εs13 * εs23s1 x41x

22 0

εs13 * εs3s12 x51x3 0

εs13 * εs2s12 x51x2 0

εs13 * εs13 x61 0

To calculate Reidemeister torsion of SU4/T we need multiplication of sixth co-homology bases elements and then we have

M6 =

0 0 0 0 0 10 0 0 0 1 00 0 0 1 0 00 0 1 0 0 00 1 0 0 0 01 0 0 0 0 0

and |det(M6)| = 1.

Hence the Reidemeister torsion of SU4/T is 1 by the Reidmeister torsion formulafor manifolds.

By Theorem 4.18 and 4.20 we reach the following result.

Theorem 4.22. The Reidemeister torsion of SUn+1/T is always 1 for any anypositive integer n with n ≥ 3.


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AIBU Izzet Baysal University Department of Mathematics Bolu / Turkey & Hacettepe

University Department of Mathematics Ankara / TurkeyE-mail address: [email protected];[email protected];[email protected];[email protected]

ON REIDEMEISTER TORSION OF FLAG MANIFOLDS OF COMPACT SEMISIMPLE LIE GROUPS

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