aa r X i v : . [ m a t h . D S ] N ov A Class of Morse Functions on Flag Manifolds
P. Duarte
Abstract
Given a positive definite symmetric matrix in one of the groups
SL( n, R ) or Sp( n, R ) , we analyse its actions on Flag Manifolds, proving these are diffeomor-phisms which admit as strict Lyapunov functions a special class of quadratic functions,all of them Z -perfect Morse functions. Stratifications on the Flag Manifolds are pro-vided which are invariant under both the diffeomorphisms and the gradient flows of theLyapunov functions. Let G ⊆ SL( n, R ) be a subgroup, ( X, µ ) be a probability space, T : X → X ameasurable transformation which preserves measure µ , and A : X → G a measur-able function. These objects, ( X, µ ), T and A , determine a linear cocycle , a namewhich is often given to the skew-product map F : X × R n → X × R n , definedby F ( x, v ) = ( T x, A ( x ) v ), whose iterates are given by F k ( x, v ) = ( T k x, A k ( x ) v ),where A k ( x ) = A ( T k − x ) · · · A ( T x ) A ( x ). This cocycle is said to be integrable when R X log + k A ( x ) k dµ ( x ) < ∞ , where where log + ( x ) = max { log x, } . A classical theo-rem of Oseledet states that if the cocycle is integrable and we assume, for the sake ofsimplicity, that T is ergodic w.r.t. µ , then there are n ∈ N , λ > λ > · · · > λ m realnumbers, and R n = E ( x ) ⊃ E ( x ) ⊃ . . . ⊃ E m ( x ) an F -invariant measurable filtra-tion such that: for µ -almost every x ∈ X (1) λ i = lim k →∞ k log (cid:13)(cid:13) A k ( x ) v (cid:13)(cid:13) , for every v ∈ E i ( x ) − E i +1 ( x ), and (2) P mi =1 λ i (dim E i ( x ) − dim E i +1 ( x )) = 0. The numbers λ , . . . , λ m are called the Lyapunov exponents of the linear cocycle. The computationof Lyapunov exponents is an active and important subject in the theory of Linear Co-cycles and Dynamical Systems. Although there are some formulas for the Lyapunovexponents, in general they can not be explicitely evaluated. One of the first formulas forthe Lyapunov exponents appeared in the seminal work of H. Furstenberg [F2]. There1he author is concerned with proving ”law of large number” theorems for products of in-dependent and identically distributed ramdom matrices, but his setting can be broughtto the linear cocycles’ framework by letting the probability space to be ( G N , µ N ), where µ is some given probability measure on G , T : G N → G N , T ( g k ) k ∈ N = ( g k +1 ) k ∈ N , be theshift transformation, and A : G N → G be the function A ( g k ) k ∈ N = g . In the languageof the Linear Cocycle’s Theory, Furstenberg gives a formula for the largest Lyapunovexponent λ , and proves that λ > G with finitecenter. Two examples of such groups are (1) G = SL( n, R ), the special linear groupof square n × n matrices with determinant one, and (2) G = Sp( n, R ), the symplecticlinear group of square (2 n ) × (2 n ) symplectic matrices. Under these assumptions thegroup G has three subgoups: K maximal compact, A abelian and N nilpotent suchthat G = K · A · N . Moreover, for each g ∈ G , there is a unique decomposition g = k a n with k ∈ K , a ∈ A and n ∈ N . The decomposition G = K · A · N is called Iwasawa’s decomposition, but it is not unique. The group S = A · N issolvable. On the examples above, the group factorization G = K · S comes fromthe QR-decomposition, which is associated with the Gram-Schmidt orthogonalizationalgorithm. Let us now identify each of these factor subgroups on the two previousexamples. On example (1), K = SO( n, R ) is the special orthogonal group, A is thegroup of determinant one positive diagonal matrices, N is the group of upper triangularmatrices with ones on the diagonal, while S is the group of upper triangular matriceswith positive diagonal and determinant one. On example (2), K = O sp ( n, R ), the groupof symplectic orthogonal matrices, which is isomorphic to U( n, C ), and whence calledthe unitary group, A is the group of symplectic positive diagonal matrices, N is thegroup of symplectic matrices of the form (cid:18) u v u − T (cid:19) where u and v are square n × n matrices, u is upper-triangular with ones on the diagonal, and v is such that v T u − T issymmetric. We shall refer to these matrices as symplectic upper-triangular. Finally, inthis case S is the group of symplectic upper-triangular matrices with positive diagonal.For a compact manifold M , P ( M ) will denote the space of probability measures on M , which is a convex compact set when equipped with the weak- ∗ topology. A keyconcept is that of boundary of a Lie Group. A boundary of G is any compact manifoldwith transitive action G × M → M with the following property: for every measure2 ∈ P ( M ) there is a sequence g k ∈ G such that g k π converges weakly to a point mass δ p , with p ∈ M . The following relation is defined among boundaries of G . Giventwo boundaries M and M ′ of G , we write M (cid:22) M ′ iff M is the surjective imageof M ′ by some G -equivariant map f : M ′ → M , where a map f : M → M ′ is saidto be G -equivariant if f ( g x ) = g f ( x ), for every x ∈ M and g ∈ G . The relation (cid:22) is a partial order on the set of G -equivariant equivalence classes of boundaries of G . The poset of G -equivariant equivalence classes of boundaries of G has a uniquemaximal element, whose representatives are referred as maximal boundaries of M .The quotient G/K ≃ S is a non-compact symmetric space with a transitive actionof G . The ”boundary” name is justified, see [F2] page 403, by the fact that everyboundary of G can be realized as part of the boundary of some compactification of thesymmetric space G/K . A class of boundaries for the group G = SL( n, R ) are the socalled flag manifolds that we now describe. See section 5 of [F1]. Given a sequenceof numbers k = ( k , k , . . . , k m ) with 1 ≤ k < k < . . . < k m ≤ n , any sequence V ∗ = ( V , . . . , V m ) of linear spaces V ⊂ V ⊂ . . . ⊂ V m ⊂ R n such that dim V i = k i forevery i = 1 , . . . , m is called a k -flag in R n , and F n,k = F k ( R n ) will denote the k -flagmanifold consisting of all k -flags in R n . There is a natural action of SL( n, R ) on F n,k defined by SL( n, R ) × F n,k → F n,k , A V ∗ = ( AV , . . . , AV m ) where V ∗ = ( V , . . . , V m ).Given A ∈ SL( n, R ) we denote by ϕ A : F n,k → F n,k the diffeomorphism ϕ A ( V ∗ ) = AV ∗ .Notice that ϕ : SL( n, R ) → Diff( F n,k ), A ϕ A , is a representation of SL( n, R ) asa group of diffeomorphisms of the flag manifold F n,k . For each symmetric matrix A ∈ SL( n, R ) with simple spectrum the diffeomorphism ϕ A : F n,k → F n,k has a uniqueattractive fixed point whose basin of attraction has full measure. This property aloneeasily implies that F n,k is a boundary of SL( n, R ). Note that the flag manifolds includethe Grassman ones G n,k = G k ( R n ) = { V ⊂ R n : V is a linear subspace, dim V = k } ,because G n,k = F n, ( k ) . The maximal boundary of SL( n, R ) is the full flag manifold F n,k determined by the sequence k = (1 , , . . . , n ). Next we define the class of isotropic flagmanifolds which are boundaries for the group G = Sp( n, R ). We consider on R n theusual symplectic structure defined by the matrix J = (cid:18) O − II O (cid:19) . Given a sequence k = ( k , k , . . . , k m ) with 1 ≤ k < k < . . . < k m ≤ n , a k -flag V ∗ in R n is called isotropic iff the largest subspace V m ⊂ R n is isotropic, i.e., u t J v = 0 forevery u, v ∈ V m . When k = (1 , , . . . , n ) the k -flag V ∗ is said to be a Lagrangian flag .We denote by F spn,k = F spk ( R n ) the submanifold of all isotropic k -flags in F k ( R n ).To each matrix B in SL( n, R ), resp. Sp( n, R ), let us associate an angle matrix ∈ O( n ), resp R ∈ U( n ), and a vector b = ( b , . . . , b n ) ∈ R n with b ≥ b ≥ . . . ≥ b n > b consists of the ordered singular values of B and R is such that forsome other orthogonal matrix S we have B = R D b S . This is the singular valuedecomposition of matrix B . We look at the angle R and the singular values b ascoordinates of B . Now, let A be another matrix in SL( n, R ), resp. Sp( n, R ), andlet R ′ be the angle of A B , while b ′ be the singular values of A B . We would liketo understand the transformation B A B in terms of these of coordinates, i.e., todescribe (
R, b ) ( R ′ , b ′ ) in terms of matrix A . We have R ′ = ϕ A ( R ) = A ∗ R .Whence this first coordinate map relates to the action of our group on the completeflag manifold, resp. Lagrangian flag manifold. About the second coordinate map,note that k A B k hs = k D b ′ k hs = Q A t A,b ( R ) / . Therefore, the function Q A t A,b givesus valuable information on the expansion of the singular values as we multiply by A .The singular values’ vector b ′ can be computed as the singular values of the nilpotentmatrix U D b , where U is the matrix in the QR-decmposition A R = K U of A R . Webelive that to analyze Lyapunov exponents it is worth to studing the geometry ofthe tranformations ϕ A and of the functions Q A t A,b on the maximal boundary. In thisarticle we only study the geometry of these objects without any reference to possibleimplications on Lyapunov exponents.We will denote by G n any of the groups SL( n, R ) and Sp( n, R ) and by g n thecorresponding Lie algebra. The first is the group of real n × n matrices with determinantone, while the second is the group of real 2 n × n symplectic matrices. Every theorembelow will hold in two different contexts: G n = SL( n, R ) and g n = sl( n, R ) on onehand, and G n = Sp( n, R ) and g n = sp( n, R ) on the other.Any sequence V ∗ = ( V , . . . , V k ) of linear spaces V ⊂ V ⊂ . . . ⊂ V k ⊂ R n suchthat dim V i = i for every i = 1 , . . . , k is called a k -flag in R n , and F n,k = F k ( R n ) willdenote the k -flag manifold consisting of all k -flags in R n . We consider on R n the usualsymplectic structure defined by the matrix J = (cid:18) O − II O (cid:19) . A k -flag V ∗ in R n is called isotropic iff the subspace V k ⊂ R n is isotropic, i.e., u t J v = 0for every u, v ∈ V k . When k = n the k -flag V ∗ is said to be a Lagrangian flag . Wedenote by F spn,k the submanifold of all isotropic k -flags in F k ( R n ). The subscript sp willbe omitted on symplectic contexts, i.e., we adopt the convention that in each contextwhere G n and F n,k appear, F n,k stands for the isotropic k -flag manifold F spn,k whenever4 n = Sp( n, R ). There is a natural action of G n on F n,k defined by G n × F n,k → F n,k , ( A, V ∗ = ( V , . . . , V k ) ) A V ∗ = ( AV , . . . , AV k ) . Given A ∈ G n we denote by ϕ A : F n,k → F n,k the diffeomorphism ϕ A ( V ∗ ) = AV ∗ .Notice that ϕ : G n → Diff( F n,k ), A ϕ A , is a representation of G n as a group ofdiffeomorphisms of the flag manifold F n,k . Given A ∈ g n , the family { ϕ e tA } t ∈ R is aflow on F n,k that we shall simply denote by ϕ tA . Note that ϕ A = ϕ e A .We introduce now a family of quadratic functions on the flag manifolds F n,k . Giventwo linear subspaces E ⊂ F denote by F ⊖ E the orthogonal complement F ⊖ E = E ⊥ ∩ F . Given a matrix A ∈ g n and a vector b ∈ R k + we define the function Q A,b : F n,k → R as follows: For each V ∗ ∈ F n,k take a vector v ∈ V with k v k = b , and a vector v i ∈ V i ⊖ V i − with k v i k = b i (2 ≤ i ≤ k ) and then set Q A,b ( V ∗ ) = 1 k k X i =1 h A v i , v i i . Because V and V i ⊖ V i − (2 ≤ i ≤ k ) are one-dimensional spaces this definition isclearly independent of the choice of vectors v i .We denote by Diag + n the subgroup of positive diagonal matrices in G n . Any sub-group of G n which is conjugated to Diag + n by some orthogonal matrix in G n will bereferred as a commutative positive symmetric subgroup . Notice that matrices in suchsubgroups are always symmetric and positive definite. The Lie algebra of a com-mutative positive symmetric subgroup will be referred as a commutative symmetricsubalgebra .We shall call stratification on a manifold F to any collection S of closed connectedsubmanifolds, that we refer as strata of F , such that1. F ∈ S is the unique stratum of dimension N = dim F ;2. There are several strata of dimension zero;3. Each stratum S ∈ S of dimension d < N is contained in some stratum S ′ ∈ S ofdimension d + 1;4. Given S, T ∈ S such that S ⊂ T and S = T then dim S < dim T ; compact without boundary
5. Given
S, T ∈ S , either S ∩ T = ∅ or else S ∩ T ∈ S .We denote by S i the union of all strata S ∈ S with dim S = i . It follows from 2. and 3.that each S i is non-empty. By 3. we have S ⊂ S ⊂ . . . ⊂ S N = F , and by 4. and 5.each S i − S i − is a (disconnected) manifold of dimension i . We say that a stratification S on F is invariant under a diffeomorphism ϕ : F → F iff ϕ ( S ) = S for every stratum S ∈ S .We prove the following results about the objects F n,k , ϕ tA and Q A,b . We say that twosymmetric matrices
A, B ∈ g n share the same ordered eigen-directions iff there is acommon basis of eigenvectors v , . . . , v n such that the eigenvalues of A , A v i = λ i ( A ) v i ,are ordered in the same way as the eigenvalues of B , B v i = λ i ( B ) v i , which means that λ i ( A ) > λ j ( A ) ⇔ λ i ( B ) > λ j ( B ), for i, j = 1 , . . . , n . For instance, A and e A share thesame ordered eigen-directions. Theorem A
Given b > b > . . . > b k > , and two symmetric matrices A, H ∈ g n sharing the same ordered eigen-directions then Q A,b : F n,k → R is a strict Lyapunovfunction for ϕ tH : F n,k → F n,k . Theorem B
Given a commutative symmetric subalgebra h ⊂ g n , there is a stratifica-tion on F n,k which is invariant under both ϕ tH : F n,k → F n,k and the gradient flow ofthe function Q H,b : F n,k → R , for each H ∈ h and each b ∈ R k + . Theorem C
Given a symmetric matrix A ∈ g n with simple spectrum and b > b >. . . > b k > , Q A,b : F n,k → R is a Z -perfect Morse function. For reader’s reference we recall the key definitions above. A matrix A ∈ g n is saidto have simple spectrum iff the eigenvalues of A are all simple. A function Q : F → R is said to be a Lyapunov function for a flow ϕ t : F → F iff Q ( ϕ t ( x )) ≥ Q ( x ), for all x ∈ F and t ≥
0. Function Q is called a strict Lyapunov function when furthermore itsatisfies Q ( ϕ t ( x )) = Q ( x ) ⇔ ϕ t ( x ) = x , for every x ∈ F and t >
0. A smooth function Q : F → R is called a Morse function iff all its critical points are non-degenerate.Function Q is said to be Z -perfect iff the Poincar´e polynomial of F with coefficients inthe field Z , P t ( F , Z ) = P dim F i =0 dim H i ( F ; Z ) t i , coincides with the Morse polynomialof Q . Finally, the Morse polynomial of Q is defined by M t ( Q ) = P dim F i =0 c i ( Q ) t i ,where c i ( Q ) is the number of critical points with index i , the index of a critical pointbeing the number of negative eigenvalues of its Hessian matrix.6 Notation
All the notation below is defined in the text’s body, but we gather it here for an easyreference.
Mat n × k ( R ) the space of real n × k matrices( X ) i the column i of matrix X [ X ] i the submatrix formed by the first i columns of X [ X ] I the submatrix formed by the columns of X with index i ∈ I h X i I the space spanned by the columns of X with index i ∈ Iδ i,j the Kronecker symbol, δ i,j = (cid:26) i = j i = jI n,k the n × k matrix with entries δ i,j X ⊳ the matrix ( x ⊳i,j ) i,j , where x ⊳i,j = x i,i if i = jx i,j + x j,i if i < j j < i for X = ( x i,j ) i,j . h X, Y i hs the Hilbert-Schmidt inner product of matrices X and Y k X k hs the Hilbert-Schmidt norm of matrix X O( n ) the real orthogonal groupU( n ) the unitary groupUT( n ) the upper triangular groupUT + ( n ) the positive diagonal upper triangular group k the signature ( k , k , . . . , k m ), with 1 ≤ k < k < . . . < k m ≤ n , of a flag F n,k the k -flag manifold F spn,k the isotropic k -flag manifoldO n,k the k -orthogonal frame manifoldO spn,k the k -unitary frame manifold 7 Flag Manifolds
A sequence of numbers k = ( k , k , . . . , k m ) with 1 ≤ k < k < . . . < k m ≤ n , iscalled a signature of length m in the set { , , . . . , n } . We denote by Λ( n ) the setof all signatures in { , , . . . , n } , which is partially ordered as follows: We say that k ≥ k ′ iff k ′ is a subsequence of k . The sequence (1 , , . . . , n ) is the unique maximalelement of Λ( n ), while every sequence of length one ( i ), with 1 ≤ i ≤ n , is a minimalelement of Λ( n ). Given a signature k ∈ Λ( n ) of length m , any sequence of linear spaces V ∗ = ( V , . . . , V m ) such that V ⊂ V ⊂ . . . ⊂ V m ⊂ R n and dim V i = k i for every i = 1 , , . . . , m is called a k -flag in R n . We denote by F n,k = F k ( R n ) the space of all k -flags in R n . Proposition 1.
The space F n,k is a compact connected manifold of dimension n ( n − − P m +1 i =1 n i ( n i − , where n = k , n i = k i − k i − for i = 2 , . . . , m and n m +1 = n − k m . The order on Λ( n ) induces a hierarchy on flag manifolds. Given k, k ′ ∈ Λ( n ), if k ≥ k ′ and k ′ i = k s i , for i = 1 , . . . , m ′ , then there is a natural projection map π : F n,k → F n,k ′ defined by π ( V , . . . , V m ) = ( V s , . . . , V s m ). The flag π ( V ∗ ) is called the restriction of V ∗ . This projection map is a smooth submersion. Note that if the signature k ′ satisfies k ′ m < n and k is obtained appending n at the end of k ′ , i.e., k = ( k ′ , . . . , k ′ m , n ), thenthe projection π : F n,k → F n,k ′ is actually a diffeomorphism. Whence, we identify thesetwo flag manifolds and restrict our attention to flag manifolds with signatures in theset Λ ′ ( n ) = { k ∈ Λ( n ) : k m < n } . The flag manifold associated with the maximumsignature k = (1 , . . . , n −
1) is called the complete flag manifold , which as explainedin the introduction, is the maximal boundary of the group SL( n, R ). Flag manifoldsinclude an important subclass of varieties called the Grassmann manifolds. For each k = 1 , . . . , n − G n,k := F n, ( k ) is the Grassmann manifold of all k -dimensional subspaces V ⊂ R n . Note that for k = 1 the Grassmann manifold G n, is the real projecticve space.The action of the group SL( n, R ) on the flag manifold F n,k is SL( n, R ) × F n,k → F n,k , g V ∗ = ( gV , . . . , gV m ) with V ∗ = ( V , . . . , V m ). Two important, but obvious, facts aboutthis action are: Proposition 2.
The action of
SL( n, R ) on F n,k is transitive, and for k ≥ k ′ , theprojection π : F n,k → F n,k ′ is SL( n, R ) -equivariant, i.e., π ( g V ∗ ) = g π ( V ∗ ) , for every g ∈ SL( n, R ) and V ∗ ∈ F n,k . Isotropic Flag Manifolds
Given a signature k ∈ Λ( n ), a k -flag V ∗ ∈ F n,k = F k ( R n ) is called isotropic iffthe largest subspace V k m is isotropic, which means that the restriction of the linearsymplectic structure to V k m is the zero 2-form. We denote by F spn,k = F spk ( R n ) thespace of all isotropic k -flags on R n . Proposition 3.
The space F spn,k is a compact connected manifold of dimension n − P m +1 i =1 n i ( n i − , where n = k , n i = k i − k i − for i = 2 , . . . , m and n m +1 = n − k m . Note that if k ≥ k ′ then the projection π : F n,k → F n,k ′ maps F spn,k onto F spn,k ′ and, therefore, restricts to a map we still denote by π : F spn,k → F spn,k ′ . Clearly, theaction of the subgroup Sp( n, R ) ⊂ SL(2 n, R ) on F n,k leaves invariant each isotropicflag manifold F spn,k , and whence induces by restriction an action Sp( n, R ) × F spn,k → F spn,k . Proposition 4.
The action of
Sp( n, R ) on F spn,k is transitive, and for k ≥ k ′ , theprojection π : F spn,k → F spn,k ′ is Sp( n, R ) -equivariant. For k = (1 , , . . . , n ) ∈ Λ( n ), the k -flags are called Lagrangian flags , and the corre-sponding manifold F spn,k , which as we have see is the maximal boundary of the groupSp( n, R ), is called the Lagrangian flag manifold . Given 1 ≤ k ≤ n , we denote by Mat n × k ( R ) the space of all real n × k matrices. A matrix X ∈ Mat n × k ( R ) is called orthogonal iff X t X = I , i.e., the k columns of X form anorthonormal family of vectors in R n . Such matrices will be referred as k -ortho-frames.We denote by O n,k the manifold of k -ortho-framesO n,k = { X ∈ Mat n × k ( R ) : X is orthogonal } . This is a compact connected manifold of dimension k (2 n − k − /
2. When k = n , O n,n isidentified with the orthogonal group O( n ). The group O( n ) acts by left multiplicationon the space O n,k and O( k ) acts by right multiplication on O n,k . Each space O n,k has a distinguished element I n,k ∈ O n,k , which is the matrix formed by the first k columns of the identity matrix I n ∈ Mat n × n ( R ). Given a matrix X ∈ O n,k let ( X ) i be its i th -column. Given a set I ⊂ { , , . . . , n } we define h X i I to be the linear span9f the columns ( X ) i with i ∈ I . Finally, given k ∈ Λ( n ), we define the projection p : O n,k m → F n,k by p ( X ) = (cid:0) h X i { ,...,k } , h X i { ,...,k } , . . . , h X i { ,...,k m } (cid:1) . We shallcall V k = p ( I n,k ) the canonical k -flag . Consider the subgroup D k of O( k m ) consistingof block diagonal matrices of the form R = diag( R , . . . , R m ) with R ∈ O( k ) and R i ∈ O( k i − k i − ) for i = 2 , . . . , m . Let O n,k / D k be the quotient orbifold, i.e., thespace of orbits of the right action of D k on O n,k . Note the projection p : O n,k m → F n,k defined above is invariant under the action of D k . Whence it induces a quotient map p : O n,k m / D k → F n,k , and it is not difficult to see that in fact Proposition 5. p : O n,k m / D k → F n,k is a diffeomorphism. Given 1 ≤ k ′ ≤ k and a matrix E ∈ Mat n × k ′ ( R ) we denote by [ E ] k ∈ Mat n × k ( R ) thematrix formed by the first k columns of E . With this notation we define a projection π : O n,k → O n,k ′ by π ( X ) = [ X ] k ′ . Later, after having defined a natural action ofSL( n, R ) on O n,k , we shall show this projection is SL( n, R )-equivariant. For now wemake two remarks about π : O n,k → O n,k ′ . First, given k ≥ k ′ in Λ( n ), the followingdiagram commutes O n,k m π −−−→ O n,k ′ m p y p y F n,k π −−−→ F n,k ′ . Let φ : D k → D k ′ be the group homomorphism which maps each k m × k m matrix inD k to the k ′ m × k ′ m submatrix with indices in the set { , . . . , k ′ m } , which clearly belongsto D k ′ . The second remark is that π : O n,k → O n,k ′ is φ -equivariant, in the sense that π ( X g ) = π ( X ) φ ( g ) for every X ∈ O n,k and g ∈ D k . Given 1 ≤ k ≤ n , a matrix X ∈ Mat n × k ( R ) is called symplectic iff X t J X = 0, i.e.,the k columns of X span an isotropic subspace of R n . A matrix X ∈ Mat n × k ( R ) iscalled unitary iff X is orthogonal and symplectic. Such matrices will be referred as k -unitary-frames. We denote by O spn,k the manifold of k -unitary-framesO spn,k = { X ∈ Mat n × k ( R ) : X is unitary } . This a compact connected manifold of dimension k (2 n − k ).10et U( n ) denote the unitary group which is formed of all matrices A ∈ Mat n × n ( R )such that A t J A = J and A t A = I . When k = n , we write O spn in stead of O spn,n .Any matrix X = (cid:18) X X (cid:19) ∈ O spn determines a unique matrix (cid:18) X − X X X (cid:19) whichbelongs to U( n ). We can therefore identify O spn with U( n ). The group U( n ) acts byleft multiplication on the manifold O spn,k and O( k ) acts by right multiplication on O spn,k .Note that if X is symplectic, resp. unitary, and R ∈ O( k ) then X R is also symplectic,resp. unitary. Whence the right multiplication action of the group O( k ) leaves O spn,k invariant. Given k ∈ Λ( n ), by restriction the projection p : O n,k m → F n,k induces aprojection p : O spn,k m → F spn,k , which is D k -invariant. Therefore, it induces a quotientmap p : O spn,k m / D k → F spn,k , which is again a diffeomorphism. We have a commutativediagram O spn,k m π −−−→ O spn,k ′ m p y p y F spn,k π −−−→ F spn,k ′ where the top map is obtained restricting the projection π : O n,k m → O n,k ′ m . We shalllater define a natural action of Sp( n, R ) on O spn,k for which this projection is Sp( n, R )-equivariant. For now we can say that the map π : O spn,k m → O spn,k ′ m in the above diagramis φ -equivariant, where φ : D k → D k ′ is the same group homomorphism introduced inthe previous section. Let UT( k, R ) be the group of upper triangular real matrices with non-zero diagonal,and UT + ( k, R ) denote the subgroup of upper triangular real matrices with positivediagonal. The action of SL( n, R ) on O n,k uses the QR-decomposition. Theorem 1 (QR-decomposition) . Given A ∈ Mat n × k ( R ) there are unique matrices K ∈ O n,k and U ∈ UT + ( k, R ) such that A = K U . The QR-decomposition is used to define the following projection mapΠ : Mat n × k ( R ) → O n,k , by Π( A ) = K , (1)11nd with it the multiplication ∗ : SL( n, R ) × O n,k → O n,k A ∗ X = Π( A X ) . (2) Proposition 6.
The operation ∗ defined above is a left action of SL( n, R ) on O n,k . Proof.
Given A ∈ SL( n, R ) and X ∈ O n,k , consider the QR-decompositions A X = K U of A X , and
B K = K U of B K . Notice that U = U U ∈ UT + ( k, R ) so that B A X = B K U = K U U = K U is the QR-decomposition of B A X . Therefore B ∗ ( A ∗ X ) = B ∗ K = Π( B K ) = K = Π( B A X ) = (
B A ) ∗ X . ⊔⊓ Given 1 ≤ k ≤ n , denote by I n,k the n × k matrix whose columns are the first k columns of the n × n identity matrix. Notice that for any matrix X ∈ Mat m × n ( R ), theproduct X I n,k is the submatrix of X formed of its first k columns. Proposition 7.
The following projections are
SL( n, R ) -equivariant:1. π : O n,k → O n,k ′ for any k ≥ k ′ ≥ , and2. p : O n,k m → F n,k for any k ∈ Λ( n ) . Proof.
Let us prove first that π : O n,k → O n,k ′ is SL( n, R )-equivariant. Take A ∈ SL( n, R ), X ∈ O n,k and let A X = K U be its QR-decomposition with K ∈ O n,k and U ∈ UT + ( k ′ , R ). Then A X I k,k ′ = K U I k,k ′ = K I k,k ′ ( I k,k ′ ) t U I k,k ′ = ( K I k,k ′ ) (( I k,k ′ ) t U I k,k ′ ) = K ′ U ′ , where K ′ = K I k,k ′ is orthogonal and U ′ = ( I k,k ′ ) t U I k,k ′ ∈ UT + ( k ′ , R ). Therefore, thisis the QR-decomposition of A X I k,k ′ . Finally, since π ( X ) = X I k,k ′ , we get A ∗ π ( X ) = A ∗ ( X I k,k ′ ) = K ′ = K I k,k ′ = ( A ∗ X ) I k,k ′ = π ( A ∗ X ) . Now, for the equivariance of p : O n,k → F n,k , first remark that given K ∈ O n,k and U ∈ UT + ( k ) we have p ( K U ) = p ( K ), and given A ∈ SL( n, R ) and X ∈ O n,k m we12ave A p ( X ) = p ( A X ). Considering then the QR-decomposition
A X = K U , we get p ( A X ) = p ( K U ) = p ( K ) = p ( A ∗ X ), and whence p ( A ∗ X ) = A p ( X ). ⊔⊓ Let us now turn to unitary frame manifolds. Consider the set Sp n,k of symplectic2 n × k matrices. Symplectic matrices, defined in section 6, are those whose columnsspan isotropic subspaces of R n . Clearly, Sp n,k is not a linear space, but it has twoimportant obvious properties whose proofs are left to the reader. Proposition 8.
The space Sp n,k is invariant under:1. the left multiplication action of Sp( n, R ) ;2. the right multiplication action of GL( k, R ) . Consider the projection Π : Mat n × k ( R ) → O n,k defined in (1). Proposition 9.
We have(a)
Π(Sp n,k ) ⊆ O spn,k , and(b) O spn,k is invariant under the left action of Sp( n, R ) . Proof.
Let us prove (a). Given A ∈ Sp n,k ⊂ Mat n × k ( R ), let A = K U be itsQR-decomposition. Then K is orthogonal and U ∈ UT + ( k, R ), and, since O spn,k =O n,k ∩ Sp n,k , we are left to prove that K is symplectic. This follows by item 2 ofproposition 8 because K = A U − . Finally, (b) follows combining (a) with item 1 ofproposition 8. ⊔⊓ Therefore, by restriction we obtain a left action of Sp( n, R ) on O spn,k . The projections π : O n,k → O n,k ′ , for k ≥ k ′ ≥
1, and p : O n,k m → F n,k , for k ∈ Λ( n ), respectivelyinduce by restriction maps π : O spn,k → O spn,k ′ and p : O spn,k m → F spn,k . It is now easy tocheck that both these projections are Sp( n, R )-equivariant. Let ut( k, R ) be the Lie algebra of upper triangular real matrices. Given a matrix X = ( x i,j ) i,j ∈ Mat k × k ( R ), denote by X ⊳ = ( x ⊳i,j ) i,j the matrix in ut( k, R ) defined by x ⊳i,j = x i,i if i = jx i,j + x j,i if i < j j < i X X ⊳ is a linear projection operator. An important property of thisoperator is that for all A ∈ Mat k × k ( R ),( A ⊳ ) t + A ⊳ = A t + A . (3)We notice that given X ∈ O n,k , T X O n,k = { V ∈ Mat n × k ( R ) : X t V + V t X = 0 } , and T U UT + ( k, R ) = ut( k, R ), for every U ∈ UT + ( k, R ), since ut( k, R ) is a linear spaceand UT + ( k, R ) an open subset of ut( k, R ). Theorem 2 (Tangent QR-decomposition) . Given
X, V ∈ Mat n × k ( R ) , let X = K U be the QR-decomposition of X , where K ∈ O n,k and U ∈ UT + ( k, R ) . Then there areunique matrices V ∈ T K O n,k and V ∈ ut( n, k ) such that V = V U + K V . V = V U − − K ( K t V U − ) ⊳ (4) V = ( K t V U − ) ⊳ U (5) Proof.
Consider the QR-decomposition X = K U . Then X is the unique point inthe transversal intersection of the two orbits O n,k · U and K · UT + ( k, R ). An easyargument shows that given V ∈ Mat n × k ( R ) there are unique vectors V ∈ T K O n,k and V ∈ T U UT + ( k, R ) = ut( k, R ) such that V = V U + K V . It follows that V satisfies K t V + V t K = 0, and V is upper triangular.We have K t V U − + ( K t V U − ) t = K t V U − + U − t V t K = K t ( V U + KV ) U − + U − t ( V U + KV ) t K = K t V + V U − + V t K + U − t V t = K t V + V t K | {z } =0 + V U − + ( V U − ) t = V U − + ( V U − ) t and since V U − is upper triangular it follows from the identity (3) that V U − =( K t V U − ) ⊳ , which proves (5). Replacing the value (5) for V in the equality V = V U + K V we get (4). ⊔⊓ This theorem gives us formulas for the derivatives of the canonical projectionsΠ : Mat n × k ( R ) → O n,k and Π ′ : Mat n × k ( R ) → UT + ( k, R ) associated with the QR-decomposition. Given matrices X, V ∈ Mat n × k ( R ), assume X = K U is the QR-decomposition of X . Then we have 14igure 1: The Tangent QR-decomposition1. D Π X ( V ) = V U − − K ( K t V U − ) ⊳ D Π ′ X ( V ) = ( K t V U − ) ⊳ U When X ∈ O n,k these formulas simplify even further1’. D Π X ( V ) = V − X ( X t V ) ⊳ D Π ′ X ( V ) = ( X t V ) ⊳ Given a matrix A ∈ sl( n, R ) and k ∈ Λ( n ), we define a flow on the k -flag manifold, ϕ tA : F n,k → F n,k by ϕ tA ( V ) = e t A V . Analogously, given k ≥ k -ortho frame manifold O n,k , ϕ tA : O n,k → O n,k by ϕ tA ( X ) = e t A ∗ X . Proposition 7shows these two flows are semiconjugate by the submersion p : O n,k m → F n,k . Wedenote by F A the vector field on O n,k associated to the flow ϕ tA , which is defined by F A ( X ) = ddt (cid:2) e t A ∗ X (cid:3) t =0 for X ∈ O n,k . We have the following explicit formula: Proposition 10.
Given A ∈ sl( n, R ) and X ∈ O n,k , F A ( X ) = A X − X ( X t A X ) ⊳ . roof. From remark 1’ above we get F A ( X ) = ddt (cid:2) e t A ∗ X (cid:3) t =0 = ddt Π (cid:0) e t A X (cid:1) t =0 = (cid:2) D Π e t A X ( A e t A X ) (cid:3) t =0 = D Π X ( A X )= A X − X ( X t A X ) ⊳ . ⊔⊓ Given A ∈ sp( n, R ), the flows ϕ tA on O n,k and F n,k respectively induce by res-triction flows on the invariant submanifolds O spn,k and F spn,k , which we still denote by ϕ tA .We also denote by F A the vector field on O spn,k associated with the flow ϕ tA : O spn,k → O spn,k . Remark 1.
Proposition 10 holds with the same expression for the induced flow ϕ tA onthe unitary frame manifold O spn,k . Assume now that A ∈ sl( n, R ) is a symmetric matrix. Proposition 11.
Given X ∈ O n,k , F A ( X ) = 0 ⇔ every column of X is an eigen-vector of A . Proof.
Assume every column of X is an eigenvector of A . Then if D ∈ Mat k × k ( R )is the diagonal matrix with the corresponding eigenvalues A X = X D , and whence F A ( X ) = A X − X ( X t A X ) ⊳ = X D − X ( X t X D ) ⊳ = X D − X D ⊳ = 0, because D ⊳ = D . Conversely, if N = ( X t A X ) ⊳ and 0 = F A ( X ) = A X − X N , it follows that h X i { ,...,i } is an invariant subspace under A , for i = 1 , . . . , k . Since A is symmetric andthe columns of X are pairwise orthogonal, every column of X must be an eigenvectorof A . ⊔⊓ Proposition 12.
Given k ∈ Λ( n ) and V ∈ F n,k , V = ϕ tA ( V ) for some t > ⇔ thereis some matrix X ∈ O n,k m such that p ( X ) = V and F A ( X ) = 0 . Proof.
Assume V = p ( X ) for some X ∈ O n,k m with F A ( X ) = 0. Then X = ϕ tA ( X )for all t , and since p : O n,k m → F n,k semiconjugates the flows ϕ tA on O n,k m and F n,k itfollows that V = ϕ tA ( V ) for all t . Conversely, assume that V = ϕ tA ( V ) for some t > V i in the flag V is invariant under A . Because A is symmetric,the same is true about the subspaces W = V W i = V i ⊖ V i − , for i = 2 , . . . , m .Whence each W i is a direct sum of eigenspaces of A , and we can find an orthonormalbasis for W i formed by eigenvectors of A . Putting these basis together, as columns ofa matrix X , we have that V = p ( X ) and F A ( X ) = 0. ⊔⊓ emark 2. Propositions 11 and 12 also hold for isotropic flags and unitary framesprovided the symmetric matrix A is chosen in the symplectic Lie algebra sp( n, R ) . Consider the Hilbert-Schmidt inner product on Mat n × k ( R ) h E, F i hs = 1 k tr ( E t F )with its associated norm k E k hs = r k tr ( E t E ) . We take the Riemannian structures induced by this inner product on the ortho frameand unitary frame manifolds O n,k and O spn,k . We shall refer to them as the
Hilbert-Schmidt metrics on these manifolds.
Lemma 1.
Given X ∈ O n,k , E ∈ Mat n × k ( R ) , R ∈ SO( n ) and S ∈ O( k ) ,1. k X k hs = 1 ,2. k R E S k hs = k E k hs . From this lemma, we see that both the left action of SO( n ) and the right actionof O( k ) on O n,k leave the Hilbert-Schmidt metric invariant, which means that boththese actions are isometric. Since O spn,k is a submanifold of O n,k invariant under boththe left action of U( n ) and the right action of O( k ), these actions too are isometricfor the Hilbert-Schmidt metric on O spn,k . We consider on F n,k and F spn,k the uniqueRiemannian metrics which respectively turn the projections p : O n,k m → F n,k and p : O spn,k m → F spn,k to Riemannian submersions, i.e., such that the tangent maps Dp X : T X O n,k m → T p ( X ) F n,k and Dp X : T X O spn,k m → T p ( X ) F spn,k are orthogonal projections. Alinear map A : E → F between Euclidean spaces E and F is said to be an orthogonalprojection iff h A X, A Y i = h X, Y i for every X, Y ∈ Ker( A ) ⊥ .The following propositions describe the tangent-normal space decompositions overthe orthogonal and unitary frame manifolds. Proposition 13.
Given X ∈ O n,k , a) T X O n,k = { S ∈ Mat n × k ( R ) : S t X + X t S = 0 } ,(b) T ⊥ X O n,k = { S ∈ Mat n × k ( R ) : S t X − X t S = 0 and ( I − X X t ) S = 0 } .The corresponding projections Π T : Mat n × k ( R ) → T X O n,k and Π ⊥ : Mat n × k ( R ) → T ⊥ X O n,k are given by :(a’) Π T ( B ) = X ( X t B − B t X ) + ( I − X X t ) B ,(b’) Π ⊥ ( B ) = X ( X t B + B t X ) . Proof.
Given X ∈ O n,k define E + n,k ( X ) = { S ∈ Mat n × k ( R ) : S t X + X t S = 0 } ,E − n,k ( X ) = { S ∈ Mat n × k ( R ) : S t X − X t S = 0 } . These two spaces are transversal, i.e., Mat n × k ( R ) = E + n,k ( X ) + E − n,k ( X ), and theirintersection is E n,k ( X ) = E + n,k ( X ) ∩ E − n,k ( X ) = { S ∈ Mat n × k ( R ) : X X t S = 0 } .Because X X t is the orthogonal projection onto the linear subspace of R n spanned bythe columns of X , it follows that the orthogonal complement of E n,k ( X ) is E n,k ( X ) ⊥ = { S ∈ Mat n × k ( R ) : ( I − X X t ) S = 0 } . This implies that E ± n,k ( X ) ⊥ = E ∓ n,k ( X ) ∩ E n,k ( X ) ⊥ . (6)Now, it is obvious that (a) T X O n,k = E + n,k ( X ), and then (6) implies (b). It is also clearthat B = Π T ( B ) + Π ⊥ ( B ). Finally, one can easily check that R = Π T ( B ) defined in(a’) satisfies R t X = − X t R , and that S = Π ⊥ ( B ) defined in (b’) satisfies S t X = X t S ,( I − X X t ) S = 0. ⊔⊓ In the symplectic case we have
Proposition 14.
Given X ∈ O spn,k ,(a) T X O spn,k = { S ∈ Mat n × k ( R ) : S t X + X t S = 0 and S t J X + X t J S = 0 } ,(b) T ⊥ X O spn,k is the sum of two subspaces:i. { S ∈ Mat n × k ( R ) : S t X − X t S = 0 and ( I − XX t ) S = 0 } ,ii. { S ∈ Mat n × k ( R ) : S t J X − X t J S = 0 and ( I + J XX t J ) S = 0 } . he corresponding projections Π T : Mat n × k ( R ) → T X O spn,k and Π ⊥ : Mat n × k ( R ) → T ⊥ X O spn,k are given by :(a’) Π T ( B ) = X ( X t B − B t X ) + J X ( B t J X − X t J B ) + ( I − XX t + J XX t J ) B ,(b’) Π ⊥ ( B ) = X ( X t B + B t X ) − J X ( X t J B + B t J X ) . Proof.
Differentiating the two defining equations of O spn,k we obtain that T X O spn,k = E + n,k ( X ) ∩ E − n,k ( J X ), which proves (a). Therefore, by (6), we get that T X O spn,k ⊥ is thesum of the subspaces i. and ii. above T X O spn,k ⊥ = ( E + n,k ( X ) ∩ E − n,k ( J X ) ) ⊥ = E + n,k ( X ) ⊥ + E − n,k ( J X ) ⊥ = E − n,k ( X ) ∩ E n,k ( X ) ⊥ | {z } i. + E + n,k ( J X ) ∩ E n,k ( J X ) ⊥ | {z } ii. . One can easily check that B = Π T ( B ) + Π ⊥ ( B ). To finish the proof we need, as inthe previous lemma, to show that R = Π T ( B ) defined in (a’) satisfies R t X + X t R = 0and R t J X + X t J R = 0, and also to show that writing S = Π ⊥ ( B ) defined in (b’) as S = S + S with S = X ( X t B + B t X ), S = − J X ( X t J B + B t J X ), then S belongsto space i. while S belongs to space ii. . Because this argument does not shade anylight on formulas (a’) and (b’), we describe now a more geometric approach on howto derive these projection formulas. First we need a couple of definitions. We denoteby P E the orthogonal projection onto a subspace E of some euclidean space. Let ussay that two subspaces E and F are perpendicular iff the inner product of any vector u ∈ ( E ∩ F ) ⊥ ∩ E with any vector v ∈ ( E ∩ F ) ⊥ ∩ F is always zero. When this occurs,the following general formulas hold P E ∩ F = P E ◦ P F and P ( E ∩ F ) ⊥ = P F ⊥ + P E ⊥ ◦ P F . Finally, formulas (a’) and (b’) can be driven from these abstract formulas applied tothe subspaces E = E − n,k ( J X ) and F = E + n,k ( X ), showing first they are perpendicular. ⊔⊓ Let ∆ k = { b = ( b , . . . , b k ) ∈ R k : b ≥ b ≥ . . . ≥ b k > } , and for each b ∈ ∆ k denote by D b the diagonal matrix D b = b · · · b · · · · · · b k . Definition 1.
Given a symmetric matrix A ∈ sl( n, R ) and b ∈ ∆ k , we define thequadratic functions Q A,b : O n,k → R and Q A : O n,k → R respectively by Q A,b ( X ) = h A X D b , X D b i hs and Q A ( X ) = h A X, X i hs . For each k ∈ Λ( n ) set∆ k = { b ∈ ∆ k m : b = · · · = b k < b k +1 = · · · = b k < . . . < b k m − +1 = · · · = b k m } ∆ k = { b ∈ ∆ k m : b = · · · = b k ≤ b k +1 = · · · = b k ≤ . . . ≤ b k m − +1 = · · · = b k m } Note that Q A = Q A,b with b = (1 , . . . , k ≥
1, the family { ∆ k } k indexedover all k ∈ Λ( n ) with last entry k m = k is a partition of ∆ k . Proposition 15.
Given a symmetric matrix A ∈ sl( n, R ) , k ∈ Λ( n ) and b ∈ ∆ k , thefunction Q A,b : O n,k m → R is invariant under the right action of D k . Proof.
Use lemma 1 and the fact that for b ∈ ∆ k the diagonal matrix D b commuteswith every matrix in D k . ⊔⊓ This proposition shows that given a symmetric matrix A ∈ sl( n, R ), k ∈ Λ( n ) and b ∈ ∆ k , Q A,b : O n,k m → R induces a quotient function Q A,b : F n,k → R . For symplecticmatrices A ∈ sp( n, R ) we denote by Q A,b : O spn,k m → R and Q A,b : F spn,k → R therestriction functions. Note that O spn,k m and F spn,k are submanifolds respectively of O n,k m and F n,k . 20 roposition 16. Given A ∈ sl( n, R ) , resp. A ∈ sp( n, R ) , and b = ( b , . . . , b k ) ∈ ∆ k , ifwe write ω i = i ( b i − b i +1 ) /k > for i = 1 , . . . , k − and ω k = b k , then the function Q A,b : O n,k → R , resp. Q A,b : O spn,k → R , is given by Q A,b ( X ) = k X i =1 ω i ( Q A ◦ π i )( X ) , (7) where each π i : O n.k → O n,i , resp. π i : O spn.k → O spn,i , denotes the canonical projection π i ( X ) = [ X ] i . Proof.
With the convention that b k +1 = 0, we have Q A,b ( X ) = h A X D b , X D b i hs = 1 k k X i =1 ( A X D b ) i · ( X D b ) i = 1 k k X i =1 b i ( A X ) i · ( X ) i = 1 k k X i =1 ( b i − b i +1 ) (( A X ) · ( X ) + · · · + ( A X ) i · ( X ) i )= k X i =1 ik ( b i − b i +1 ) h [ A X ] i , [ X ] i i hs = k X i =1 ik ( b i − b i +1 ) h A [ X ] i , [ X ] i i hs = k X i =1 ω i Q A ( π i ( X ) ) . ⊔⊓ If b ∈ ∆ k with k = ( k , . . . , k m ) then ω k i > i = 1 , . . . , m , and ω j = 0 for all j = k i . This shows that Q A,b : F n,k → R is a sum of pullbacks of functions defined overthe Grassmannian manifolds G n,k i = F n, ( k i ) (cid:22) F n,k . Lemma 2.
The gradient of Q A : O n,k → R at X is given by ▽ ( Q A )( X ) = 2 ( I − X X t ) A X .
Proof.
Consider the natural quadratic extension of Q A to the linear space of allmatrices: Q A ( X ) = tr ( X t A X ). The gradient of this extension at X is 2 AX . Then21y proposition 13, ▽ ( Q A )( X ) = Π T ( 2 AX )= X ( X t A X − ( A X ) t X ) | {z } =0 +2 ( I − X X t ) A X = 2 ( I − X X t ) A X .
In the symplectic case we compute the following expression for the gradient ▽ ( Q A )( X ) = 2 ( J X )( J X ) t AX + 2 ( I − X X t + J XX t J ) A X which simplified gives exactly the same expression 2 ( I − XX t ) AX . ⊔⊓ Remark 3.
Let P X be the orthogonal projection onto h X i { ,...,k } . Then we can write ▽ ( Q A )( X ) = 2 ( I − P X ) A X . Next we derive an explicit formula for the gradient of Q A,b . Given X ∈ O n,k , wedenote by P Xi = P i (1 ≤ i ≤ k + 1) the following orthogonal projection matrices: For k ≤ i , P i is the projection onto the linear span h X i { i } , while for i = k + 1, P k +1 is theorthogonal projection onto ( h X i { ,...,k } ) ⊥ . Notice that P + . . . + P k +1 = I . We write asabove ω i = i ( b i − b i +1 ) /k >
0. Notice that for i ≤ j we have ki ω i + . . . + kj ω j = b i − b j +1 .Let E i ( b ) ∈ Mat k × k ( R ) denote the matrix E i ( b ) = b − b i +1 · · · · · · b − b i +1 · · · · · · · · · b i − b i +1 · · ·
00 0 · · · · · · · · · · · · , (8)and let E i ∈ Mat k × k ( R ) be the matrix representing the orthogonal projection onto thelinear span h e , . . . , e i i . 22 roposition 17. The gradient of Q A,b : O n,k → R at X is given by ▽ ( Q A,b )( X ) = 2 k +1 X i =1 P Xi A X E i − ( b ) . Proof.
Notice that I k,i I i k = E i . The projection π i can be written as π i ( X ) = X I k,i .Its derivative ( Dπ i ) X V = V I k,i has adjoint ( Dπ i ) ∗ X W = ki W I i,k . Therefore, usinglemma 2, we get that ▽ ( Q A ◦ π i )( X ) = ( Dπ i ) ∗ X ( ▽ Q A )( π i ( X ))= ki ( ▽ Q A )( XI k,i ) I i × k = 2 ki ( I − ( P + · · · + P i )) A X E i = 2 ki ( P i +1 + · · · + P k +1 ) A X E i . Thus ▽ ( Q A,b )( X ) = k X i =1 ω i ▽ ( Q A ◦ π i )( X )= 2 k X i =1 ki ω i k +1 X j = i +1 P j ! A X E i = 2 k +1 X j =2 P j A X j − X i =1 ki ω i E i ! = 2 k +1 X j =2 P j A X E j − ( b ) . ⊔⊓ Proposition 18.
Given A ∈ g sym n , b > b > . . . > b k > , and X ∈ O n,k , X is acritical point of Q A,b iff every column of X is an eigenvector of A . roof. It is clear that if each column of X is an eigenvector of A then X is a criticalpoint of Q A ◦ π i , for every i = 1 , . . . , k . Therefore, X is a critical point of Q A,b . Assumenow that ▽ ( Q A,b )( X ) = 0. Fix j = 1 , . . . , k . Since0 = 12 ▽ ( Q A,b )( X ) e j = k +1 X i = j +1 ( b j − b i ) P i A X e j = k +1 X i = j +1 ( > z }| { b j − b i ) P i A ( X ) j , it follows that P i A ( X ) j = 0 for every i = j + 1 , . . . , k + 1, which implies that A ( X ) j ∈h [ X ] j i . By induction we derive (see proof of lemma 4) that ( X ) j is an eigenvector of A . ⊔⊓
11 Lyapunov functions
We shall denote by G sym n, + the space of symmetric positive definite matrices in G n andby g sym n the space of symmetric matrices in g n . We write SL sym n, + , Sp sym n, + , sl sym n or sp sym n to emphasize that G n = SL( n, R ), G n = Sp( n, R ), g n = sl( n, R ) or g n = sp( n, R ).To establish Theorem A it is enough proving that the lift Q A,b : O n,k → R is aLyapunov function for lifted flow ϕ tH : O n,k → O n,k when A, H ∈ g n are symmetricmatrices sharing the same ordered eigen-directions. Proposition 19.
Given B ∈ SL( n, R ) and X ∈ O n,k , let B X = R D R ′ be the singularvalue decomposition of B X , with R ∈ O n,k , R ′ ∈ O k orthogonal, and D a diagonalpositive k × k matrix. Then there is an orthogonal matrix S ∈ O k such that B ∗ X = R S . Proof.
Let
B X = K U be the QR-decomposition of
B X . We have B ∗ X = K = R S with S = D R ′ U − ∈ SL( k, R ). Since K and R are both orthogonal, it follows I = K t K = ( R S ) t ( R S ) = S t ( R t R ) S = S t S , which proves that S ∈ O k . ⊔⊓ Remark 4.
With the notation of the previous proposition it follows that Q A ( B ∗ X ) = h A R, R i hs . emma 3. Given matrices
E, F ∈ Mat n × n ( R ) , if E and F are symmetric positivesemi-definite, i.e., E, F ≥ , then tr ( E F ) ≥ , with tr ( E F ) = 0 iff E F = 0 . Proof.
We can write E = R t D R and F = R t D R with D i diagonal positivesemi-definite and R i orthogonal, for i = 1 ,
2. Thentr (
E F ) = tr ( ( R t D R ) ( R t D R ) )= tr ( p D R R t D R R t p D )= tr ( p D ( R R t ) t D ( R R t ) p D )= n (cid:13)(cid:13)(cid:13)p D ( R R t ) p D (cid:13)(cid:13)(cid:13) ≥ . Since the matrices
E F and √ D ( R R t ) √ D are conjugated, tr ( E F ) = 0 implies √ D ( R R t ) √ D = 0, which in turn forces E F = 0. ⊔⊓ Lemma 4.
Given B ∈ G sym n, + and X ∈ O n,k , B ∗ X = X iff every column of X is aneigenvector of B . Proof.
Assume every column of X is an eigenvector of B . Then there is some positivediagonal matrix D such that B X = X D , which implies that B ∗ X = Π( B X ) =Π(
X D ) = X . Conversely, if B ∗ X = Π( B X ) = X there is some U ∈ UT + ( k, R ) suchthat B X = X U . Denote by X i the i th -column of X and let V i = h X , . . . , X i i be thelinear span of the first i columns of X . Then { } = V ⊂ V ⊂ V ⊂ . . . ⊂ V k is a flaginvariant by B . Since X i ∈ V i ∩ V ⊥ i − , for each i = 1 , . . . , k , it follows that X i must bean eigenvector of B . ⊔⊓ Remark 5.
Let B ∈ G sym n, + , A ∈ g sym n , X ∈ O n,k and S ∈ O k .1. If B ∗ X = X then in general B ∗ ( X S ) = X S , but2. if Q A ( B ∗ X ) = Q A ( X ) then Q A ( B ∗ ( X S )) = Q A ( X S ) . Let P ∈ gl( n, R ) be an orthogonal projection matrix, i.e., P = P t = P . We definea bilinear form ξ P : g sym n × g sym n → R by ξ P ( A, H ) = tr (
P A ( I − P ) H ). Proposition 20.
Given symmetric matrices
A, H ∈ g n sharing the same ordered eigen-directions, ξ P ( A, H ) ≥ , and ξ P ( A, H ) = 0 ⇔ A P = P A and
H P = P H . roof. We can with no loss of generality assume that A and H are diagonal matrices.Denote by D x the diagonal matrix with diagonal entries x = ( x , . . . , x n ) ∈ R n . Thenthe bilinear form ξ P can be seen as the bilinear form ξ P : R n × R n → R defined by ξ P ( x, y ) = tr ( P D x ( I − P ) D y ). Consider the cone Γ = { x ∈ R n : x ≥ . . . ≥ x n } .We want to see that given a, h ∈ Γ, ξ P ( a, h ) ≥
0, and ξ P ( a, h ) = 0 ⇔ D a P = P D a and D h P = P D h . We define the vectors v i = (1 , . . . , , , . . . , ∈ Γ with the first i coordinates equal to 1 for 1 ≤ i ≤ n , which form a basis of R n . Notice that because D v n = I is the identity matrix we get that ξ P ( v n , x ) = 0 = ξ P ( x, v n ) for every x ∈ R n .If i ≤ j ≤ n − ξ P ( v i , v j ) = i X k =1 p k,k − i X k =1 j X r =1 ( p k,r ) = i X k =1 p k,k − j X r =1 ( p k,r ) ! = i X k =1 n X r = j +1 ( p k,r ) ≥ , and otherwise, if j < i ≤ n − ξ P ( v i , v j ) = j X k =1 p k,k − i X r =1 j X k =1 ( p r,k ) = j X k =1 p k,k − j X r =1 ( p r,k ) ! = j X k =1 n X r = i +1 ( p r,k ) ≥ . Now, given a, h ∈ Γ we can write a = a n v n + n − X i =1 ( a i − a i +1 ) v i and h = h n v n + n − X i =1 ( h i − h i +1 ) v i . Whence ξ P ( a, h ) = n − X i,j =1 ( a i − a i +1 ) | {z } ≥ ( h j − h j +1 ) | {z } ≥ ξ P ( v i , v j ) | {z } ≥ ≥ . Assume now that ξ P ( a, h ) = 0, and a i > a j ⇔ h i > h j , for every 1 ≤ i < j ≤ n . Thenfor every i = 1 , . . . , n such that a i > a i +1 we have 0 = ξ P ( v i , v i ) = P ik =1 P nr = i +1 ( p r,k ) ,which implies that p r,k = p k,r = 0 for all k ≤ i and r > i . This shows that P is ablock-diagonal matrix whose blocks correspond in A = D a and H = D h to multiplesof identity matrices. Therefore P commutes with A = D a and H = D h . ⊔⊓ emma 5. Given
A, B ∈ gl( k, R ) , if A t = A and B t = B then tr ( B A ⊳ ) = tr ( B A ) . Proof.
Let S = A ⊳ − A . The entriy s i,j of S is s i,j = a i,j + a j,i − a i,j = a i,j when i < j , s i,j = a i,i − a i,i = 0 when i = j , and s i,j = 0 − a i,j = − a i,j if i > j . Therefore S is skew-symmetric andtr ( B S ) = tr (
S B ) = tr ( − S t B t ) = − tr (( B S ) t ) = − tr ( B S ) , which implies tr ( B ( A ⊳ − A )) = tr ( B S ) = 0 and proves the lemma. ⊔⊓ Lemma 6.
Given H ∈ g sym n and X ∈ O n,k , F H ( X ) = 0 iff every column of X is aneigenvector of H . Proof.
Assume every column of X is an eigenvector of H . Then there is some positivediagonal matrix D such that H X = X D , which implies that F H ( X ) = H X − X ( X t H X ) ⊳ = X D − X ( X t X D ) ⊳ = X D − X D ⊳ = 0 . Conversely, assume 0 = F H ( X ) = H X − X U with U = ( X t H X ) ⊳ ∈ UT + ( k, R ), sothat H X = X U . Denote by X i the i th -column of X and let V i = h X , . . . , X i i be thelinear span of the first i columns of X . Then { } = V ⊂ V ⊂ V ⊂ . . . ⊂ V k is a flaginvariant by H . Since X i ∈ V i ∩ V ⊥ i − , for each i = 1 , . . . , k , it follows that X i must bean eigenvector of H . ⊔⊓ Remark 6.
Given
A, H ∈ g sym n , X ∈ O n,k and S ∈ O k .1. If F H ( X ) = 0 then in general F H ( X S ) = 0 , but2. if Q A ( ϕ tH ( X )) = Q A ( X ) then Q A ( ϕ tH ( X S )) = Q A ( X S ) . Proposition 21.
Given
A, H ∈ g sym n sharing the same ordered eigen-directions, wehave for every X ∈ O n,k and t > , Q A ( ϕ tH ( X )) ≥ Q A ( X ) , with equality iff for some S ∈ O k , F H ( X S ) = 0 . roof. Given X ∈ O n,k , using lemma 5 and proposition 20 we get D ( Q A ) X F H ( X ) = 2 h A X, F H ( X ) i hs = 2 k − tr ( X t A F H ( X ))= 2 k − tr ( X t A ( H X − X ( X t H X ) ⊳ ))= 2 k − tr ( X t A H X ) − k − tr (( X t A X ) ( X t H X ) ⊳ ))= 2 k − tr ( X t A H X ) − k − tr ( X t A X X t H X )= 2 k − tr ( X t A ( I − X X t ) H X )= 2 k − tr ( X X t A ( I − X X t ) H )= 2 k − tr ( P A ( I − P ) H ) = 2 k − ξ P ( A, H ) ≥ , where P = X X t is the orthogonal projection onto the subspace h X i spanned bythe columns of X . Whence Q A is a Lyapunov function for the flow ϕ tH of F H . Tosee it is a strict Lyapunov function assume Q A ( ϕ tH ( X )) = Q A ( X ) for some t > D ( Q A ) X F H ( X ) = 0 which by proposition 20 implies that H P = P H . Thiscommutativity shows that the subspace h X i is H -invariant. Therefore, there is anorthogonal matrix S ∈ O k such that every column of X S is an eigenvector of H , andby lemma 6 it follows that F H ( X S ) = 0. ⊔⊓ Remark 7.
The argument above also proves that Q A ( ϕ tH ( X )) = Q A ( X ) iff h X i isan H -invariant subspace. Corollary 1.
Given
A, H ∈ g sym n sharing the same ordered eigen-directions, the func-tion Q A : O n,k → R is a Lyapunov function for ϕ A : O n,k → O n,k , but it is only a strictLyapunov function when k = 1 . Theorem 3.
Given
A, H ∈ g sym n sharing the same ordered eigen-directions and b >b > . . . > b k > , the function Q A,b : O n,k → R is a strict Lyapunov function for themap ϕ A : O n,k → O n,k . Proof.
By propositions ?? , 21 and 16, Q A,b is a sum of k Lyapunov functions forthe flow ϕ tH , and, therefore, it is also a Lyapunov function of ϕ tH . Assume now that Q A,b ( ϕ tH ( X ) ) = Q A,b ( X ) for some t >
0. Then Q A ( π i ◦ ϕ tH ( X ) ) = Q A ( π i ( X ) ),for every i = 1 , . . . , k . By remark 7 this implies that h [ X ] i i is H -invariant for all i = 1 , . . . , k . By induction we can easily prove that each column of X is an eigenvectorof H . Therefore, by lemma 6, F H ( X ) = 0. ⊔⊓ Fix a commutative positive symmetric subgroup H n ⊂ SL( n, R ), and let E , E , . . . , E n be its eigen-directions. Each E i is a one dimensional subspace such that A E i = E i for all A ∈ H n , and we have R n = E ⊕ E ⊕ . . . ⊕ E n . We also assume that acommutative positive symmetric subgroup H n ⊂ Sp( n, R ) is fixed. In this case H n has2 n eigen-directions denoted by E , E , . . . , E n , and we assume the directions E i and E i + n to be conjugate in the sense that J E i = E i + n , for every i = 1 , . . . , n . Given a set I ⊂ { , , . . . , n } we shall write E I := L i ∈ I E i . In all examples given we shall assumethat H n = Diag + n .In this section we define the stratifications associated to H n whose existence isasserted in Theorem B. Definition 2.
We call ( n, k )-tree to any sequence P = ( P , P , . . . , P k ) of subsets P i ⊂ { , , . . . , n } such that for every ≤ i < j ≤ k either P i ∩ P j = ∅ or P i ⊆ P j .In other words, ( n, k ) -trees are just trees in a set of k nodes consisting of subsets of { , , . . . , n } . We say that k is the length of the ( n, k ) -tree P and write | P | := k . Theunion ∪ kj =1 P j is called the support of P and denoted by supp( P ) . We shall borrow the following terminology from Graph Theory. We say that P j is a parent of P i iff P i ⊆ P j . We say that P i is the mother of P j iff j < i , P j ⊆ P i and there is no j < s < i with P j ⊆ P s ⊆ P i . We say that P i is a leaf iff there isno 1 ≤ s < i with P s ⊆ P i . In other words, P i is a node with no daughters. We saythat P i is a root iff there is no i < s ≤ k such that P i ⊆ P s , which means that P i is a node with no mother. Given an increasing map π : { , . . . , r } → { , . . . , k } anda ( n, k )-tree Q we define a new ( n, r )-tree setting π ∗ Q = ( Q π (1) , . . . , Q π ( r ) ). We shallrefer to it a subtree of Q . A connected component subtree of Q means a ( n, r )-tree P = π ∗ Q where π : { , . . . , r } → { , . . . , k } is an increasing map such that Q π ( r ) isa root of Q and { π (1) , . . . , π ( r ) } = { j ∈ { , . . . , k } : Q j ⊆ Q π ( r ) } . The number ofconnected components of P is the number of roots of P . Given a tree P we define thetwo functions card P , depth P : { , , . . . , k } → N bycard P ( i ) = P i anddepth P ( i ) = { s ∈ { , . . . , i − } : P s ⊆ P i } . The second one gives the depth of a node in the tree P , which can be recursively definedas follows: leaves have depth zero. For all other nodes, the depth is the numberof its daughter nodes plus the sum of all its daughter nodes’ depths. Alternatively,29epth P ( i ) + 1 is the length of the subtree of P consisting of all nodes P j such that P j ⊆ P i ( P i included).Given a ( n, k )-tree P we defineS P = { V ∈ F n,k : V i ⊆ E P ∪···∪ P i , ∀ ≤ i ≤ k } , e S P = { X ∈ O n,k : ( X ) i ∈ E P i , ∀ ≤ i ≤ k } . It follows easily that e S P = p − (S P ) for each ( n, k )-tree P , where p is the canonicalprojection from O n,k onto F n,k defined in section ?? . These sets will be referred as( n, k )-strata. Proposition 22.
Given a ( n, k ) -tree P , the following are equivalent:1. the ( n, k ) -stratum S P is non-empty ,2. depth P ( i ) + 1 ≤ card P ( i ) for every i = 1 , . . . , k . Proof.
Assume first S P is non-empty and take X ∈ e S P . We have card P ( i ) = dim E P i .Denote by C i ( P , X ) linear span of the columns of ( X ) j of X with j < i and P j ⊂ P i .Then depth P ( i ) = dim C i ( P , X ), and because E P i must contain C i ( P , X ) ⊕ h ( X ) i i item 2. follows. The converse is proved recursively constructing (column by column)an orthogonal matrix X ∈ e S P . ⊔⊓ This motivates the definition
Definition 3.
We say that a ( n, k ) -tree P is consistent iff depth P ( i ) + 1 ≤ card P ( i ) for every i = 1 , . . . , k . We say that a node P i is full in P iff depth P ( i ) + 1 = card P ( i ) . Given a ( n, k )-tree P = ( P , . . . , P k ) and 1 ≤ i ≤ k we define π i ( P ) := ( P , . . . , P i ),which clearly is a ( n, i )-tree. Consider the projection π i : F n,k → F n,i defined in theprevious section. One can easily check that the ( n, k )-strata are preserved by theseprojections in the following sense: Proposition 23.
For any ( n, k ) -tree P and ≤ i ≤ k , π i (S P ) = S π i ( P ) . We are now going to prove that strata are always closed manifolds.30 roposition 24.
Given a consistent ( n, k ) -tree P , the ( n, k ) -stratum S P is a compactconnected manifold (without boundary) of dimension dim(S P ) = − k − κ ( P ) + k X i =1 P i , (9) where κ ( P ) := { ( i, j ) : 1 ≤ i < j ≤ k, P i ⊆ P j } . The stratum e S P is also acompact manifold without boundary of the same dimension, which in general may bedisconnected. Proof.
Consider the linear space b S n,k ( P ) = { B ∈ Mat n × k ( R ) : ( B ) i ∈ E P i , ∀ i = 1 , . . . , k } , which has dimension dim b S n,k ( P ) = P ki =1 P i ). Consider the setΣ = { ( i, j ) : 1 ≤ i < j ≤ k, P i ⊆ P j } , and define Φ : b S n,k ( P ) → R k × R Σ byΦ( X ) = (cid:0) { k ( X ) i k − } ≤ i ≤ k , { ( X ) i · ( X ) j } ( i,j ) ∈ Σ (cid:1) Notice that e S P = b S n,k ( P ) ∩ O n,k = Φ − (0 , X ∈ e S P is a regular point of Φ, in the sense that the derivative D Φ X : b S n,k ( P ) → R k × R Σ is surjective. Then e S P being a regular level set of Φ will bea closed manifold of the claimed dimension. The derivative of Φ is explicitly given by D Φ X ( Y ) = (cid:0) { X ) i · ( Y ) i } ≤ i ≤ k , { ( X ) i · ( Y ) j + ( Y ) i · ( X ) j } ( i,j ) ∈ Σ (cid:1) . Given ( d, c ) = (cid:0) { d i } ≤ i ≤ k , { c i,j } ( i,j ) ∈ Σ (cid:1) ∈ R k × R Σ , we shall prove by induction thatthere is a matrix Y ∈ Mat n × k ( R ) such that D Φ X ( Y ) = ( d, c ). Fix X ∈ e S P = Φ − (0 , i : there are vectors Y , . . . , Y i in R n such that1. X ) j · Y j = d j , for all j = 1 , . . . , i ,2. ( X ) s · Y j + Y i · ( X ) j = d s,j , for all ( s, j ) ∈ Σ with j ≤ i ,3. Y i ∈ E P i , for all i = 1 , . . . , k . i = 1 this is true: condition (2) is empty, and condition(1) is obvious because( X ) = 0. Assume this property holds for i − Y , . . . , Y i − . To find Y i such that (1)-(3) above hold, we have to solve the following system of equations in theunknown Y i ∈ E P i . (cid:26) X ) i · Y i = d i ( X ) j · Y i = c ′ j,i for every ( j, i ) ∈ Σ , where c ′ j,i = d j,i − ( X ) i · Y j is already determined. This system is determined becausethe columns of X are orthonormal. In particular, the columns ( X ) i and ( X ) j with( j, i ) ∈ Σ form an orthonormal system in E P i . Therefore property above holds for i ,which completes the induction, and proves D Φ X is surjective.The strata e S P are not connected in general. For instance if P = ( { } ) then e S P consists of two points, and if P = ( { } , { , } ) then e S P consists of two circles. Theproof that every strata S P is connected goes by induction on k . For k = 1, every stratumis diffeomorphic to some projective space and is therefore connected. In general theprojection π k − : F n,k → F n,k − preserves strata. For each consistent ( n, k )-tree P let P ′ be the ( n, k − P ′ := ( P , P , . . . , P k − ). Then π k − (S P ) = S P ′ and theprojection π k − : S P → S P ′ is a fibration with connected fibers. Knowing (by inductionhypothesis) that S P ′ is connected it follows that S P is also connected. Notice thateach fiber of π k − : S P → S P ′ is the submanifold of all k -dimensional vector subspaces V ∈ G m,k ( m = ∪ ki =1 P i )) that contain a given k − ⊔⊓ Corollary 2.
Let P be a consistent ( n, k ) -tree and ≤ i ≤ k . Then the node P i is fullin P iff dim S π i ( P ) = dim S π i − ( P ) . Given two ( n, k )-trees P = ( P , P , . . . , P k ) and Q = ( Q , Q , . . . , Q k ), we say that Q is contained in P , and write Q ≤ P , iff Q i ⊆ P i for every i = 1 , . . . , k . The relation ≤ is a partial order on the set of all ( n, k )-trees. The following monotony is obvious. Proposition 25.
Given ( n, k ) -trees P , Q , if P ≤ Q then S P ⊆ S Q . In particular, dim S P ≤ dim S Q . Consider two trees: a ( n, r )-tree P = ( P , . . . , P r ) and a ( n, s )-tree Q = ( Q , . . . , Q s ).We say they are disjoint iff supp( P ) ∩ supp( Q ) = ∅ . In this case we define their con-catenation to be ( n, r + s )-tree P ⋄ Q = ( P , . . . , P r , Q , . . . , Q s ).32 roposition 26. Given two disjoint trees P and Q , e S P ⋄ Q ≃ e S P × e S Q . Proof.
The diffeomorphism takes a matrix X ∈ e S P ⋄ Q to the pair of matrices([ X ] r , [ X ] s ) ∈ e S P × e S Q where [ X ] r is formed by the first r columns of X and [ X ] s is formed by the last s columns of X . ⊔⊓ Corollary 3.
For any tree P , e S P is diffeomorphic to the cartesian product of itsconnected component strata. In particular, dim S P is the sum of dimensions of theconnected component strata of P . Definition 4.
We say that a ( n, k ) -tree P is irreducible iff for each full node in P isa root of P . Given a ( n, k )-tree P , we denote by P ∗ = ( P ∗ , . . . , P ∗ k ) the tree obtained from P asfollows: if P is irreducible then we set P ∗ := P . Otherwise take the first i = 1 , . . . , k such that depth P ( i ) + 1 = card P ( i ) and P i is not a root of P . Define P ′ = ( P ′ , . . . , P ′ k )where P ′ j = P j when j ≤ i , and P ′ j = P j − P i for all other j . It is not difficult to checkthat P ′ is still a ( n, k )-tree with P ′ ≤ P . If it is irreducible we set P ∗ := P ′ . Otherwisewe repeatedly apply this procedure until we reach an irreducible tree, which we setto be P ∗ . We denote by K n,k the set of all consistent irreducible ( n, k )-trees. Then P P ∗ is a projection operator mapping consistent ( n, k )-trees onto the set K n,k . Proposition 27.
For any consistent ( n, k ) -tree P , S P = S P ∗ . Proof.
The inclusion S P ∗ ⊆ S P follows because P ∗ ≤ P . For the reverse inclusion itis enough proving that S P ⊆ S P ′ for each one step operation P P ′ in the reductionprocedure. Take X ∈ e S P . We shall use the notation introduced in the proof ofproposition 22. Let i = 1 , . . . , k be the first index such that depth P ( i ) + 1 = card P ( i ).Then E P i = C i ( P , X ) ⊕ h ( X ) i i and for every j > i , because X is orthogonal, we have( X ) j ∈ E P j − P i = E P ′ j . Therefore X ∈ e S P ′ . ⊔⊓ Therefore, restricting to irreducible trees we still get all ( n, k )-strata .
Lemma 7.
Given a consistent tree P ,1. P ) ≥ | P | , . P ) ≥ | P | + 1 when P has no full nodes in P . Proof.
There is an easy proof by induction in the length of P . ⊔⊓ Proposition 28. If P , Q ∈ K n,k , P ≤ Q and dim S P = dim S Q then P = Q . Proof.
Because strata are connected manifolds, it is clear that P ≤ Q and dim S P =dim S Q imply that S P = S Q . The proof goes by induction in k . It is trivial for k = 1. Assume it holds for k −
1, and take
P , Q ∈ K n,k , such that P ≤ Q anddim S P = dim S Q . Then π k − ( P ) ≤ π k − ( Q ) and π k − (S P ) = π k − (S P ). Thereforedim S π k − ( P ) = dim π k − (S P ) = dim π k − (S P ) = dim S π k − ( Q ) , and by induction hypothesis π k − ( P ) = π k − ( Q ). It remains to prove that P k = Q k .Let T ( i ) (1 ≤ i ≤ ℓ ) be the (distinct) connected component subtrees of P such thatsupp( T ( i ) ) ⊆ Q k − P k . Because their supports are disjoint, using lemma 7, we have0 = dim S Q − dim S P = Q k − P k ) − ℓ X j =1 (cid:12)(cid:12)(cid:12) T ( j ) (cid:12)(cid:12)(cid:12) ≥ ℓ X j =1 T ( j ) ) − (cid:12)(cid:12)(cid:12) T ( j ) (cid:12)(cid:12)(cid:12) ≥ ℓ ≥ . Whence ℓ = 0, i.e., there is no connected component subtree of P with support con-tained in Q k − P k . Therefore, Q k − P k ) = 0 and P k = Q k . ⊔⊓ There is a unique maximum tree in K n,k , which corresponds to take P i = { , . . . , n } for every i = 1 , . . . , k . We have S P = F n,k and e S P = O n,k for the maximum stratum.The dimension of S P is of course k (2 n − k − / F n,k .There are many minimal trees in ( K n,k , ≤ ). They are the trees where each P i is asingular set, P i = { n i } , i.e., P = ( { n } , { n } , . . . , { n k } ). The set { n , n , . . . , n k } mustbe formed by k -distinct elements from { , , . . . , n } . The minimal trees correspond tozero dimensional strata: S P is a one point set, while e S P is a set with 2 k points. Proposition 29.
Given trees
P , Q ∈ K n,k , then either S P ∩ S Q = ∅ or else the infimum P ∧ Q exists in ( K n,k , ≤ ) and S P ∩ S Q = S P ∧ Q . roof. Given two ( n, k )-trees P and Q define P ∩ Q := ( P ∩ Q , . . . , P k ∩ Q k ).This may not be a tree because P i ∩ Q i = ∅ for some i = 1 , . . . , k , but if all theseintersections are non-empty then it is straightforward checking that P ∩ Q is a tree.We claim that S P ∩ S Q = S P ∩ Q and e S P ∩ e S Q = e S P ∩ Q , which follow easily from thefact E P i ∩ E Q i = E P i ∩ Q i . Now, if S P ∩ S Q is non-empty then the ( n, k )-tree P ∩ Q isconsistent, and the tree P ∧ Q := ( P ∩ Q ) ∗ is both consistent and irreducible. This treeis the infimum of P and Q in the partially ordered set ( K n,k , ≤ ). By proposition 27,S P ∧ Q = S P ∩ Q = S P ∩ S Q . ⊔⊓ Proposition 30.
Given P ∈ K n,k with dim S P < dim F n,k there is P ′ ∈ K n,k such that P ≤ P ′ and dim S P ′ = 1 + dim S P . Proof.
The proof goes by induction in k . It is obvious for k = 1 since in this case thedimension of a stratum corresponds to the cardinal of the unique tree node. Assumethe statement of this proposition holds for k −
1, and let us prove it holds for k too.Along the proof we use the following definition: we say a tree P = ( P , . . . , P k ) is a fullchain when P = . . . = P k . We shall consider four exhausting cases: Case 1. supp( P ) $ { , , . . . , n } . In this case we take α ∈ { , , . . . , n } − supp( P )and define P ′ = ( P ′ , . . . , P ′ k ) setting P ′ j = (cid:26) P j for 1 ≤ j < kP i ∪ { α } for j = k . It is then clear that dim S P ′ = 1 + dim S P . Case 2. supp( P ) = { , , . . . , n } and all the tree connected components of P are fullchains. Notice that in this case P must have more than one connected components,because otherwise P would be the maximum ( n, k )-tree whose dimension equals thatof F n,k . Assume P i and P k ( i < k ) are two distinct roots of P . Notice that P i ∩ P k = ∅ .Choose any element α ∈ P k and define P ′ = ( P ′ , . . . , P ′ k ) setting P ′ j = P j if 1 ≤ j < iP i ∪ { α } if j = iP j if i < j ≤ k and P j = P k P i ∪ P k if i < j ≤ k and P j = P k . n i = P i and m = { i < j ≤ k : P j = P k } . Thendim S P ′ − dim S P = k X j =1 P ′ j ) − P j ) − κ ( P ′ ) + κ ( P )= 1 + m n i − { ( j, j ′ ) : j ≤ i < j ′ , P j = P i and P j ′ = P k } = 1 + m n i − m n i = 1 . The next two cases make use of the induction hypothesis.
Case 3. supp( P ) = { , , . . . , n } and the tree P is connected. In this case P k = { , , . . . , n } . Take Q = π k − ( P ). By induction hypothesis there is a ( n, k − Q ′ such that Q ≤ Q ′ and dim S Q ′ = 1 + dim S Q . Set P ′ = ( Q ′ , . . . , Q ′ k − , P k ). Then,because P ′ k = P k = { , , . . . , n } ,dim S P ′ − dim S P = dim S Q ′ − dim S Q = 1 . Case 4. supp( P ) = { , , . . . , n } , the tree P is disconnected, and at least one of itsconnected components is a not a full chain. We apply the induction hypothesis to theconnected component of the tree P which is not a full chain. Keeping all other treeconnected components of P unchanged we obtain a new tree P ′ . Using proposition 26we get that dim S P ′ = 1 + dim S P . ⊔⊓ We define S n,k := { S P : P ∈ K n,k } and e S n,k := { e S P : P ∈ K n,k } . Collectinginformation above we see that Proposition 31. S n,k is a stratification on F n,k , while e S n,k is a stratification on O n,k with possibly disconnected strata. In the rest of this section we discuss the symplectic case. Some notation is needed:Given a subset P ⊂ { , , . . . , n } , let P be the conjugate set P := { i = i + n (mod 2 n ) : i ∈ P } , P be the reduced set P := { i (mod n ) : i ∈ P } , and b P be the conjugationsaturated set b P := P ∪ P . Definition 5.
We call symplectic ( n, k )-tree to any sequence P = ( P , P , . . . , P k ) ofsubsets P i ⊂ { , , . . . , n } such that for every ≤ i < j ≤ k
1. either P i ∩ P j = ∅ or P i ⊆ P j , . either P i ∩ P j = ∅ or P i ⊆ P j . The full support of a symplectic ( n, k )-tree P is the set [ supp( P ) := \ ∪ kj =1 P j . Given asymplectic ( n, k )-tree P define S spP := S P ∩ F spn,k and e S spP := e S P ∩ O spn,k . We shall refer tothese as symplectic ( n, k )-strata. Consider the conjugate-depth function co-depth P : { , , . . . , k } → N defined byco-depth P ( i ) = { j < i : P j ⊆ P i } . Proposition 32.
Given a symplectic ( n, k ) -tree P , the following are equivalent:1. the symplectic ( n, k ) -stratum S spP is non-empty ,2. depth P ( i ) + co-depth P ( i ) + 1 ≤ card P ( i ) for every i = 1 , . . . , k . Proof.
The proof is analogous to that of proposition 22. Denote by C i ( P , X ) thelinear span of the columns of ( X ) j of X with j < i and P j ⊂ P i , and by C i ( P , X )the linear span of the columns of ( X ) j of X with j < i and P j ⊂ P i . Assume S spP isnon-empty and take X ∈ e S spP . The key point is that E P i must contain the subspace C i ( P , X ) ⊕ C i ( P , X ) ⊕ h ( X ) i i , whose dimension is depth P ( i ) + co-depth P ( i ) + 1. ⊔⊓ This motivates the definition
Definition 6.
We say that a symplectic ( n, k ) -tree P is symplectic consistent iffdepth P ( i ) + co-depth P ( i ) + 1 ≤ card P ( i ) for every i = 1 , . . . , k . When depth P ( i ) + co-depth P ( i ) + 1 = card P ( i ) we shall say that P i is a full isotropic node of P . We are now going to prove that symplectic strata are always closed manifolds.
Proposition 33.
Given a symplectic consistent ( n, k ) -tree P , the symplectic ( n, k ) -stratum S spP is a compact connected manifold (without boundary) of dimension dim( P ) = − k − κ ( P ) − µ ( P ) + k X i =1 card P ( i ) , where (10) κ ( P ) := { ( i, j ) : 1 ≤ i < j ≤ k, P i ⊆ P j } , and µ ( P ) := { ( i, j ) : 1 ≤ i < j ≤ k, P i ⊆ P j } . The stratum e S spP is also a compact manifold without boundary of the same dimension,which in general may be disconnected. roof. Consider the linear space b S n,k ( P ) defined in the proof of proposition 24, andthe sets Σ, Γ where Σ = { ( i, j ) : 1 ≤ i < j ≤ k, P i ⊆ P j } , Γ = { ( i, j ) : 1 ≤ i < j ≤ k, P i ⊆ P j } . Then define Φ : b S n,k ( P ) → R k × R Σ × R Γ byΦ( X ) = (cid:0) { k ( X ) i k − } ≤ i ≤ k , { ( X ) i · ( X ) j } ( i,j ) ∈ Σ , { J ( X ) i · ( X ) j } ( i,j ) ∈ Γ (cid:1) Notice that e S spP = b S n,k ( P ) ∩ O spn,k = Φ − (0 , X ∈ e S spP is a regular point of Φ, in the sense that the derivative D Φ X : b S n,k ( P ) → R k × R Σ × R Γ is surjective. Then e S spP being a regular level set of Φwill be a closed manifold of the claimed dimension. The derivative of Φ is explicitlygiven by D Φ X ( Y ) = ( { X ) i · ( Y ) i } ≤ i ≤ k , { ( X ) i · ( Y ) j + ( Y ) i · ( X ) j } ( i,j ) ∈ Σ , { J ( X ) i · ( Y ) j + J ( Y ) i · ( X ) j } ( i,j ) ∈ Γ (cid:1) . Given ( d, c, e ) = (cid:0) { d i } ≤ i ≤ k , { c i,j } ( i,j ) ∈ Σ , { e i,j } ( i,j ) ∈ Γ (cid:1) ∈ R k × R Σ × R Γ , we shallprove by induction that there is a matrix Y ∈ Mat n × k ( R ) such that D Φ X ( Y ) = ( d, c, e ).Fix X ∈ e S spP = Φ − (0 ,
0) and consider the following property in the index i : there arevectors Y , . . . , Y i in R n such that1. X ) j · Y j = d j , for all j = 1 , . . . , i ,2. ( X ) s · Y j + Y i · ( X ) j = d s,j , for all ( s, j ) ∈ Σ with j ≤ i ,3. J ( X ) s · Y j + J Y i · ( X ) j = e s,j , for all ( s, j ) ∈ Γ with j ≤ i ,4. Y i ∈ E P i , for all i = 1 , . . . , k . For i = 1 this is true: condition (2) and (3) are empty, and condition(1) is obviousbecause ( X ) = 0. Assume this property holds for i − Y , . . . , Y i − .To find Y i such that (1) and (2) above hold, we have to solve the following system ofequations in the unknown Y i ∈ E P i . X ) i · Y i = d i ( X ) j · Y i = c ′ j,i for every ( j, i ) ∈ Σ J ( X ) j · Y i = e ′ j,i for every ( j, i ) ∈ Γ , c ′ j,i = d j,i − ( X ) i · Y j and e ′ j,i = e j,i + J ( X ) i · Y j are already determined. Thissystem has a solution because the columns of X and J X are orthogonal. In particular,the columns ( X ) i , ( X ) j with ( j, i ) ∈ Σ and J ( X ) j with ( j, i ) ∈ Γ form an orthonormalsystem in E P i . Therefore property above holds for i , which completes the inductionand proves that D Φ X is surjective.The connectedness of strata S spP is proved as in proposition 24. ⊔⊓ Corollary 4.
Let P be a symplectic consistent ( n, k ) -tree and ≤ i ≤ k . Then thenode P i is full isotropic in P iff dim S spπ i ( P ) = dim S spπ i − ( P ) . Definition 7.
We say that a symplectic consistent ( n, k ) -tree P is symplectically irre-ducible iff each full isotropic node P i is a root of P . The partial order among symplectic ( n, k )-trees is the same Q ≤ P iff Q i ⊆ P i forevery i = 1 , . . . , k . Then the following monotony is obvious. Proposition 34.
Given symplectic ( n, k ) -trees P , Q , if P ≤ Q then S spP ⊆ S spQ . Inparticular, dim S spP ≤ dim S spQ . Concatenation of symplectic trees P ⋄ Q is defined when their full supports aredisjoint, i.e., [ supp( P ) ∩ [ supp( Q ) = ∅ . Proposition 35.
Given two symplectic trees P and Q with disjoint full supports, e S spP ⋄ Q ≃ e S spP × e S spQ . Corollary 5.
For any symplectic tree P whose roots satisfy b P i ∩ b P j = ∅ for every pairof distinct roots P i , P j , e S spP is diffeomorphic to the cartesian product of its connectedcomponent strata. In particular, dim S spP is the sum of dimensions of the connectedcomponent strata of P . Given a symplectic ( n, k )-tree P , we denote by P ∗ = ( P ∗ , . . . , P ∗ k ) the tree obtainedfrom P as follows: if P is symplectically irreducible then we set P ∗ := P . Otherwisetake the first i = 1 , . . . , k such that depth P ( i ) + co-depth P ( i ) + 1 = card P ( i ) and P i isnot a root of P . Define P ′ = ( P ′ , . . . , P ′ k ) setting P ′ = ( P ′ , . . . , P ′ k ) setting P ′ j = (cid:26) P j if 1 ≤ j ≤ iP j − b P i otherwise .
39t is not difficult to check that P ′ is still a symplectic ( n, k )-tree with P ′ ≤ P . Ifit is symplectically irreducible we set P ∗ := P ′ . Otherwise we repeatedly apply thisprocedure until we reach a symplectically irreducible tree, which we set to be P ∗ .We denote by K spn,k the set of all symplectic consistent and symplectically irreducible( n, k )-trees. Then P P ∗ is a projection operator mapping symplectic consistent( n, k )-trees onto the set K spn,k . Proposition 36.
For any symplectic consistent ( n, k ) -tree P , S spP = S spP ∗ . Proof.
The inclusion S spP ∗ ⊆ S spP follows because P ∗ ≤ P . For the reverse inclusion itis enough proving that S spP ⊆ S P ′ for each one step operation P P ′ in the symplecticreduction procedure. Take X ∈ e S spP . We shall use the notation introduced in theproof of proposition 32. Let i = 1 , . . . , k be the first index such that depth P ( i ) +co-depth P ( i ) + 1 = card P ( i ). Then C i ( P , X ) ⊕ C i ( P , X ) ⊕ h ( X ) i i = E P i and for every j such that j > i , because X is unitary, we have ( X ) j ∈ E P j − c P i = E P ′ j . Therefore X ∈ e S spP ′ . ⊔⊓ This means that restricting to symplectic irreducible trees we still get all symplectic( n, k )-strata .
Proposition 37. If P , Q ∈ K spn,k , P ≤ Q and dim S P = dim S Q then P = Q . Because this section is already long we omit the proof of this and of the followingpropositions, which are simple adaptations of the general case arguments.There is a unique maximum symplectic tree in K spn,k , which corresponds to take P i = { , . . . , n } for every i = 1 , . . . , k . We have S spP = F spn,k and e S spP = O spn,k for themaximum stratum. The dimension of S spP is k (2 n − k ).There are many minimal trees in ( K spn,k , ≤ ). They are the symplectic trees whereeach P i is a singular set, P i = { n i } , i.e., P = ( { n } , { n } , . . . , { n k } ). The set { n , n , n , n , . . . , n k , n k } must be formed by 2 k -distinct elements from { , , . . . , n } .The minimal symplectic trees correspond to zero dimensional strata: S spP is a one pointset, while e S spP is a set with 2 k points. Proposition 38.
Given trees
P , Q ∈ K spn,k , then either S spP ∩ S spQ = ∅ or else the infimum P ∧ Q exists in ( K spn,k , ≤ ) and S spP ∩ S spQ = S spP ∧ Q . roposition 39. Given P ∈ K spn,k with dim S spP < dim F spn,k there is P ′ ∈ K spn,k such that P ≤ P ′ and dim S spP ′ = 1 + dim S spP . We define e S n,k := { S spP : P ∈ K spn,k } and e S n,k := { e S spP : P ∈ K spn,k } . Collectinginformation above we see that Proposition 40. S spn,k is a stratification on F spn,k , while e S spn,k is a stratification on O spn,k with possibly disconnected strata.
13 Invariance of the Stratifications
The aim of this section is to establish the invariance of the previous section’ strati-fications under both the diffeomorphisms ϕ A and the gradient flows of the functions Q A,b . Proposition 41.
The stratification S n,k is invariant under ϕ A for any given A ∈ H n . Proof.
Given P ∈ K n,k it is enough seeing that e S P = ϕ A ( e S P ) in order to prove thatS P is invariant under ϕ A . Take X ∈ e S P . We need to prove that ( A ∗ X ) i ∈ E P i , for every i = 1 , . . . , k , which is done by induction. We shall write x . = y to express colinearity ofthe vectors x and y . For i = 1 we have ( A ∗ X ) . = A ( X ) ∈ E P because A E P = E P .Assume now that ( A ∗ X ) j ∈ E P j for every j < i , and let I = { j < i : P j ⊆ P i } .Using this assumption we get( A ∗ X ) i . = ( A X ) i − X j
41e consider the manifolds O n,k and O spn,k with Riemannian metrics induced fromthe Hilbert-Schmidt inner product on the respective matrix spaces, Mat n × k ( R ) andMat n × k ( R ). Then, using the covering mappings p : O n,k → F n,k and p : O spn,k → F spn,k ,we project these Riemannian structures onto the flag manifolds F n,k and F spn,k . Proposition 42.
The stratification S n,k is invariant under the gradient flow of thefunction Q A,b : F n,k → R for any A ∈ H n and b ∈ R k . Proof.
We have only to prove that that the gradient of Q A,b is tangent to F n,k , andwe shall do it on the covering space O n,k . It is enough to prove this proposition forthe function Q A : O n,k → R , because of (7) and the fact that the derivatives of the theprojections π i are self-adjoint operators (see the proof of proposition 17). The matrix P X = I − X X t ∈ Mat n × n ( R ) represents the orthogonal projection onto the orthogonalcomplement h X i ⊥ of the linear space spanned by the columns of X . Consider a ( n, k )-stratum P and let b S n,k ( P ) denote the linear space defined in proposition 24. Then e S P = b S n,k ( P ) ∩ O n,k and T X e S P = b S n,k ( P ) ∩ T X O n,k . By lemma 2, where we havecomputed the gradient of Q A , we only need to prove that X ∈ e S P ⇒ P X A X ∈ b S n,k ( P ) . Take X ∈ e S P so that ( X ) i ∈ E P i for every i = 1 , . . . , k . Because the linear spaces E i are eigen-directions of the symmetric matrix A it follows that A X ∈ b S n,k ( P ). Whenceit is enough to show that P X E P i ⊂ E P i , for every i = 1 , . . . , k . Let us say that a basis( w , . . . , w n ) of R n is adapted to P when each space E P i is spanned by the vectors w j (1 ≤ j ≤ n ) with w j ∈ E P i . By induction in k we can easily prove that the columns of X ∈ e S P can always be extended to form a basis ( w , . . . , w n ) adapted to P . Thereforefor every j = 1 , . . . , n , either P w j = 0 (when w j is a column of X ), or else P w j = w j ,which implies that P X E P i ⊂ E P i .The invariance in the symplectic case follows from the general invariance, becausethe submanifold F spn,k of F n,k is invariant under the gradient flow of Q A,b : F n,k → R and by definition of symplectic strata S spP = S P ∩ F spn,k . ⊔⊓
14 The Stratifications Skeleton Graphs
In this section we describe the structure of zero and one dimensional strata.42iven 1 ≤ k ≤ n we denote by S n,k the set of all permutations π = ( π , . . . , π k )with length k of the set { , , . . . , n } . S n,k is a set with n ! / ( n − k )! elements. When k = n we write S n instead of S n,n . This set has a group structure: it is the so called n th symmetric group. Given a permutation π ∈ S n,k we denote by π the ( n, k )-stratum π := ( { π } , { π } , . . . , { π k } ), we denote by V π the flag in F n,k defined by V π = ( E π , E π ⊕ E π , . . . , E π ⊕ · · · ⊕ E π k ) , and we denote by X π any matrix in O n,k whose i th -column belongs to E π i (1 ≤ i ≤ k ).There are 2 k such matrices, all them in the coset X π D k . We have p ( X π ) = V π ∈ S π . Proposition 43.
A stratum P ∈ K n,k has dimension zero iff there is a permutation π ∈ S n,k such that P = π . This shows that the correspondence π π is one-to-one from S n,k onto the set ofzero dimensional strata in K n,k .For the symplectic case we make the following definition: A permutation π ∈ S n,k iscalled isotropic iff the set { π i (mod n ) : i ∈ { , . . . , k } } has k distinct elements in Z n .We denote by S spn,k the set of all (2 n )(2 n − · · · (2 n − k + 2) isotropic permutationsof length k . We say that π ∈ S n is an isotropic permutation iff π i = π i for everyevery i = 1 , , . . . , n , where i = i + n (mod 2 n ). We denote by S spn the subgroup of allisotropic permutations in S n . This group has order 2 n n !. When k = n the set S spn,n canbe identified with the group S spn because each permutation of length n in S spn,n can beuniquely extended to an isotropic permutation in S spn . Given an isotropic permutation π ∈ S spn,k , the flag V π is isotropic, i.e., V π ∈ F spn,k , the matrix X π ∈ O n,k is unitary, i.e., X π ∈ O spn,k , and the same relation p ( X π ) = V π ∈ S spπ holds. Proposition 44.
A symplectic stratum P ∈ K spn,k has dimension zero iff there is anisotropic permutation π ∈ S spn,k such that P = π . Therefore π π is a one-to-one correspondence from S spn,k onto the set of zerodimensional strata in K spn,k .We now turn our attention to one-dimensional strata. Proposition 45.
A stratum P ∈ K n,k has dimension one iff either1. supp( P ) has k + 1 elements, all but one node has exactly one element, while theexceptional node has two elements, or else . supp( P ) has k elements, all but two nodes have exactly one element, while thetwo exceptional nodes share the same two elements.In any case S P is diffeomorphic to a circle and contains exactly two zero dimensionalstrata. Proof.
Assume dim S P = 1 and look at the dimensions dim S π i ( P ) (1 ≤ i ≤ k ).We see at once that there is exactly one index i for which P i is not a full node of P .We have dim S π j ( P ) = 1 for all j ≥ i and dim S π j ( P ) = 0 for all j < i . Recall thatfull nodes are roots because we have assumed that P ∈ K n,k . For j < i the node P j is root corresponding to a zero dimensional connected component subtree, whenceit has exactly one element. The node P i must have exactly two distinct elements,different from all elements in the previous nodes, because dim S π i ( P ) = 1. We have twopossibilities: either for some j > i , P j is a root with the same two elements in P i , andwe are in case 1., or else every P j with j > i is a root with exactly one element (distinctfro the previous ones), and we are in case 2..The stratum S P is diffeomorphic to S π i ( P ) through the projection π i , but this isclearly diffeomorphic to the (2 , Q = ( { , } ), and a simplecomputation shows S Q is the projective line, therefore a circle. The fact that there isonly room for two zero dimensional strata inside S P is obvious. ⊔⊓ We say that two distinct permutations π, π ′ ∈ S n,k are linked iff there is a one-dimensional stratum P ∈ K n,k such that P contains π and π ′ . With this linkingrelation S n,k becomes a graph that we refer as the skeleton graph of the stratification S n,k . We shall use the notation R( π ) = { π , . . . , π k } and C( π ) = { , . . . , n } − R( π ). Proposition 46.
Given two permutations π, π ′ ∈ S n,k , π and π ′ are linked iff either(1) there is some ( i, j ) ∈ { , . . . , k } × C( π ) such that π ′ is obtained from π replacing π i by j , or else(2) there is a pair ( i, j ) with ≤ i < j ≤ k such that π ′ is obtained from π switchingthe entries π i and π j of π . Proof.
Assume π, π ∈ S n,k are linked, and let P ∈ K n,k be the one-dimensional treewhose stratum contains both π and π ′ . Case 1. of proposition 45 implies case (1) here,while case 2. of proposition 45 implies case (2) here. The converse is also clear. ⊔⊓ Next we prove that the skeleton graph of the stratification S n,k is a regular graphwhose degree is the dimension of the flag manifold F n,k .44 roposition 47. S n,k is a regular graph of degree d = k (2 n − k − / . Proof.
By proposition 46, the degree of graph S n,k at π is the number of pairs ( i, j ) incase (2), which is k ( k − /
2, plus the number of pairs ( i, j ) in case (1), which is k ( n − k ).Therefore the degree of graph S n,k at π is d = k ( k − / k ( n − k ) = k (2 n − k − / ⊔⊓ With a straightforward adaptation of the argument used in the proof of proposi-tion 45 we can show that:
Proposition 48.
A symplectic stratum P ∈ K spn,k has dimension one iff either1. supp( P ) has k + 1 elements, all but one node P i has exactly one element andis a full isotropic node in P , while the exceptional node P i has two elements P i = { a, b } , or else2. supp( P ) has k elements, all but two nodes P i , P j ( i < j ) have exactly one elementand are full isotropic nodes in P , while the two exceptional nodes P i , P j share thesame two elements P i = P j and P j is a full isotropic node in P , or else3. supp( P ) has k + 2 elements, all but two nodes P i , P j ( i < j ) have exactly oneelement and are full isotropic nodes in P , while the two exceptional nodes P i , P j are such that P i ∩ P j = ∅ , P i = P j and P j is a full isotropic node in P .In any case S P is diffeomorphic to a circle and contains exactly two zero dimensionalsymplectic strata. We say that two distinct isotropic permutations π, π ′ ∈ S spn,k are linked iff there isa one-dimensional stratum P ∈ K spn,k such that P contains π and π ′ . With this linkingrelation S spn,k becomes a graph that we refer as the skeleton graph of the stratification S spn,k . As before we write R( π ) = { π , . . . , π k } and i = i + n (mod 2 n ). Proposition 49.
Given two permutations π, π ′ ∈ S spn,k , π and π ′ are linked iff one ofthe following cases occurs:(1) there is ≤ i ≤ k such that π ′ is obtained from π replacing π i by π i ;(2) there are ≤ i ≤ k and j / ∈ R( π ) ∪ R( π ) such that π ′ is obtained from π replacing π i by j .
3) there are ≤ i < j ≤ k such that π ′ is obtained from π switching π i with π j ;(4) there are ≤ i < j ≤ k such that π ′ is obtained from π replacing π i by π j and π j by π i ; Proof.
Assume π, π ∈ S spn,k are linked, and let P ∈ K spn,k be the one-dimensional treewhose symplectic stratum contains both π and π ′ . Case 1. of proposition 48 with P i = { a, b } and b = a implies case (1) here; case 1. of proposition 48 with P i = { a, b } and b = a implies case (2) here; case 2. of proposition 48 implies case (3) here, andfinally case 3. of proposition 48 implies case (4) here. ⊔⊓ Proposition 50. S spn,k is a regular graph of degree d = k (2 n − k ) . Proof.
By proposition 49, the degree of graph S spn,k at π is the sum of the numbers ofpairs ( i, j ) in each of the four cases (1)-(4) above. These numbers are: k ( k − / k in case (1), k ( k − / k (2 n − k ) in case (2). Thereforethe degree of graph S spn,k at π is d = k ( k − / k + k ( n − k ) = k (2 n − k ) = 2 nk − k . ⊔⊓
15 Oriented Skeleton Graphs
In the sequel we use the order of the eigendirections E = { E i } i of the group H n fixed inthe beginning of section 12. Given a positive definite matrix A ∈ H n , we say that E is A -ordered iff denoting by λ i ( A ) the eigenvalue of A associated with the eigenspace E i we have λ m ( A ) ≥ λ m − ( A ) ≥ . . . ≥ λ ( A ) ≥ λ ( A ), where m = n or m = 2 n (in thesymplectic case). We denote by H n ( E ) the subset of all matrices A ∈ H n such that E is A -ordered. In all statements of this and the following section where a matrix A ∈ H n plays a role we shall assume that A ∈ H n ( E ).Next we are going to introduce orientations on S n,k and S spn,k which will make themoriented graphs with no cycles. Definition 8.
Given two permutations π, π ′ ∈ S n,k , we say that π leads to π ′ , andwrite π π ′ , iff either
1) there is ( i, j ) ∈ { , . . . , k } × C( π ) with π i < j such that π ′ is obtained from π replacing π i by j , or else(2) there is ( i, j ) with ≤ i < j ≤ k , π i < π j and π ′ is obtained from π switching theentries π i and π j . Definition 9.
Given two permutations π, π ′ ∈ S n,k , we say that π precedes π ′ iffeither(1) there is ( i, j ) ∈ { , . . . , k }× C( π ) with j = π i +1 , and π ′ obtained from π replacing π i by j , or else(2) there is a pair ( i, j ) with ≤ i < j ≤ n , π i < π j , π ′ obtained from π switchingthe entries π i and π j , for which no s ∈ { , . . . , k } exists with π i < π s < π j . Comparing the definition of the ”leads-to” relation with proposition 46 we see thatgiven π, π ′ ∈ S n,k , π and π ′ are linked iff either π leads to π ′ or else π ′ leads to π . Inother words, the ”leads-to” relation orients each and every edge of S n,k , making it anoriented graph, with an oriented edge from π to π ′ whenever π π ′ . The ”precede”relation determines a subgraph of S n,k with an an oriented edge from π to π ′ whenever π precedes π ′ . Lemma 8.
Given π, π ′ ∈ S n,k , if π leads to π ′ then there is a sequence of permutations π (0) , π (1) , . . . , π ( m ) in S n,k such that1. π = π (0) and π ′ = π ( m ) ,2. π ( i ) precedes π ( i +1) , for i = 0 , , . . . , m − . This lemma proves that the transitive closures of relations ”leads-to” and ”precede”do coincide. We shall denote this transitive closure by ≺ , and write π (cid:22) π ′ iff π = π ′ or π ≺ π ′ .Given a permutation π ∈ S n,k , consider the set Σ n,k ( π ) of all pairs ( i, j ) satisfyingeither one of the following conditions:(1) 1 ≤ i ≤ k , j / ∈ C( π ) and π i > j ;(2) 1 ≤ i < j ≤ k and π i > π j , 47nd define the function H : S n,k → N by H ( π ) := n,k ( π ). Note that H ( π ) countsthe number of oriented edges of S n,k which arrive at the vertex π . Later we shall see(c.f. propositions 55 and 56) that H ( π ) is exactly the dimension of the stable manifoldat V π of the diffeomorphism ϕ A : F n,k → F n,k and of the gradient flow of the function Q A,b : F n,k → R , for any matrix A ∈ H n ( E ) with simple spectrum. Lemma 9.
Given π, π ′ ∈ S n,k , if π precedes π ′ then H ( π ′ ) = H ( π ) + 1 . Corollary 6.
Given π, π ′ ∈ S n,k , if π ≺ π ′ then H ( π ) < H ( π ′ ) . Corollary 7.
The oriented graph S n,k has no cycles. Therefore, (cid:22) is a partial orderon S n,k . Figure 2: The graphs S , and S , We now turn our attention to the symplectic case. Let R( π ) = { π , . . . , π k } , and b C( π ) := { , . . . , n } − (cid:16) R( π ) ∪ R( π ) (cid:17) . Consider on Z n = { , , . . . , n } the followingorder 1 ⊳ ⊳ · · · ⊳ n ⊳ n ⊳ n − ⊳ · · · ⊳ n + 1 . efinition 10. Given two permutations π, π ′ ∈ S spn,k , we say that π leads to π ′ , andwrite π π ′ , iff one of the following cases occurs:(1) there is ≤ i ≤ k such that π i ⊳ π i and π ′ is obtained from π replacing π i by π i ;(2) there are ≤ i ≤ k and j ∈ b C( π ) such that π i ⊳ j and π ′ is obtained from π replacing π i by j ;(3) there are ≤ i < j ≤ k such that π i ⊳ π j and π ′ is obtained from π switching π i with π j ;(4) there are ≤ i < j ≤ k such that π i ⊳ π j and π ′ is obtained from π replacing π i by π j and π j by π i . Definition 11.
Given two isotropic permutations π, π ′ ∈ S spn,k , we say that π precedes π ′ iff any one of the following situations occurs:(1) there is ≤ i ≤ k such that π i and π i are consecutive elements for ⊳ (whichimplies π i = n and π i = 2 n ), and π ′ is obtained from π replacing π i by π i ;(2) there are ≤ i ≤ k and j ∈ C( π ) such that π i and j are consecutive elements for ⊳ , and π ′ is obtained from π replacing π i by j ;(3) there are ≤ i < j ≤ k such that π i ⊳ π j , no s ∈ { , . . . , k } exists with π i ⊳ π s ⊳ π j ,and π ′ is obtained from π exchanging π i with π j ;(4) there are ≤ i < j ≤ k such that π i and π j are consecutive elements for ⊳ , and π ′ is obtained from π replacing π i by π j and π j by π i . Comparing the definition of the ”leads-to” relation with proposition 49 we see thatgiven π, π ′ ∈ S spn,k , π and π ′ are linked iff either π leads to π ′ or else π ′ leads to π . Inother words, the ”leads-to” relation orients each and every edge of S spn,k , making it anoriented graph, with an oriented edge from π to π ′ whenever π π ′ . The ”precede”relation determines a subgraph of S spn,k with an an oriented edge from π to π ′ whenever π precedes π ′ . Lemma 10.
Given π, π ′ ∈ S spn,k , if π leads to π ′ then there is a sequence of permutations π (0) , π (1) , . . . , π ( m ) in S n,k such that1. π = π (0) and π ′ = π ( m ) , . π ( i ) precedes π ( i +1) , for i = 0 , , . . . , m − . This lemma proves that the transitive closures of relations ”leads-to” and ”precede”do coincide. We shall denote this transitive closure by ≺ , and write π (cid:22) π ′ iff π = π ′ or π ≺ π ′ .Given a isotropic permutation π ∈ S spn,k , consider the set Σ n,k ( π ) of all pairs ( i, j )satisfying either one of the following conditions:(1) 1 ≤ i ≤ k , j = i + n and π i ⊲ π i + n ;(2) 1 ≤ i ≤ k , j / ∈ b C( π ) and π i ⊲ j ;(3) 1 ≤ i < j ≤ k and π i ⊲ π j ;(4) 1 ≤ i < j ≤ k and π i ⊲ π j + n ,and define the function H sp : S spn,k → N by H sp ( π ) := n,k ( π ). As in the generalcontext, H sp ( π ) counts the number of oriented edges of S spn,k which arrive at the vertex π . We shall see (c.f. propositions 57 and 58) that H sp ( π ) is exactly the dimension ofthe stable manifold at V π of the diffeomorphism ϕ A : F spn,k → F spn,k and of the gradientflow of the function Q A,b : F spn,k → R , for any matrix A ∈ H n ( E ) with simple spectrum. Lemma 11.
Given π, π ′ ∈ S spn,k , if π precedes π ′ then H sp ( π ′ ) = H sp ( π ) + 1 . Corollary 8.
Given π, π ′ ∈ S spn,k , if π ≺ π ′ then H sp ( π ) < H sp ( π ′ ) . Corollary 9.
The oriented graph S spn,k has no cycles. Therefore (cid:22) is a partial order on S spn,k . The partial ordered sets (S n,k , (cid:22) ) and (S spn,k , (cid:22) ) are not lattices for k >
1. Forinstance, π = (4 ,
1) and π ′ = (2 ,
3) do not have a greatest lower bound (g.l.b.), nor aleast upper bound (l.u.b.), neither in S , nor in S sp , . However, many subsets of S n,k and S spn,k have g.l.b. and l.u.b. Proposition 51.
Given a tree P ∈ K n,k , resp. an isotropic tree P ∈ K spn,k , the set ofall permutations π ∈ S n,k , resp. π ∈ S spn,k , such that π ≤ P has a g.l.b. and l.u.b. in (S n,k , (cid:22) ) , resp. (S spn,k , (cid:22) ) . sp , Proof.
The proof goes by induction in k . Let A P := { π ∈ S n,k : π ≤ P } . For k = 1 P = ( P ), and the permutations π = g . l . b . A P and π ′ = l . u . b . A P are simply definedby π := min P and π ′ := max P . In the symplectic case we must use the order ⊳ .Assume now that the statement of this proposition holds for k −
1, and take P ∈ K n,k .By induction hypothesis there exist π = g . l . b . A π k − ( P ) and π ′ = l . u . b . A π k − ( P ) . Wedefine π k := min P k − R( π ) and π ′ k := max P k − R( π ), and set π := ( π , . . . , π k − , π k ) and π ′ := ( π ′ , . . . , π ′ k − , π ′ k ). Then it is easily checked that π = g . l . b . A P and π ′ = l . u . b . A P .In the symplectic case we proceed as above but taking π k := min P k − (R( π ) ∪ R( π ))and π ′ k := max P k − (R( π ) ∪ R( π )). ⊔⊓ We shall use the following notation ∧ P = g . l . b . { π ∈ S n,k : π ≤ P } , ∨ P = l . u . b . { π ∈ S n,k : π ≤ P } . Proposition 52.
Given A ∈ H n ( E ) with simple spectrum, b ∈ R k + with b k > . . . > b > , and a tree P ∈ K n,k , resp. an isotropic tree P ∈ K spn,k , let π = ∧ P and π ′ = ∨ P .Then denoting by leb P the Lebesgue measure on S P ,(a) V π is the unique attractive fixed point for both ϕ A : F n,k → F n,k and the gradientflow of Q A,b : F n,k → R over the invariant stratum S P . Moreover, for both thesedynamical systems, V π is the ω -limit of leb P -almost every flag V ∈ S P ; b) V π ′ is the unique repeller fixed point for both ϕ A : F n,k → F n,k and the gradientflow of Q A,b : F n,k → R over the invariant stratum S P . Moreover, for both thesedynamical systems, V π ′ is the α -limit of leb P -almost every flag V ∈ S P .
16 Morse Functions
In this final section we prove that Q A,b : F n,k → R are Z -perfect Morse functions. Proposition 53 (Fixed Points) . Given A ∈ H n with simple spectrum and V ∈ F n,k ,resp. V ∈ F spn,k , V = ϕ A ( V ) iff V = V π for some permutation π ∈ S n,k , resp. isotropicpermutation π ∈ S spn,k . Proof.
Follows from lemma 4. ⊔⊓ In particular ϕ A : F n,k → F n,k has n ! / ( n − k )! fixed points, while ϕ A : F spn,k → F spn,k has (2 n )(2 n − · · · (2 n − k + 2) fixed points. Proposition 54 (Critical Points) . Given A ∈ H n with simple spectrum and V ∈ F n,k ,resp. V ∈ F spn,k , V is a critical point of Q A,b iff V = V π for some permutation π ∈ S n,k , resp. isotropic permutation π ∈ S spn,k . Proof.
Follows from lemma 18. ⊔⊓ In particular Q A,b : F n,k → R has n ! / ( n − k )! critical points, while Q A,b : F spn,k → R has (2 n )(2 n − · · · (2 n − k + 2) critical points.We use abstract lemmas to compute the eigenvalues at the fixed and critical points.Assume E ⊂ R n is an A -invariant 2-dimensional plane, and define S ( E ) = { x ∈ E : k x k = 1 } , and φ A : S ( E ) → S ( E ) by φ A ( x ) = A x/ k A x k . Denote by λ i ( i = 1 , A on E , and by v i ( i = 1 ,
2) the corresponding eigenvectors of A ,i.e., A v i = λ i v i , with v i ∈ E and k v i k = 1. Lemma 12.
If the eigenvalues are distinct, λ = λ , then φ A : S ( E ) → S ( E ) hasfour fixed points: ± v and ± v . The eigenvalues of φ A are: λ /λ at ± v , and λ /λ at ± v . q : S ( E ) → R , q ( x ) = c k A x k . Lemma 13.
If the eigenvalues are distinct, λ = λ , then q : S ( E ) → R has fourcritical points: ± v and ± v . The eigenvalues of the Hessian of q are: c (( λ ) − ( λ ) ) at ± v , and c (( λ ) − ( λ ) ) at ± v . Consider now the function q : O(2 , E ) → R , q ( X ) = k A X D c k , where O(2 , E )is the space of orthogonal linear maps X : R → E onto the 2-plane E , and D c = (cid:18) c c (cid:19) with c > c > Lemma 14.
If the eigenvalues are distinct, λ = λ , then q : O(2 , E ) → R has fourcritical points: ± v and ± v . The eigenvalues of the Hessian of q are: c (( λ ) − ( λ ) ) at ± v , and c (( λ ) − ( λ ) ) at ± v , where c = ( c ) − ( c ) > . Given two linked permutations π, π ′ ∈ S n,k there is a unique one-dimensionalstratum containing π and π ′ . We shall denote it by E ( π, π ′ ). By proposition 46in case (1) E i ( π, π ′ ) = { π i , j } and E s ( π, π ′ ) = { π s } for s = i , while in case (2) E i ( π, π ′ ) = E j ( π, π ′ ) = { π i , π j } and E s ( π, π ′ ) = { π s } for s / ∈ { i, j } . π and π ′ arethe unique zero dimensional strata contained in the circle stratum E ( π, π ′ ). These cor-respond to attractive and repelling fixed points on E ( π, π ′ ) for both ϕ A and the gradientflow of Q A,b . In the following propositions λ i will always denote the eigenvalue of thegiven matrix A ∈ H n associated with the eigendirection E i .Given a permutation π ∈ S n,k we call eigenvector basis at V π ∈ F n,k to any basisof T V π F n,k whose vectors are tangent to the one dimensional strata S E ( π,π ′ ) containing π = { V π } . A similar definition is given for every isotropic permutation π ∈ S spn,k . Proposition 55 (Jacobian at the Fixed Points) . Given π ∈ S n,k , the Jacobian matrix D ( ϕ A ) V π is a diagonal matrix w.r.t. any eigenvector basis at V π with the followingeigenvalues:1. λ j /λ π i for every pair ≤ i ≤ k , j ∈ { , . . . , n } − R( π ) ,2. λ π j /λ π i for every pair ≤ i < j ≤ k .The Jacobian D ( ϕ A ) V π has H ( π ) eigenvalues λ with λ > and d − H ( π ) eigenvalues λ with ≤ λ < , where d = k (2 n − k − / n,k . roof. We have to compute the eigenvalue of the Jacobian matrix D ( ϕ A ) V π along theeach eigen-direction tangent to an invariant one-dimensional stratum S E ( π,π ′ ) with π ′ linked to π . There are two cases according to proposition 46. Consider the A -invariant2-plane: E = E { π i ,j } in case (1), and E = E { π i ,π j } in case (2), and the projection p : e S E ( π,π ′ ) → S ( E ) defined by p ( X ) = ( X ) i in both cases. We can easily check that p ◦ ϕ A = φ A ◦ p . This shows that the eigenvalue of ϕ A at the fixed point X π (along e S E ( π,π ′ ) ) coincides with the eigenvalue of φ A at p ( X π ), and the claimed eigenvaluesfollow from lemma 12. ⊔⊓ Proposition 56 (Hessian at the Critical Points) . Given π ∈ S n,k , the Hessian matrix Hess( Q A,b ) V π is a diagonal matrix w.r.t. any eigenvector basis at V π with the followingeigenvalues:1. k − b i ( ( λ j ) − ( λ π i ) ) for every pair ≤ i ≤ k , j ∈ { , . . . , n } − R( π ) ,2. k − ( b i − b j ) ( ( λ π j ) − ( λ π i ) ) for every pair ≤ i < j ≤ k .The Hessian Hess( Q A,b ) V π has H ( π ) positive eigenvalues and d − H ( π ) negative eigen-values, where d = k (2 n − k − / n,k . Proof.
We have to compute the eigenvalue of the Hessian matrix Hess( Q A,b ) V π alongthe each eigen-direction tangent to an invariant one-dimensional stratum S E ( π,π ′ ) with π ′ linked to π . There are two cases according to proposition 46. Let E be the 2-planedefined in the proof of proposition 55. In case (2) consider the projection p : e S E ( π,π ′ ) → O(2 , E ) defined by p ( X )( u , u ) = ( X ) i u + ( X ) j u . We can easily check that for X over e S E ( π,π ′ ) Q A,b ( X ) = 1 k k X i =1 b i k ( A X ) i k = const + b i k k ( A X ) i k + b j k k ( A X ) j k = const + 12 k A p ( X ) D c k = const + ( q ◦ p )( X )where c i := 2 b i /k > c j := 2 b j /k >
0, and q ( Y ) = k A Y D c k , for Y ∈ O(2 , E ). Thisshows that the eigenvalue of Hess( Q A,b ) X along e S E ( π,π ′ ) coincides with the correspondingeigenvalue of Hess( q ) p ( X ) , and the claimed eigenvalues follow from lemma 14.54n case (1) consider the projection p : e S E ( π,π ′ ) → S ( E ) defined by p ( X ) = ( X ) i .We can check that for X over e S E ( π,π ′ ) Q A,b ( X ) = 1 k k X i =1 b i k ( A X ) i k = const + b i k k ( A X ) i k = const + c k A p ( X ) k = const + ( q ◦ p )( X )where c := 2 b i /k >
0, and q ( y ) = c k A y k , for y ∈ S ( E ). This shows that theeigenvalue of Hess( Q A,b ) X along e S E ( π,π ′ ) coincides with the corresponding eigenvalue ofHess( q ) p ( X ) , and the claimed eigenvalues follow from lemma 13. ⊔⊓ In the symplectic case, given linked isotropic permutations π, π ′ ∈ S spn,k we havefour cases by proposition 49. In case (1) E i ( π, π ′ ) = { π i , π i } and E s ( π, π ′ ) = { π s } for s = i . In case (2) E i ( π, π ′ ) = { π i , j } and E s ( π, π ′ ) = { π s } for s = i . In case (3) E i ( π, π ′ ) = E j ( π, π ′ ) = { π i , π j } and E s ( π, π ′ ) = { π s } for s / ∈ { i, j } . Finally, in case (4) E i ( π, π ′ ) = { π i , π j } , E j ( π, π ′ ) = { π i , π j } and E s ( π, π ′ ) = { π s } for s / ∈ { i, j } . Proposition 57 (Symplectic Jacobian at the Fixed Points) . Given π ∈ S spn,k , the Ja-cobian matrix D ( ϕ A ) V π of ϕ A : F spn,k → F spn,k is a diagonal matrix w.r.t. any eigenvectorbasis at V π with the following eigenvalues:1. λ π i /λ π i for every ≤ i ≤ k ,2. λ j /λ π i for every pair ≤ i ≤ k , j ∈ { , . . . , n } − b R( π ) ,3. λ π j /λ π i for every pair ≤ i < j ≤ k ,4. λ π j /λ π i for every pair ≤ i < j ≤ k .The Jacobian D ( ϕ A ) V π has H sp ( π ) eigenvalues λ with λ > and d − H sp ( π ) eigen-values λ with ≤ λ < , where d = k (2 n − k ) = dim O spn,k . roposition 58 (Symplectic Hessian at the Critical Points) . Given π ∈ S spn,k , theHessian matrix Hess( Q A,b ) V π of Q A,b : F spn,k → R is a diagonal matrix w.r.t. anyeigenvector basis at V π with the following eigenvalues:1. k − b i ( ( λ π i ) − ( λ π i ) ) for every ≤ i ≤ k ,2. k − b i ( ( λ j ) − ( λ π i ) ) for every pair ≤ i ≤ k , j ∈ { , . . . , n } − b R( π ) ,3. k − ( b i − b j ) ( ( λ π j ) − ( λ π i ) ) for every pair ≤ i < j ≤ k ,4. k − ( b i − b j ) ( ( λ π j ) − ( λ π i ) ) for every pair ≤ i < j ≤ k .The Hessian Hess( Q A,b ) V π has H sp ( π ) positive eigenvalues and d − H sp ( π ) negativeeigenvalues, where d = k (2 n − k ) = dim O spn,k . Proposition 59.
Given A ∈ H n ( E ) with simple spectrum and b = ( b , . . . , b k ) ∈ R k with b > b > . . . > b k > , the functions Q A,b : F n,k → R and Q A,b : F spn,k → R areMorse functions. Proof.
By proposition 18 the critical points of Q A,b are exactly the points V π with π ∈ S n,k , or π ∈ S spn,k . Then propositions 56 and 58 show that every critical point isnon-degenerate. Therefore, Q A,b is a Morse function. ⊔⊓ Proposition 60.
The Poincar´e polynomial of F n,k with coefficients in Z is P n,k ( t ) = (1 − t n ) (1 − t n − ) · · · (1 − t n − k +1 )(1 − t ) k . (11) The Poincar´e polynomial of F spn,k with coefficients in Z is P spn,k ( t ) = (1 − t n ) (1 − t n − ) · · · (1 − t n − k +2 )(1 − t ) k . (12) Proof.
See [B] or [MN]. ⊔⊓ We define the following coefficients: Given 1 ≤ k ≤ n , 0 ≤ s ≤ d = k (2 n − k − / p n,ks := { ( s , . . . , s k ) : s + · · · + s k = s and 0 ≤ s i ≤ n − i, ∀ i } . (13)56nalogously, given 1 ≤ k ≤ n , 0 ≤ s ≤ d = k (2 n − k ), sp p n,ks := { ( s , . . . , s k ) : s + · · · + s k = s and 0 ≤ s i ≤ n − i, ∀ i } . (14) Lemma 15.
The Poincar´e polynomials given in (11) and (12) are respectively equalto P n,k ( t ) = k (2 n − k − / X i =0 p n,ki t i and P spn,k ( t ) = k (2 n − k ) X i =0 sp p n,ki t i . Proof.
Follows easily from the relations P n,k ( t ) = k Y i =1 (1 − t n − i +1 )(1 − t ) = k Y i =1 (1 + t + t + · · · + t n − i ) P spn,k ( t ) = k Y i =1 (1 − t n − i +2 )(1 − t ) = k Y i =1 (1 + t + t + · · · + t n − i +1 ) ⊔⊓ Proposition 61.
The Morse polynomial of Q A,b : F n,k → R is equal to P n,k ( t ) . Inparticular, this function is Z -perfect. Proof.
We claim that for every 1 ≤ k ≤ n and 0 ≤ i ≤ d = k (2 n − k − / p n,ki = { π ∈ S n,k : H ( π ) = i } . Thus, by proposition 56 and lemma 15, P n,k ( t ) is the Morse polynomial of Q A,b . Letus now prove the claim. For 1 ≤ k ≤ n , letΓ n,k = { ( s , . . . , s k ) : 0 ≤ s i ≤ n − i, ∀ ≤ i ≤ k } . We shall define a mapping φ : Γ n,k → S n,k such that H ◦ φ ( s , . . . , s k ) = s + · · · + s k ,for every ( s , . . . , s k ) ∈ Γ n,k , and prove that φ is one-to-one. Given π ∈ S n,k writeC i ( π ) := { , . . . , n } − { π , . . . , π i − } . Define ψ : S n,k → Γ n,k by ψ ( π ) := ( s , . . . , s k ),where s i := { j ∈ C i ( π ) : π i > j } . By definition we have 0 ≤ s i ≤ n − i , sothat ψ ( π ) = ( s , . . . , s k ) ∈ Γ n,k . Conversely, π = φ ( s , . . . , s k ) is recursively defined asfollows: π i is taken to be the ( s i + 1) th -element of C i ( π ). In this way, there are exactly s i indices j = i + 1 , . . . , n such that j ∈ C i ( π ) and π i > j , and this shows that φ is theinverse of ψ . ⊔⊓ roposition 62. The Morse polynomial of Q A,b : F spn,k → R is equal to P spn,k ( t ) . Inparticular, this function is Z -perfect. Proof.
We claim that for every 1 ≤ k ≤ n and 0 ≤ i ≤ d = k (2 n − k ), sp p n,ki = { π ∈ S n,k : H sp ( π ) = i } . Thus, by proposition 58 and lemma 15, P n,k ( t ) is the Morse polynomial of Q A,b . Letus now prove the claim. For 1 ≤ k ≤ n , letΓ spn,k = { ( s , . . . , s k ) : 0 ≤ s i ≤ n − i, ∀ ≤ i ≤ k } . We shall define a mapping φ : Γ spn,k → S spn,k such that H sp ◦ φ ( s , . . . , s k ) = s + · · · + s k , forevery ( s , . . . , s k ) ∈ Γ spn,k , and prove that φ is one-to-one. Given π ∈ S spn,k write b C i ( π ) := { , . . . , n }−{ π , . . . , π i − , π , . . . , π i − } . Define ψ : S spn,k → Γ spn,k by ψ ( π ) := ( s , . . . , s k ),where s i := { j ∈ b C i ( π ) : π i > j } . By definition we have 0 ≤ s i ≤ n − i , so that ψ ( π ) = ( s , . . . , s k ) ∈ Γ spn,k . Conversely, π = φ ( s , . . . , s k ) is recursively defined asfollows: π i is taken to be the ( s i + 1) th -element of b C i ( π ). In this way, there are exactly s i indices j = i + 1 , . . . , n such that j ∈ b C i ( π ) and π i > j , and this shows that φ is theinverse of ψ . ⊔⊓ References [B] Borel, Armand
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