We find explicit miniversal deformations of flags in the generalized flag manifolds,
with regard a natural equivalence relation defined by the group action that keeps
fixed the reference flag

Puerta Coll, Xavier

English

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Versal deformations in generalized flagmanifoldsX. PuertaDepartament de Matema`tica Aplicada IEscola Te`cnica Superior d’Enginyers Industrials de Barcelona, UPCAv. Diagonal, 647 08028 Barcelona, SpainAbstractWe find explicit miniversal deformations of flags in the generalized flag manifolds,with regard a natural equivalence relation defined by the group action that keepsfixed the reference flag.Keywords: Versal deformation, orbit space, Grassmann manifold, generalized flagmanifold.AMS classification: 93B29,93C73IntroductionGiven a smooth manifold M and an element of x ∈ M, a versal deformation of xwith regard an equivalent relation defined by a group action is a concept introducedby Arnold in [1] and subsequently developed in the study of local perturbations.Although Arnold’s theory has been stated in a very general formulation, in practice,almost all explicit versal deformations have been calculated in open sets of Rn. In [4],versal deformations in orbit spaces are studied and a description of them in terms ofa system of equations is obtained ([4]-Theorem 2.6). Nevertheless, these equationsare, in general, unable to be solved explicitly. In other terms, it is needed for eachorbit space, a particular treatment of these equations.In this paper we apply the results of [4] to generalized flag manifolds, which are aparticular kind of orbit space that generalize the Grassmann manifold. More pre-cisely, we fix a reference flag (a chain of subspaces F1 ⊂ · · · ⊂ Fk of a F -vectorspace F ) and we consider the set of subflags of it whose elements have fixed dimen-sion. This set is a smooth manifold with an orbit space structure similar to thatPreprint submitted to Elsevier Science 19 November 2004of the Grassmann manifold (see, for example, [2]). The generalized flag manifoldarises in a natural way while studying the topology of the set of A-invariant and(C,A)-invariant subspaces. More precisely in [2] and [3] it is proved the existence ofdeformation retracts of the set of A-invariant subspaces and (C,A)-invariant sub-spaces, respectively, on a generalized flag manifold (when the discrete invariants ofthe restrictions are fixed). The natural equivalence relations defined in the set of Aand in the set of (C,A)-invariant subspaces induce an equivalence relation in thegeneralized flag manifold as follows: two flags are equivalent if there exists an auto-morphism keeping fixed the reference flag such that sends the elements of the firstflag to the elements of the second one.The organization of this paper is the following one. In section 1 we recall the basicdefinitions and results on the generalized flag manifold and versal deformations inorbit spaces. In section 2 we study the classification of subflags of a given flag.Finally, in section 3 we find an explicit miniversal deformation of a subflag withregard to the above classification.In this paper we make use of the following notation. F is the field of either the com-plex or the real numbers. If E is an F -vector space, Grd(E) denotes the Grassmannmanifold of d-dimensional subspaces of E. We say that a basis is adapted to a setof subspaces if one can obtain bases of those subspaces taking subsets of this basis.In particular, a basis adapted to a chain of subspaces V1 ⊂ · · · ⊂ Vk is a basis ofVk obtained by extending successively bases of V1, V2, . . .. We say that an n-tuple ofintegers (k1, . . . , kn) is a partition if 0 ≤ ki ≤ ki+1 for all i = 1, . . . , n− 1. Ia meansthe a× a-identity matrix and 0a,b the a× b-zero matrix.1 PreliminariesLet F be a fixed F -vector space of dimension n. A flag of F is a chain of subspacesV1 ⊂ · · · ⊂ Vk. We are given a reference flag, F1 ⊂ · · · ⊂ Fk, that is fixed all alongthis note. A flag V1 ⊂ · · · ⊂ Vk such that Vi ⊂ Fi, i = 1, . . . , k, is said to be a subflagof F1 ⊂ · · · ⊂ Fk.Let s = (s1, . . . , sk) be a partition with s1 ≤ · · · ≤ sk ≤ n. We say that V1 ⊂ · · · ⊂ Vkis an s-flag if dimVi = si, i = 1, . . . , k. Assume that F1 ⊂ · · · ⊂ Fk is an r-flag withr = (r1, . . . , rk). If si ≤ ri for i = 1, . . . , k, the generalized flag manifold, Flag(r, s),is the set of s-subflags of the above reference r-flag. Flag(r, s) is diffeomorphic to theorbit space M(r, s)/M(s, s), where M(r, s) is the set of full rank matrices of theblock form X = [Xi,j], being Xi,j an (ri − ri−1)× (sj − sj−1)-matrix (r0 := 0) withXi,j = 0 if j < i, and defining in M(r, s) the action of M(s, s) by (X,P ) 7→ XP .We refer the reader to [2] for the corresponding developments.We define in Flag(r, s) the following equivalence relation.2Definition 1.1 (V1, . . . , Vk) ∼ (V ′1 , . . . , V ′k) if there exist P ∈ Gl(n) such thatV ′i = P (Vi) and P (Fi) = Fi for all i = 1, . . . , k.Let G = {P ∈ Gl(n)|P (Fi) = Fi}. It is clear that G is a subgroup of Gl(n) andthat the classes of the above equivalence relation are the orbits of the subflags of(F1, . . . , Fk) with regard to the action of G on Flag(r, s) defined by(P, (V1, . . . , Vk)) 7→ (P (V1), . . . , P (Vk)).Taking a basis of F adapted to (F1, . . . , Fk), one can check that G is M(r, r). Fornotational convenience, let us denote M(s, s) by Γ and M(r, s) by M. It is clearthat there exists a bijection between the orbits G(XΓ), XΓ ∈M/Γ, and the orbitsGXΓ corresponding to the action on M defined by ((P,Q), X) 7→ PXQ, where inG × Γ we define the group product (P,Q)(P ′, Q′) := (P ′P,QQ′).In the next sections we exhibit a miniversal deformation of any given element XΓ ofthe orbit space M/Γ. Our notation here is especially adapted to the notation andcontext of the previous work [4]. The reader may find there a definition and resultsrelevant for our present purposes. The key result that we apply in this note is thefollowing theorem of [4].Theorem 1.2 A miniversal deformation of an orbit XΓ inM/Γ is (X+W )Γ whereW ∈ W, being W a neighbourhood of the origin of the set of matrices W ∈M suchthattrace (PXW ∗) = trace (XQW ∗) = 0for all P ∈ G and Q ∈ Γ.In section 3 we solve explicitly the equations trace (PXW ∗) = trace (XQW ∗) = 0 inthe context of the generalized flag manifold. For this purpose it is needed a canonicalrepresentant of the orbit GXΓ in terms of a complete set of invariants of XΓ. Thisis the goal of section 2.2 Classification of the subflags of a reference flagNow we will find a canonical element of the orbit GXΓ. One can easily prove thatby means of the action of G and Γ, one can reduce the matrix X to a permutationmatrix ofM. However, we are specially interested in describing the positions of the1’s of this permutation matrix in terms of geometric invariants of the flag representedby X. In order to do that, we interpret the columns of X as a basis of F adaptedto the chain of subspaces of the corresponding flag. This leads us to consider thefollowing diagram3(1)V1 = V1 ∩ F1 ⊂ V2 ∩ F1 ⊂ · · · ⊂ Vk ∩ F1 ⊂ F1∩ ∩ ∩ ∩V2 = V2 ∩ F2 ⊂ · · · ⊂ Vk ∩ F2 ⊂ F2∩ ∩ ∩V3 = · · · ⊂ Vk ∩ F3 ⊂ F3. . .......Vk ⊂ FkDefining Fk+1 = Vk+1 = F and F0 = V0 = {0}, we can write for k ≥ i ≥ j ≥ 0Vi ∩ Fj ⊂ Vi+1 ∩ Fj∩ ∩Vi ∩ Fj+1 ⊂ Vi+1 ∩ Fj+1where, obviously, Vi∩Fj = (Vi+1∩Fj)∩ (Vi∩Fj+1). Let Ei,j be a subspace such thatVi ∩ Fj = Ei,j ⊕ (Vi−1 ∩ Fj + Vi ∩ Fj−1)and let ei,j be a basis of Ei,j. Arranging the vectors of⋃i,j ei,j in order to obtainbases of Vk and Fk we have that(2){e1,1; e2,1, e2,2; . . . ; ek,1, . . . , ek,k} is a basis of Vk adapted to (V1, . . . , Vk){e1,1, . . . , ek,1; e2,2, . . . , ek,2; . . . ; ek+1,k+1} is a basis of Fk adapted to (F1, . . . , Fk)Arranging in columns the coefficients of the elements of the first basis with regardthe second one can check that we obtain the following matrix(3) X = [Xj,i],0 < j ≤ i ≤ k + 1, with Xj,i = 0 if i < j, and for i ≥ j,Xj,i =0βi,j ,αi,j 0 00 Isi,j 00 0 0βi,j ,αi,j4wheresi,j := dimEi,j = dimVi ∩ Fj − dimVi−1 ∩ Fj − dimVi ∩ Fj−1 + dimVi−1 ∩ Fj−1andαi,j := si−1,j + si−2,j + · · · sj,j or 0 if i = jαi,j := sk,j + · · ·+ si+1,jβi,j := si,j−1 + · · ·+ si,1 or 0 if j = 0βi,j := si,i + · · ·+ si,j+1This proves the following theoremTheorem 2.1 Let (V1, . . . , Vk) be a subflag of (F1, . . . , Fk) represented by X ∈ M.With the above notations we have that(i) The matrix defined in (3) is a canonical form of the orbit GXΓ.(ii) si,j for 0 < j ≤ i ≤ k, is a complete set of invariants of GXΓ.(iii) dimVi ∩ Fj for 0 < j ≤ i ≤ k, is a complete set of invariants of GXΓ.Moreover, there exists a bijection between Flag(r, s)/G and the set of integers si,jsatisfying for 0 ≤ i ≤ k the conditionssi − si−1 = si,1 + · · ·+ si,iri − ri−1 = si,i + · · ·+ sk,iBesides last theorem, the construction of the canonical bases (2), has the followingconsequence.Theorem 2.2 Let (V1(t), . . . , Vk(t)) be a smooth family of flags of the same classparameterized by a contractible manifold M . Then, there exists a smooth family ofbases of Vi(t), t ∈M , defined as in (2). In particular, if X(t) is a smooth family ofmatrices of M of the same class parameterized by M , there exist smooth familiesP (t) and Q(t) of G and Γ, respectively such that P (t)X(t)Q(t) is the matrix (3).3 Versal deformations in the generalized flag manifoldIn this section we apply theorem 1.2 in the context of the generalized flag manifold.For this purpose, we solve the equations trace (PXW ∗) = trace (XQW ∗) = 0 whereX ∈ M, P ∈ G and Q ∈ Γ. Since the map X 7→ PXQ is a diffeomorphism, we canassume that X is a particular element of the orbit GXΓ. So, we assume that X isthe canonical matrix (3) described in the last section.5We have that XQ = [Qi,j] whereQi,j =Q1i,jQ2i,j. . . ,with Qhi,j being the corresponding rows of Qi,j if h ≤ j, and Qhi,j = 0 if h > j.On the other side, PX = [Pi,j] where Pi,j =[P 1i,j P2i,j . . .], with P hi,j being thecorresponding columns of Pi,j if h ≥ j, and P hi,j = 0 if h < j.Notice that the set of matrices Y with a fixed zero structure and having in therest of the entries different, independent parameters is a vector space of dimensionthe number of parameters, and its orthogonal is the set of matrices of the samesize, having different, independent parameters in the zero entries of Y and 0 in thenonzero entries of Y .Therefore, taking into account the form of the matrices PX and XQ, we have thattrace (PXW ∗) = trace (XQW ∗) = 0 implies that(4) W = [Wj,i],0 < j ≤ i ≤ k+1, withWj,i = 0 if i < j, and for i ≥ j, decomposinginto blocks Wj,i as in (3),Wj,i =0βi,j ,αi,j 0 00 0si,j 0Yβi,j ,αi,j 0 0βi,j ,αi,jwith Yβi,j ,αi,j a full parameter βi,j × αi,j-matrix.This proves the following theoremTheorem 3.1 With the above notation, the set of flags (X +W )Γ where X and Ware as in (3) and (4), respectively, is a miniversal deformation of the flag XΓ.Last theorem allows us to compute the dimension of the orbit of a given flag. Inparticular, stable flags (flags having the codimension of its orbit equal to 0) arecharacterized through the following corollary.Corollary 3.2 The flag (V1, . . . , Vk) is stable if and only if βi,j = 0 for all 0 < j ≤i ≤ k. In particular, all the stable flags are equivalent.Moreover, the versal deformation obtained in the last theorem, allows us to computebifurcation diagrams as it is shown in the following exampleExample 3.3 Let r = (2, 3, 4) and s = (1, 2, 3). We have four possible flag classes6represented by the canonical matrices(A)1 0 00 0 01 01(B)1 0 00 1 00 01(C)1 0 00 0 11 00(D)1 0 00 1 00 10The stable flags are the flags of the class of (A) and, on the other hand, a miniversaldeformation of the element (D) is, according to the last theorem,X + TX(GXΓ)⊥ = {1 0 00 1 0x 1y}Therefore, the bifurcation diagram of (D) is given by the following partition of thespace of parameters, according to his adjacent classes.for x = 0 and y 6= 0, class of (B)for x 6= 0 and y = 0, class of (C)for x 6= 0 and y 6= 0, class of (A)References[1] V.I. Arnold, On Matrices Depending on Parameters, Uspekhi Mat. Nauk., 26(1971), p. 101–114.[2] U.Helmke, M.Shayman, The biflag manifold and the fixed point set of a nilpotenttransformation on the flag manifold Linear Algebra Appl. 92 (1987), 125-159[3] X.Puerta, U.Helmke, The topology of the set of conditioned invariant subspacesSystem and Control Letters, 40 (2000), 97–105[4] F. Puerta, X. Puerta and S. Tarragona, Versal deformations in orbit spaces LinearAlgebra Appl., In Press, Corrected Proof, Available online 18 July 2003,7

Versal deformations in generalized flag manifolds

We find explicit miniversal deformations of flags in the generalized flag manifolds,

with regard a natural equivalence relation defined by the group action that keeps

fixed the reference flag

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Versal deformations in generalized flag manifolds

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