A correspondence between boundary coefficients of real flag manifolds and height of roots
aa r X i v : . [ m a t h . A T ] S e p A CORRESPONDENCE BETWEEN BOUNDARY COEFFICIENTS OF REALFLAG MANIFOLDS AND HEIGHT OF ROOTS
JORDAN LAMBERT AND LONARDO RABELO
Abstract.
In this paper we prove a new formula for the coefficients of the cellular homology of realflag manifolds in terms of the height of certain roots. In particular, for flag manifolds of type A, we geta very simple formula for these coefficients and an explicit expression for the first and second homologygroups with integer coefficients. Introduction
In this work we revisit the determination of the coefficient of the boundary map for the cellular ho-mology of real flag manifolds, a problem which is equivalent to that of finding the incidence coefficientsof the differential map for the cohomology. It follows a series of previous work on this subject likeKocherlakota [5] with its Morse Theory approach, Casian-Stanton [2] by a representation theoreticalview, Rabelo-San Martin [12] within the frame of the cellular homology and, lastly, Matszangosz [8]throughout the cohomology of the Vassiliev complex. We prove a new formula for these coefficients interms of the height of certain roots.A generalized flag manifold F is a homogeneous space G { P , where G is a real noncompact semisimpleLie Group and P is a parabolic subgroup. It admits a cellular decomposition called Bruhat decom-position, where the cells are the Schubert cells and parametrized by the Weyl group W . Considera pair of Schubert cells S w , S w such that w covers w , i.e., w ď w by the Bruhat-Chevalley orderand ℓ p w q “ ℓ p w q ´
1. In this case, there is a root β such that w “ s β ¨ w . According to both [5]and [12], we may summarize how to compute the coefficient c p w, w q as it follows: Consider the setΠ w of positive roots sent to negative by w ´ and denote by φ p w q the sum of all such roots. Hence, c p w, w q “ ˘p ` p´ q κ p w,w q q , where κ is obtained from the equation φ p w q ´ φ p w q “ κ p w, w q ¨ β . Thepapers [11] and [6] apply this procedure in the context of the Isotropic Grassmannians. The resultsobtained in these works (for instance, see [7], Theorem 3.12) suggest a formula for the coefficients interms of the height of some root. This is our first Theorem, which is remarkable by its simplicity. Theorem.
If we write w “ w ¨ s γ , it follows that κ p w, w q “ ht p γ _ q , where ht p γ _ q is the height of thecorresponding dual coroot γ _ with respect to the dual system of roots. In other words, the parity of ht p γ _ q determines whether the coefficient c p w, w q is zero or ˘ β and γ or, equivalently, by the right and left actions in theunderstanding of this topic.In the context of flags of type A n ´ , this theorem simplifies a lot the task of computing the coefficient.Schubert cells are parametrized by the symmetric group S n . If we denote a permutation by the one-linenotation as w “ w ¨ ¨ ¨ w n then w covers w if and only if there is i ă j such that w “ w ¨ p i, j q , w i ă w j and there is no i ă k ă j such that w i ă w k ă w j . Since the simple system of roots is equal to itsdual, it is immediate to conclude that κ p w, w q “ j ´ i (see Proposition 3.13).It provides a very nice link between combinatorics and topology. As a consequence, we retrieve theorientability condition for this class of flags according to Patr˜ao-San Martin-Santos-Seco [9] (Proposi-tion 3.14).We may also go further and derive an explicit formula for 1, 2-homology of any partial flag manifoldof type A. Mathematics Subject Classification.
Primary: 05A05, 14M15, 57T15.
Key words and phrases.
Real flag manifolds, Symmetric Group, Schubert cell, Homology.This work was supported by the Coordination for the Improvement of Higher Level Personnel – Capes.
Theorem.
Given any partial flag manifold F Θ of type A : (1) For n ě , the 1-homology is given by H p F Θ , Z q – p Z q n ´| Θ |´ . (2) For n ě , the 2-homology is given by H p F Θ , Z q – p Z qp n ´| Θ |´ q ` r Θ ´ , where r Θ is the number of connected components of the Dynkin diagram of Θ . With respect to the 1-homology group, as the abelianization of the fundamental group, it may bealso derived from the work of Wiggerman [14] which gives a presentation of Π p F Θ q with generatorsin the set Σ z Θ subject to some relations. In this sense, although it is not a kind of new result, ourcombinatorial approach presents a direct and simple computation for such abelianization. A similarresult that should be mentioned is that in the context of the 2-homotopy of the complex flag manifolds,Grama-Seco ([4], Theorem 2.4) obtained by geometric methods that it is generated by 2-spheres whichare also enumerated by the complement of Θ inside Σ. With respect to the 2-homology group, DelBarco-San Martin ([3], Theorem 4.1) has shown that it is a torsion group Z N without providing anydescription for N .The article is organized as follows. In the Section 2, we introduce the main definitions about flagmanifolds, root and coroot systems, Bruhat decomposition, and combinatorics of the symmetric group.In the next Section 3, we prove the height formula for the coefficients and derive some consequences.Finally, in the Section 4 we point some further directions out.2. Preliminaries
Let N “ t , , , . . . u and Z be the set of integers. For n, m P Z , where n ď m , denote the set r n, m s “ t n, n ` , . . . , m u . For n P N , denote r n s “ r , n s .2.1. Flag Manifolds.
We will begin by defining all required structure to deal with flag manifolds.We define the flag manifolds as homogeneous spaces G { P where G is a non-compact semi-simpleLie group and P is a parabolic subgroup of G . The flag manifolds for the several groups G withnon-compact real semi-simple Lie algebra g are the same.If g “ k ‘ s is a Cartan decomposition, let a be a maximal abelian sub-algebra contained in s . Asub-algebra h Ă g is said to be a Cartan sub-algebra if h C is a Cartan sub-algebra of g C . If h “ a is aCartan sub-algebra of g , we say that g is a split real form of g C .Let Π be the set of roots of the pair p g , a q and fix a simple system of roots Σ Ă Π. Denote by Π ˘ respectively the set of positive and negative roots and by a ` the Weyl chamber a ` “ t H P a : α p H q ą α P Σ u . The direct sum of root spaces corresponding to the positive roots is denoted by n “ ř α P Π ` g α . The Iwasawa decomposition of g is given by g “ k ‘ a ‘ n . The notations K and N refer respectively to the connected subgroups whose Lie algebras are k and n .A minimal parabolic sub-algebra of g is given by p “ m ‘ a ‘ n where m is the centralizer of a in k .Let P be the minimal parabolic subgroup with Lie algebra p . Note that P is the normalizer of p in G .We call F “ G { P the maximal flag manifold of G and denote by b the base point 1 ¨ P in G { P .Now, assume that Θ Ă Σ is any subset of simple roots. Such a choice provides a very interestingway to obtain several flag manifolds, called partial flag manifolds as we now explain. Denote by g p Θ q the semi-simple Lie algebra generated by g ˘ α , α P Θ. Let G p Θ q be the connected group with Liealgebra g p Θ q . Yet, let n Θ be the sub-algebra generated by the roots spaces g ´ α , α P Θ and consider p Θ “ n Θ ‘ p . The normalizer P Θ of p Θ in G is a standard parabolic subgroup which contains P .Finally, the corresponding flag manifold F Θ “ G { P Θ is called a partial flag manifold of G of type Θ.We denote by b Θ the base point 1 ¨ P Θ in G { P Θ . CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 3
Root Systems and Coroots.
In this section, we highlight some results about the dual systemof a root system which will play a key role in the sequel. We follow closely the book of Perrin [10].Let E be a finite dimensional vector space. A set Π Ă E is an abstract system of roots if it is finite,spans E and does not contain 0 and satisfies(1) for every α P Π, there exists a reflection s α with respect to α such that s α p Π q “ Π;(2) for every α, β P Π, s α p β q ´ β is an integer multiple of α .The Weyl group W of the root system Π is the group generated by reflections s α , α P Π. It ispossible to show that there exists an inner product x¨ , ¨y in E which is invariant by W . It follows that s α is the corresponding orthogonal reflection s α p β q “ β ´ x β, α yx α, α y α. From the identification of E with E ˚ by x¨ , ¨y , for each root α P Π, let us denote by α _ “ α x α, α y .We call α _ the coroot of α . It follows that if β P Π is another root,(1) s α p β q “ β ´ x α _ , β y α. We know that if we choose x¨ , ¨y to be the Killing form, then the set of roots Π associated to the Liealgebra g is an abstract root system. Furthermore, the set of coroots Π ˚ is also a root system in E ˚ which is called the dual root system. Proposition 2.1 ([10], Proposition 11.4.1) . Let α, β P Π . Then, p s α β q _ “ s α _ p β _ q . Hence, we derive the following corollary that will be very useful.
Corollary 2.2.
Suppose that α P Π is given by α “ s ¨ ¨ ¨ s m ´ p δ m q such that s i “ s δ i is the simplereflection associated to δ i P Π . Then, the coroot α _ can be written as follows (2) α _ “ s δ _ ¨ ¨ ¨ s δ _ m ´ p δ _ m q . A system of simple roots Σ Ă Π is a basis of E such that every root α P Π is written as a linearcombination with integer coefficients with the same signal, i.e., all of them either non-negative or non-positive. A root system is called reduced when for α P Π, R α X Π “ t´ α, α u . Such a reduced rootsystem is always the set of roots of a Cartan subalgebra over an algebraically closed field. Non-reducedroot systems appear in real semi-simple Lie algebras. However, split real forms correspond to reducedroot systems. Proposition 2.3 ([10], Prop. 11.6.13) . If Π is reduced then Σ ˚ “ t α _ : α P Σ u is a simple root systemof Π ˚ . By this Proposition, if α P Π is given in terms of the system Σ of simple roots as(3) α “ ÿ δ P Σ d δ δ we should obtain an analogous expression for α _ with respect to the simple root system Σ ˚ . Indeed,it follows that(4) α _ “ ÿ δ P Σ d δ x δ, δ yx α, α y δ _ As a consequence, if α _ “ ř δ _ P Σ ˚ d ˚ δ δ _ is the coroot of α given by (3), the relationship between itscoefficients is given by d ˚ δ “ d δ x δ, δ yx α, α y . By Equation 4, we also conclude that if g is of type A, D, E then its dual root system is isomorphicto itself while the dual root system of a lie algebra of type B is isomorphic to the root system of typeC (and vice-versa).
CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 4
The height of the root α , denoted by ht p α q is the sum of the coefficients that appear in the decom-position of α in Equation (3) ht p α q “ ÿ δ P Σ d δ . Bruhat decomposition.
If we consider the elements of W as product of simple reflections s i “ s α i , α i P Σ, it is defined the length ℓ p w q of w P W as the number of simple reflections in anyreduced decomposition of w .There is a partial order ď in the Weyl group called the Bruhat-Chevalley order: we say that w ď w if given a reduced decomposition w “ s j ¨ ¨ ¨ s j r then w “ s j i ¨ ¨ ¨ s j ik for some 1 ď i ď ¨ ¨ ¨ ď i r ď r .When there exists w, w P W such that w ď w and ℓ p w q “ ℓ p w q ` w covers w (alternatively, w, w is a covering pair). If w covers w and given a reduced decomposition w “ s ¨ ¨ ¨ s ℓ then we will denote by I the integer in r ℓ s such that w “ s ¨ ¨ ¨ p s I ¨ ¨ ¨ s ℓ , where the integer I dependson w and the choice of the reduced decomposition of w . For convenience, we will sometimes refer tothis decomposition of w as p w I .For the subset Θ Ă Σ, we define the subgroup W Θ generated by the reflections with respect to theroots α P Θ. We denote by W Θ the subset of minimal representatives of the cosets of W Θ in W .The Bruhat decomposition presents flag manifolds as union of N -orbits, namely, F Θ “ ž w P W Θ N ¨ wb Θ . Each orbit N ¨ wb Θ , w P W , is called a Bruhat cell. It is diffeomorphic to a euclidean space and, inthe case of a split real form, its dimension coincides with the length of w , i.e., dim p N ¨ wb Θ q “ ℓ p w q . A Schubert variety S w is the closure of a Bruhat cell. The Bruhat-Chevalley order also characterizesa partial order between the corresponding Schubert varieties. It also endows the flag manifolds with acellular structure where S w “ Ť u ď w N ¨ ub Θ .2.4. Recursive formula for the roots of Π w . A very important role is developed by the set Π w “ Π ` X w Π ´ composed of the positive roots sent to negative roots by w ´ . If w “ s ¨ ¨ ¨ s m is a reduceddecomposition of w then Π w “ t β , . . . , β m u where β k “ s ¨ ¨ ¨ s k ´ p δ k q , for 1 ď k ď m. (5)In particular, we have that ℓ p w q equals the cardinality of Π w . In this section, we consider a slightlymore general setting in which we derive a recursive formula for a root that is obtained after a finitesequence of composition of reflections over a simple root. As a consequence, we obtain a formula forthose roots of Π w .Consider a finite ordered sequence of simple roots p δ , . . . , δ m q , eventually with repetition. We areinterested in writing s ¨ ¨ ¨ s m ´ p δ m q in terms of δ , . . . , δ m . When m “
1, it is trivial. However, as m increases, we may observe the occurrence of a pattern. Let us show what happens for m “ , , s p δ q “ δ ´ x δ _ , δ y δ s s p δ q “ δ ´ x δ _ , δ y δ ` p´ x δ _ , δ y ` x δ _ , δ y x δ _ , δ yq δ s s s p δ q “ δ ´ x δ _ , δ y δ ` p´ x δ _ , δ y ` x δ _ , δ y x δ _ , δ yq δ `` p´ x δ _ , δ y ` x δ _ , δ y x δ _ , δ y ` x δ _ , δ y x δ _ , δ y ´ x δ _ , δ y x δ _ , δ y x δ _ , δ yq δ . By these equations, it turns out that s ¨ ¨ ¨ s m ´ p δ m q is a combination of some integer coefficientsin terms of the roots δ , . . . , δ m . The coefficients are given as alternating sums of products of theKilling numbers of the roots that belong to a specific interval. Besides, as m increases, the sum isgiven by products of a greater number of factors. We now proceed to obtain a general formula for thisexpression. CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 5
Let us define a sum formula in terms of the ordered sequence of roots p δ , . . . , δ m q : given integers x, y such that 1 ď x ă y ď m and 0 ď l ă y ´ x , we define(6) P lx,y p δ , . . . , δ m q “ ÿ x ă j 㨨¨ă j l ă y x δ _ x , δ j y x δ _ j , δ j y ¨ ¨ ¨ x δ _ j l ´ , δ j l y x δ _ j l , δ y y , which the sum runs among all choices of l -uples x ă j ă ¨ ¨ ¨ ă j l ă y of the following product x δ _ x , δ j y x δ _ j , δ j y ¨ ¨ ¨ x δ _ j l , δ y y . In particular, for l “ P x,y p δ , . . . , δ m q “ x δ _ x , δ y y as the Killing number of the x -th coroot with the y -th root of the sequence p δ , . . . , δ m q .Notice that this definition depends on the order of the roots and the integers x and y correspondsrespectively to the position in the sequence that gives the first coroot and the last root of the factorsof the product.By definition of P , we have the following property: suppose that x ą a , for some integer a . Then(8) P lx,y p δ , . . . , δ m q “ P lx ´ a,y ´ a p δ a ` , . . . , δ m q . In particular, if x ą P lx,y p δ , . . . , δ m q “ P lx ´ ,y ´ p δ , . . . , δ m q . The next lemma presents a recursive formula for P . Lemma 2.4.
Given x, y, l such that ď x ă y ď m and ď l ă y ´ x , we have that P l ` x,y p δ , . . . , δ m q “ y ´ l ´ ÿ k “ x ` x δ _ x , δ k y P lk,y p δ , . . . , δ m q Proof. If l “
0, it follows that y ´ ÿ k “ x ` x δ _ x , δ k y P k,y p δ , . . . , δ m q “ y ´ ÿ k “ x ` x δ _ x , δ k y x δ _ k , δ y y “ ÿ x ă k ă y x δ _ x , δ k y x δ _ k , δ y y “ P x,y p δ , . . . , δ m q . For l ą
0, it follows that y ´ l ´ ÿ k “ x ` x δ _ x , δ k y P lk,y p δ , . . . , δ m q “ y ´ l ´ ÿ k “ x ` x δ _ x , δ k y ÿ k ă j 㨨¨ă j l ă y x δ _ k , δ j y x δ _ j , δ j y ¨ ¨ ¨ x δ _ j l ´ , δ j l y x δ _ j l , δ y y“ ÿ x ă k ă j 㨨¨ă j l ă y x δ _ x , δ k y x δ _ k , δ j y x δ _ j , δ j y ¨ ¨ ¨ x δ _ j l ´ , δ j l y x δ _ j l , δ y y“ P l ` x,y p δ , . . . , δ m q . (cid:3) It motivates the following general result.
Proposition 2.5.
Let δ , . . . , δ m , with m ą , be an ordered sequence of simple roots whose simplereflections are, respectively, s , . . . , s m . Then (10) s ¨ ¨ ¨ s m ´ p δ m q “ δ m ` m ´ ÿ i “ ˜ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , . . . , δ m q ¸ ¨ δ i . Proof.
We will prove by induction in the number m of roots. For m “
2, we have that s p δ q “ δ ´ x δ _ , δ y δ which satisfies Equation (10).For m ą
2, denote s ¨ ¨ ¨ s m ´ p δ m q “ s p δ q such that δ “ s ¨ ¨ ¨ s m ´ p δ m q . If we consider the orderedsequence of roots δ “ δ , . . . , δ m ´ “ δ m which has p m ´ q elements, it is possible to apply theinductive hypothesis in δ such that δ “ s ¨ ¨ ¨ s m ´ p δ m q “ δ m ´ ` p m ´ q´ ÿ i “ ¨˝ p m ´ q´ i ´ ÿ l “ p´ q l ´ P li,m ´ p δ , . . . , δ m ´ q ˛‚ ¨ δ i CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 6 “ δ m ` m ´ ÿ i “ ˜ m ´ i ´ ÿ l “ p´ q l ´ P li ´ ,m ´ p δ , . . . , δ m q ¸ ¨ δ i . For every 2 ď i ď m ´ ď l ď m ´ i ´
1, by Equation (9), we have that P li ´ ,m ´ p δ , . . . , δ m q “ P li,m p δ , δ , . . . , δ m q . Then, s p δ q “ s ˜ δ m ` m ´ ÿ i “ ˜ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , δ , . . . , δ m q ¸ ¨ δ i ¸ “ δ m ` m ´ ÿ i “ ˜ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , δ , . . . , δ m q ¸ ¨ δ i ´ x δ _ , δ m ` m ´ ÿ i “ ˜ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , δ , . . . , δ m q ¸ ¨ δ i y ¨ δ All coefficients which accompany δ , . . . , δ m are equal to the respective coefficients in Equation (10).It remains to verify that it is also true for the coefficients that accompany the root δ in both equations,i.e., m ´ ÿ l “ p´ q l ´ P l ,m p δ , . . . , δ m q “ ´ x δ _ , δ m y ´ m ´ ÿ i “ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , δ , . . . , δ m q ¨ x δ _ , δ i y . We will show that the right side is equal to the left one. Firstly, by a change in the order of thesummation, we have that m ´ ÿ i “ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , δ , . . . , δ m q ¨ x δ _ , δ i y“ m ´ ÿ l “ p´ q l ´ m ´ l ´ ÿ i “ x δ _ , δ i y P li,m p δ , δ , . . . , δ m q“ m ´ ÿ l “ p´ q l ´ P l ` ,m p δ , . . . , δ m q p by Lemma 2.4 q Finally, by definition, ´ x δ _ , δ m y “ ´ P ,m p δ , . . . , δ m q . Hence, ´ x δ _ , δ m y ´ m ´ ÿ i “ m ´ i ´ ÿ l “ p´ q l ´ P li,m p δ , δ , . . . , δ m q ¨ x δ _ , δ i y“ ´ P ,m p δ , . . . , δ m q ` m ´ ÿ l “ p´ q l P l ` ,m p δ , . . . , δ m q “ m ´ ÿ l “ p´ q l ´ P l ,m p δ , . . . , δ m q . (cid:3) A new formula for coefficient of the boundary map
In this section we present a formula for the coefficients of the boundary map of real flag manifoldsby means of the height of some roots in the Lie algebra. Our main applications will be given alone inthe context of split real forms.3.1.
Algebraic Expression for the coefficients.
Let us begin by reviewing some main results aboutthe determination of the cellular homology coefficients following [12]. We start in the context of themaximal flag manifold F . Given a Schubert variety S w , we fix once and for all reduced decompositions w “ s ¨ ¨ ¨ s ℓ as a product of simple reflections, for each w P W , with ℓ “ ℓ p w q . Let C be the Z -module freelygenerated by S w , w P W . The boundary map B defined over C is given by B S w “ ř w c p w, w q S w , CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 7 where c p w, w q P Z in such way that non-trivial coefficients may occur when w covers w . Furthermore,the non-trivial coefficients must be equal to ˘ w, w to be a covering pair is equivalentto say that if w “ s ¨ ¨ ¨ s ℓ is a reduced decomposition of w P W as a product of simple reflections,then w “ s ¨ ¨ ¨ p s I ¨ ¨ ¨ s ℓ is a uniquely defined reduced decomposition with g p α I q – sl p , R q .It will also be useful to denote by v “ s ¨ ¨ ¨ s I ´ and u “ s I ` ¨ ¨ ¨ s ℓ such that w “ v ¨ u . There areroots not necessarily simple β “ v p α I q and γ “ u ´ p α I q such that w “ s β ¨ w and w “ w ¨ s γ . Let us determine if c p w, w q is either 0 or ˘
2. We define(11) σ p w, w q “ ÿ δ P Π u x α _ I , δ y ¨ dim g δ For w P W , let φ p w q “ ÿ δ P Π w dim g δ ¨ δ The following results show how we determine when c p w, w q is either 0 or ˘ Proposition 3.1 ([12], Proposition 2.7) . Let β the unique root such that w “ s β w . Then φ p w q ´ φ p w q “ κ p w, w q ¨ β where κ p w, w q “ ´ σ p w, w q . Theorem 3.2 ([12], Thm. 2.8, [5], Thm. 1.1.4) . Suppose that w covers w . Then the coefficient c p w, w q is given as follows: c p w, w q “ ˘ ´ ` p´ q κ p w,w q ¯ Now, we address the question about the determination of the signal. This method is developed in[12] whose argument is based on the reduced decompositions of the elements w P W .For each w fix a reduced decomposition. There is a first p´ q I ingredient which is related with thedeleted position. The second component appears when the fixed reduced decomposition for w is notequal to that p w I . According to [12] Proposition 1.9, there are characteristic maps for S w given byΦ w : B w Ñ S w and Φ p w I : B p w I Ñ S w , where B w and B p w I are balls of dimension ℓ p w q . The first mapis obtained by the choice of reduced decomposition for w whereas the latter follows from the deletionoperation. In addition, there is a property where both maps Φ w and Φ p w I are diffeomorphisms whenrestricted to the interior of the respective balls. Theorem 3.3 ([12], Theorem 2.8) . c p w, w q “ p´ q I ¨ deg ` Φ ´ w ˝ Φ p w I ˘ ¨ p ` p´ q κ p w,w q q . Remark 3.4.
When both reduced decompositions w and p w I are equal then deg ` Φ ´ w ˝ Φ p w I ˘ “ .We compute the degree of the composition Φ ´ w ˝ Φ p w I considered as map between spheres in whichthe boundaries of the balls are collapsed to points. B w Φ w / / (cid:15) (cid:15) S w (cid:15) (cid:15) B p w I Φ p wI o o (cid:15) (cid:15) B w {Bp B w q / / σ w B p w I {Bp B p w I q o o We denote by σ w “ S w {p S w z N ¨ w b q the space obtained by identifying the complement of the Bruhatcell N ¨ w b to a point. Remark 3.5.
Let Θ Ă Σ and consider the partial flag manifold F Θ . The coefficients c p w, w q , where w, w P W Θ , are obtained in the same way as a restriction of the boundary operator to the Z -modulegenerated by the corresponding Schubert cells (for details, see [12] Theorem 3.4).
CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 8
Height formula.
We now seek to find a formula for the coefficients of the cellular homology interms of the height of the roots.This formula previously established for κ p w, w q takes the relationship of w and w by a left action( w “ s β ¨ w ) into account. We will show how to obtain an equivalent expression by exploring the rightaction, i.e, w “ w ¨ s γ , which becomes impressively simple for split real forms.Assume that w “ s ¨ ¨ ¨ s ℓ and w “ s ¨ ¨ ¨ p s I ¨ ¨ ¨ s ℓ “ v ¨ u , with v “ s ¨ ¨ ¨ s I ´ and u “ s I ` ¨ ¨ ¨ s ℓ , bereduced decompositions such that p δ , . . . , δ ℓ q is the corresponding sequence of simple roots associatedwith this decomposition. Our strategy consists on finding an explicit computation of σ p w, w q as definedin Equation (11) in terms of the root u . When u is non-trivial, by Equation (5), recall that the rootsof Π u “ Π ` X u Π ´ are given by β j “ s I ` ¨ ¨ ¨ s j ´ p δ j q , j P r I ` , ℓ s . Lemma 3.6.
For every j P r I ` , ℓ s , x δ _ I , β j y “ j ´ I ´ ÿ l “ p´ q l P lI,j p δ , . . . , δ ℓ q . Proof. If j “ I `
1, by the definition of P , it follows that j ´ I ´ ÿ l “ p´ q l P lI,j p δ , . . . , δ ℓ q “ p´ q P I,I ` p δ , . . . , δ ℓ q “ x δ _ I , δ I ` y . Now, suppose that j ą I ` I ă ℓ ´ β j may be written as β j “ s I ` ¨ ¨ ¨ s j ´ p δ j q “ s ¨ ¨ ¨ s j ´ I ´ p δ j ´ I q , where s k “ s I ` k for k P r j ´ I s with its respective roots. By Proposition 2.5, β j “ δ j ´ I ` j ´ I ´ ÿ k “ ˜ j ´ I ´ k ´ ÿ l “ p´ q l ´ P lk,j ´ i p δ , . . . , δ j ´ I q ¸ ¨ δ k “ δ j ` j ´ I ´ ÿ k “ ˜ j ´ I ´ k ´ ÿ l “ p´ q l ´ P lk,j ´ I p δ I ` , . . . , δ j q ¸ ¨ δ I ` k . By Equation (8), P lk,j ´ I p δ I ` , . . . , δ j q “ P lk ` I,j p δ , . . . , δ j q “ P lk ` I,j p δ , . . . , δ j , . . . , δ ℓ q . We may write the roots β j as β j “ δ j ` j ´ I ´ ÿ k “ ˜ j ´ I ´ k ´ ÿ l “ p´ q l ´ P lk ` I,j p δ , . . . , δ ℓ q ¸ ¨ δ I ` k “ δ j ` j ´ ÿ k “ I ` ˜ j ´ k ´ ÿ l “ p´ q l ´ P lk,j p δ , . . . , δ ℓ q ¸ ¨ δ k . Hence, x δ _ I , β j y “ x δ _ I , δ j y ` j ´ ÿ k “ I ` j ´ k ´ ÿ l “ p´ q l ´ P lk,j p δ , . . . , δ ℓ q x δ _ I , δ k y . By a change in order of the summation, we have that x δ _ I , β j y “ P I,j p δ , . . . , δ ℓ q ` j ´ I ´ ÿ l “ j ´ l ´ ÿ k “ I ` p´ q l ´ P lk,j p δ , . . . , δ ℓ q x δ _ I , δ k y“ P I,j p δ , . . . , δ ℓ q ` j ´ I ´ ÿ l “ p´ q l ´ j ´ l ´ ÿ k “ I ` x δ _ I , δ k y P lk,j p δ , . . . , δ ℓ q CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 9 “ P I,j p δ , . . . , δ ℓ q ` j ´ I ´ ÿ l “ p´ q l ´ P l ` I,j p δ , . . . , δ ℓ q p by Lemma 2.4 q“ j ´ I ´ ÿ l “ p´ q l P lI,j p δ , . . . , δ ℓ q . (cid:3) Corollary 3.7. κ p w, w q “ ` ℓ ÿ j “ I ` ˜ j ´ I ´ ÿ l “ p´ q l ´ P lI,j p δ , . . . , δ ℓ q ¸ ¨ dim ` g β j ˘ .Proof. It follows as a direct application of the Lemma 3.6 on Equation 11. (cid:3)
The expression of κ in Corollary 3.7 becomes plausible when the Lie algebra is a split real form. Theorem 3.8.
Assume that g is a split real form. Let γ “ u ´ p δ I q be the root for which w “ w ¨ s γ .Then κ p w, w q “ ht p γ _ q , where ht p γ _ q is the height of the coroot γ _ in the dual root system Π ˚ .Proof. By hypothesis, g is split real form, which means that dim g α “
1, for every root α P Π. First ofall, if I “ ℓ then γ “ δ ℓ is a simple root and γ _ “ δ _ ℓ . Hence, κ p w, w q “ “ ht p γ _ q .Suppose that I ă ℓ , the root γ can be written as γ “ s ℓ s ℓ ´ ¨ ¨ ¨ s I ` p δ I q . By Corollary 2.2, thecoroot of γ is γ _ “ s δ _ ℓ s δ _ ℓ ´ ¨ ¨ ¨ s δ _ I ` p δ _ I q . Consider the sequence of coroots δ , . . . , δ ℓ ´ I ` given by δ j “ δ _ ℓ ´ j ` , for j P r ℓ ´ I ` s , and their simple reflections s “ s δ _ ℓ , . . . , s ℓ ´ I ` “ s δ _ I . By Proposition2.5, γ _ “ s . . . s ℓ ´ I p δ ℓ ´ I ` q “ δ ℓ ´ I ` ` ℓ ´ I ÿ j “ ˜ ℓ ´ I ´ j ÿ l “ p´ q l ´ P lj,ℓ ´ I ` p δ , . . . , δ ℓ ´ I ` q ¸ ¨ δ j “ δ _ I ` ℓ ´ I ÿ j “ ˜ ℓ ´ I ´ j ÿ l “ p´ q l ´ P lj,ℓ ´ I ` p δ _ ℓ , . . . , δ _ I q ¸ ¨ δ _ ℓ ´ j ` “ δ _ I ` ℓ ÿ j “ I ` ˜ j ´ I ´ ÿ l “ p´ q l ´ P lℓ ´ j ` ,ℓ ´ I ` p δ _ ℓ , . . . , δ _ I q ¸ ¨ δ _ j , where we have replaced j by ℓ ´ j ` j P r I ` , ℓ s , P ℓ ´ j ` ,ℓ ´ I ` p δ _ ℓ , . . . , δ _ I q “ P ℓ ´ j ` ,ℓ ´ I ` p δ , . . . , δ ℓ ´ I ` q “ x δ ℓ ´ j ` , δ ℓ ´ I ` y “ x δ __ j , δ _ I y“ x δ _ I , δ j y “ P I,j p δ , . . . , δ ℓ q since δ j “ δ __ j . For j P r I ` , ℓ s and l P r j ´ I ´ s , P lℓ ´ j ` ,ℓ ´ I ` p δ _ ℓ , . . . , δ _ I q “ P lℓ ´ j ` ,ℓ ´ I ` p δ , . . . , δ ℓ ´ I ` q“ ÿ ℓ ´ j ` ă j 㨨¨ă j l ă ℓ ´ I ` x δ ℓ ´ j ` , δ j y x δ j , δ j y ¨ ¨ ¨ x δ j l , δ ℓ ´ I ` y“ ÿ ℓ ´ j ` ă j 㨨¨ă j l ă ℓ ´ I ` x δ __ j , δ _ ℓ ´ j ` y ¨ ¨ ¨ x δ __ ℓ ´ j l ` , δ _ I y“ ÿ j ą ℓ ´ j ` ą¨¨¨ą ℓ ´ j l ` ą I x δ _ ℓ ´ j ` , δ j y ¨ ¨ ¨ x δ _ I , δ ℓ ´ j l ` y“ ÿ I ă k 㨨¨ă k l ă j x δ _ I , δ k y ¨ ¨ ¨ x δ _ k l , δ j y “ P lI,j p δ , . . . , δ ℓ q . Hence, γ _ “ δ _ I ` ℓ ÿ j “ I ` ˜ j ´ I ´ ÿ l “ p´ q l ´ P lI,j p δ ℓ , . . . , δ I q ¸ ¨ δ _ j . CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 10
Therefore, by Corollary 3.7, the height of the coroot γ _ isht p γ _ q “ ` ℓ ÿ j “ I ` ˜ j ´ I ´ ÿ l “ p´ q l ´ P lI,j p δ ℓ , . . . , δ I q ¸ “ κ p w, w q . (cid:3) Corollary 3.9.
Let g be a Lie algebra of type A n , D n , E , E or E . Then κ p w, w q “ ht p γ q Proof.
In the context of a split real form of a Lie algebra of type A n , D n , E , E or E , all roots havethe same length. Then, ht p α q “ ht p α _ q for all α P Π. (cid:3) Remark 3.10.
We can use Theorem 3.8 to get the formula for F and G Lie algebras. If g is a Liealgebra of type F , suppose that the simple roots are ordered canonically as follows a a a a . Then, κ p w, w q “ ht p r γ q where r γ is the root obtained from γ by reversing the simple roots Σ “ t a , a , a , a u to Σ _ “t a , a , a , a u , respectively, since the dual simple roots correspond to reverse roots in the Dynkin dia-gram, i.e., a _ “ a , a _ “ a , a _ “ a , a _ “ a . The same idea applies to the Lie algebra G . Remark 3.11.
We also can use Theorem 3.8 to get the formula for B n and C n Lie algebras from theisomorphism Π ˚ B – Π C . For instance, in the context of isotropic and odd orthogonal Grassmannians,it coincides with Theorem 3.12 of [7] . Type A case.
In this section, we present some immediate consequences of Theorem 3.8 for flagsassociated with type A Lie algebras. It emphasizes the convenience of the permutation model thesymmetric group provides parametrizing the Schubert cells.Let G “ Sl p n, R q be a Lie group of type A and Σ “ t a , . . . , a n ´ u the simple roots ordered as itfollows: a a a n ´ The respective Weyl group W is the symmetric group S n . We denote a permutation w P S n in theone-line notation by w “ w w ¨ ¨ ¨ w ℓ where w i “ w p i q for all i “ , . . . , ℓ .The following lemma provides a characterization of the covering relation of two permutations usingthe one-line notation. Lemma 3.12 ([1], Lemma 2.1.4) . Let w, w P S n . Then, w covers w in the Bruhat order if and onlyif w “ w ¨ p i, j q for some transposition p i, j q with i ă j such that w p i q ă w p j q and there does not existany k such that i ă k ă j , w p i q ă w p k q ă w p j q . The lemma says that if w “ w ¨ ¨ ¨ w n is the one-line notation of w P S n then w is covered by w ifand only if the one-line notation of w is obtained from w by switching the values in position i and j ,for some pair i ă j and such that no value between positions i and j lies in r w p j q , w p i qs .The next proposition follows from the covering relation stated in Lemma 3.12. Proposition 3.13.
Let w, w P S n such that w is covered by w , i.e., w “ w ¨ p i, j q for some i ă j .Then, the coefficient κ p w, w q is given by κ p w, w q “ j ´ i. In particular, c p w, w q “ if, and only if, j ´ i is odd.Proof. Since γ is the root such that w “ w s γ , the transposition p i, j q is the reflection s γ through γ . Considering the simple reflections s i “ s a i , for a i P Σ, a reduced decomposition for p i, j q is s j ´ ¨ ¨ ¨ s i ` s i s i ` ¨ ¨ ¨ s j ´ . Using the fact that s w p α q “ ws α w ´ we have p i, j q “ s s j ´ ¨¨¨ s i ` p a i q and,then, γ “ s j ´ ¨ ¨ ¨ s i ` p a i q . Applying Proposition 2.5, the root γ is the sum of simple roots a i ` a i ` `¨ ¨ ¨ ` a j ´ . Therefore, by Corollary 3.9, κ p w, w q “ ht p γ q “ j ´ i . (cid:3) CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 11
This proposition simplifies a lot the task to compute the boundary coefficient. For instance, given w “ w “ w “ w ¨ p , q . Hence, κ p w, w q “ ´ “ c p w, w q “ ˘p ` p´ q κ p w,w q q “ ˘ Proposition 3.14.
Let the complement of Θ with respect to Σ be the set of roots t a d , a d , . . . , a d k u where d “ ă d ă ¨ ¨ ¨ ă d k ă n “ d k ` . Then, the flag manifold F Θ is orientable if and only if d j ` ´ d j have the same parity, for every j P r k s .Proof. We will establish a criterion for orientability by seeking in which condition the top dimensionalhomology group is Z . Let us begin with the (unique) Schubert top cell S w Θ . The associated permutation w Θ is the longest permutation (with respect to the Bruhat order) with descents at positions d , . . . , d k .In one-line notation, w Θ “ p d k ` q ¨ ¨ ¨ n |p d k ´ ` q ¨ ¨ ¨ d k | ¨ ¨ ¨ | ¨ ¨ ¨ d There are k Schubert cells S p w Θ q j , j P r k s , covered by S w Θ . For each j P r k s , the correspondingpermutations of w Θ and p w Θ q j differ only by the values at positions d j ´ ` d j ` , where weconsider d “ d k ` “ n , i.e., w Θ “ p w Θ q j ¨ p d j ´ ` , d j ` q . By Proposition 3.13, it follows that κ p w Θ , p w Θ q j q “ d j ` ´ p d j ´ ` q , i.e., c p w Θ , p w Θ q j q “ p d j ` ´ d j ´ q is even. Since d j ` ´ d j ´ “ p d j ` ´ d j q`p d j ´ d j ´ q , B S w Θ “ d j ` ´ d j have the same parity, for every j . (cid:3) Low-dimensional integral homology.
We now seek a formula for the 1,2-homology for partialflag manifolds of type A. This require to introduce a few combinatorial notations.The code (also called Lehmer code) of a permutation w P S n is an integer sequence α with α i “ t k ą i | w k ă w i u and it will be denoted by code p w q . In other word, each entry of the codecorresponds to the number of inversions to the right of w i . It’s clear that 0 ď α i ď n ´ i . The codeprovides a bijection between S n and the set r , n ´ s ˆ r , n ´ s ˆ ¨ ¨ ¨ ˆ r , s .Given w P S n , the code spectrum of w is the unique partition 0 ă b ď b ď ¨ ¨ ¨ ď b l ă n suchthat the code α of w is given by α i “ t j : b j “ i u . We will denote by J b , ¨ ¨ ¨ , b ℓ K the permutation w given by such code spectrum to distinguish it from the other notations. For instance, w P S suchthat code p w q “ p , , , q then its code spectrum is p , , q , i.e, w “ J , , K .This notation allows us to easily describe permutations with some choose length. Let us describeall permutations w P S n with length up to three: ‚ s i “ J i K for i P r n ´ s ; ‚ s i s j “ J i, j K for i ă j and i, j P r n ´ s ; ‚ s i ` s i “ J i, i K for i P r n ´ s ; ‚ s i s j s k “ J i, j, k K for i ă j ă k and i, j, k P r n ´ s ; ‚ s i ` s i s k “ J i, i, k K for i ă k and i, k P r n ´ s ; ‚ s i s j ` s j “ J i, j, j K for i ă j and i, j P r n ´ s ; ‚ s i ` s i ` s i “ J i, i, i K for i P r n ´ s .When we require to fix a reduced decomposition, we will choose the ones as above.The following lemma provides us all the boundary maps required to compute 1,2-homology of anypartial flag manifold of type A. Lemma 3.15. (1) B S J i K “ , for i P r n ´ s ; (2) B S J i,i K “ ´ S J i K , for i P r n ´ s ; (3) B S J i,i ` K “ ´ S J i ` K , for i P r n ´ s ; CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 12 (4) B S J i,j K “ , for i P r n ´ s and j P r i ` , n ´ s . (5) B S J i,j ´ ,j K “ S J i,j K , for i P r n ´ s and j P r i ` , n ´ s ; (6) B S J i,i ` ,i ` K “ S J i,i ` K ´ S J i ` ,i ` K for i P r n ´ s ;Proof. Proposition 3.13 gives us when the coefficient is ˘
2. Recall that p w I is the reduced decompositionof w obtained by removing the I -th simple reflection of w . To get the sign we can observe that if wechoose to fix the reduced decomposition for w and w as given above, both reduced decompositions w and p w I are exactly the same. Then, Φ w “ Φ p w I and deg ` Φ ´ w ˝ Φ p w I ˘ “
1. Then, the sign of thecoefficient is given by p´ q I . Therefore,(1) For i P r n ´ s , c p J i K , e q “ i P r n ´ s , c p J i, i K , J i K q “ ´ c p J i, i K , J i ` K q “ i P r n ´ s , c p J i, i ` K , J i ` K q “ ´ c p J i, i ` K , J i K q “ i P r n ´ s and j P r i ` , n ´ s , c p J i, j K , J i K q “ c p J i, j K , J j K q “ i P r n ´ s and j P r i ` , n ´ s , c p J i, j ´ , j K , J j ´ , j K q “ c p J i, j ´ , j K , J i, j K q “
2, and c p J i, j ´ , j K , J i, j ´ K q “ i P r n ´ s , c p J i, i ` , i ` K , J i ` , i ` K q “ ´ c p J i, i ` , i ` K , J i, i ` K q “
2, and c p J i, i ` , i ` K , J i, i ` K q “ (cid:3) Denote by r Θ the number of connected components of the Dynkin diagram of Θ. Theorem 3.16.
Consider a partial flag manifold F Θ of type A , where Θ Ă Σ . (1) For n ě , the 1-homology is given by H p F Θ , Z q – p Z q n ´| Θ |´ . (2) For n ě , the 2-homology of is given by H p F Θ , Z q – p Z q N Θ where N Θ “ ` n ´| Θ |´ ˘ ` r Θ ´ . Proof.
To compute the 1-homology, suppose that Σ ´ Θ “ t a d , a d , . . . , a d k u where 0 ă d ă ¨ ¨ ¨ ă d k ă n . All permutations of length 1 in W Θ are in the form J d i K . By Lemma 3.15, the kernel ker pB Θ q is generated by S J d i K , i P r k s .For every i P r k s such that d i ă n ´
1, we have that B S J d i ,d i K “ ´ S J d i K . The only exception is when d k “ n ´
1. In this case, B S J d k ´ ,d k K “ ´ S J d k K . We conclude that H p F Θ , Z q has no free part and theset t J d i K : i P r k su generates the torsion, i.e, H p F Θ , Z q – p Z q k .To compute the 2-homology, let us prove that it has no free part, i.e., H p F Θ , Z q – p Z q x for someinteger x . Consider the maximal flag manifold F .By Lemma 3.15, the kernel ker pBq is generated by ‚ X i,j “ S J i,j K , for i P r n ´ s and j P r i ` , n ´ s ; ‚ X i,i ` “ S J i,i ` K ´ S J i ` ,i ` K , for i P r n ´ s .Notice that we do not allow X n ´ ,n ´ since J n ´ , n ´ K R S n . Also by Lemma 3.15, we have thateach X i,j is image through B of the following Schubert cells: B S J i,j ´ ,j K “ X i,j , for i P r n ´ s and j P r i ` , n ´ s , B S J i,i ` ,i ` K “ X i,i ` , for i P r n ´ s . These generators X i,j are indexed by the set I of pairs p i, j q given by I “ tp i, j q : i P r n ´ s and j P r i ` , n ´ su “ tp i, j q P r n ´ s : i ă j u ´ tp n ´ , n ´ qu . Hence, H p F , Z q – p Z q | I | “ p Z qp n ´ q ´ since I counts the number of 2-combinations in n ´ F Θ , consider the following cases: CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 13 ‚ if J i, j K P W Θ such that i P r n ´ s and j P r i ` , n ´ s then J i, j ´ , j K P W Θ . Hence, X i,j isa generator of the torsion of H p F Θ , Z q ; ‚ if J i, i ` K P W Θ for i P r n ´ s then J i ` , i ` K P W Θ and J i, i ` , i ` K P W Θ . Hence, X i,i ` is a generator of the torsion of H p F Θ , Z q .Therefore, the 2-homology has no free part and the torsion is generated by all X i,j such that J i, j K P W Θ , i.e., H p F Θ , Z q – p Z q |tp i,j qP I : J i,j K P W Θ u| .Finally, we will prove that |tp i, j q P I : J i, j K P W Θ u| “ N Θ by induction on the cardinality of Θ. If | Θ | “ | I | “ ` n ´ ˘ ´ “ N H . Suppose, by induction, that |tp i, j q P I : J i, j K P W ∆ u| “ N ∆ forany strict subset ∆ of Θ.Assume that ∆ “ Θ ´ t a k u , for k P r n ´ s , where a k is the greatest simple root (with respect to Σ)in Θ. Since W Θ Ă W ∆ , by induction(12) |tp i, j q P I : J i, j K P W Θ u| “ N ∆ ´ |tp i, j q P I : J i, j K P W ∆ z W Θ u| Let us compute |tp i, j q P I : J i, j K P W ∆ z W Θ u| . Given p i, j q P I such that J i, j K P W ∆ , considerthe following cases: ‚ if i ‰ k and j ‰ k then J i, j K P W Θ ; ‚ if i “ k and j “ k ` J i, j K P W Θ ; ‚ if i “ k and j ą k ` J i, j K R W Θ ; ‚ if j “ k then J i, j K R W Θ .Thus, |tp i, j q P I : J i, j K P W ∆ z W Θ u| “|tp i, j q P I : i “ k, j ą k ` , and J i, j K P W ∆ u|`` |tp i, j q P I : j “ k and J i, j K P W ∆ u| . Since a k is the greatest root of Θ then |tp i, j q P I : i “ k, j ą k ` , and J i, j K P W ∆ u| “ |t j : j P r k ` , n ´ su|“ " n ´ k ´ k ă n ´ , k “ n ´ . On the other hand, |tp i, j q P I : j “ k and J i, j K P W ∆ u| ““ |t i : i P r k ´ s X r n ´ s and a i R ∆ u| ` " a k ´ P ∆ , “ |t i : i P r k ´ s X r n ´ s and a i R ∆ u| ` p ´ p r Θ ´ r ∆ qq“ ´ p r Θ ´ r ∆ q ` " k ´ ´ | ∆ | if k ă n ´ ,n ´ ´ | ∆ | if k “ n ´ . “ k ´ | ∆ | ´ r Θ ` r ∆ ` " k ă n ´ , ´ k “ n ´ . Hence, |tp i, j q P I : J i, j K P W ∆ z W Θ u| “ n ´ | ∆ | ´ ´ r Θ ` r ∆ and, by Equation (12), |tp i, j q P I : J i, j K P W Θ u| “ ˆ n ´ | ∆ | ´ ˙ ` r ∆ ´ ´ p n ´ | ∆ | ´ ´ r Θ ` r ∆ q“ p n ´ | ∆ | ´ qp n ´ | ∆ | ´ q ` r Θ ´ “ ˆ n ´ p| ∆ | ` q ´ ˙ ` r Θ ´ “ N Θ . (cid:3) Final comments and further directions
We would like to highlight that, although this is a classical theme – topology of real flag manifoldssuch as Projective spaces and Grassmannian manifolds – we have not found in the literature such simpleformula to compute its homology groups. Its remarkable how simple are the formulas for 1,2-homologyof partial flag manifolds of type A.
CORRESPONDENCE BETWEEN COEFFICIENTS OF FLAG MANIFOLDS AND HEIGHT OF ROOTS 14
With the results obtained in this paper, we are able to visualize other directions to research as listedbelow. ‚ For type A flag manifolds, it seems possible to get a formula for 3, 4-homology. It will requireto get a better understanding of the coefficient since the degree in Theorem 3.3 are not aseasy to compute. A forthcoming paper will deal with the combinatorics involved to explicitlycompute the sign in the type A case. ‚ Theorem 3.8 provides a formula of the boundary coefficient for split real forms. It is reasonableto ask about low dimensional homology of other types of flag manifolds. This would require toget a nicer combinatorial model for the Weyl group.
Acknowledgments.
We thank to San Martin and Lucas Seco for helpful suggestions and valuable dis-cussions. This research was motivate by computer investigation using the open-source mathematicalsoftware Sage [13].
References [1] A. Bj¨orner and F. Brenti.
Combinatorics of Coxeter groups . Springer-Verlag, 2005.[2] L. Casian and R. J. Stanton. Schubert cells and representation theory.
Invent. Math. , 137(3):461–539, 1999.[3] V. del Barco and L.A.B. San Martin. De Rham 2-Cohomology of Real Flag Manifolds.
SIGMA , 15:51, 2019.[4] L. Grama and L. Seco. Second Homotopy Group and Invariant Geometry of Flag Manifolds.
Results Math , 75:94,2020.[5] R. R. Kocherlakota. Integral homology of real flag manifolds and loop spaces of symmetric spaces.
Adv. Math. ,110:1–46, 1995.[6] J. Lambert and L. Rabelo. Covering relations of k-grassmannian permutations of type b.
Australas. J. Combin ,75(1):73–95, 2019.[7] J. Lambert and L. Rabelo. Integral homology of real isotropic and odd orthogonal grassmannians.
ArXiv e-prints ,2019. arXiv:1604.02177v2 .[8] A. K. Matszangosz. On the cohomology ring of real flag manifods: Schubert cycles.
ArXiv e-prints , 2019. arXiv:1910.11149 .[9] M. Patr˜ao, L. A. B. San Martin, L. J. dos Santos, and L. Seco. Orientability of vector bundles over Real flagmanifolds.
Topol. Its Appl. , 159 (10-11):2774–2786, 2012.[10] N. Perrin. Introduction to lie algebras. 2015. https://lmv.math.cnrs.fr/wp-content/uploads/2019/09/lie-alg-2.pdf.[11] L. Rabelo. Cellular Homology of Real Maximal Isotropic Grassmannians.
Adv. in Geometry , 16(3):361–380, 2016.[12] L. Rabelo and L. A. B. San Martin. Cellular homology of real flag manifolds.
Indag. Math , 30(5):745–772, 2019.[13] SageMath. the Sage Mathematics Software System (Version 9.1).
The Sage Developers
Indag. Mathem. N.S. , 1:141–153, 1998.(Jordan Lambert)
Department of Mathematics – ICEx, Universidade Federal Fluminense, Volta Redonda27213-145, Rio de Janeiro, Brazil
E-mail address : [email protected] (Lonardo Rabelo) Department of Mathematics, Federal University of Juiz de Fora, Juiz de Fora 36036-900, Minas Gerais, Brazil
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