Bruhat Order in the Full Symmetric $\mathfrak{sl}_n$ Toda Lattice on Partial Flag Space

In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric $\mathfrak{sl}_n$ Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when $t\to\pm\infty$ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations.


Introduction
The present paper is devoted to the study of the Full Symmetric sl n Toda lattice which can be considered as a straightforward generalization of the non-periodic Toda lattice. Let us briefly remind that non-periodic Toda lattice (Toda chain) is the dynamical system of n particles on a straight line with interactions between neighbours. This system was first considered in [1,2]; in paper [3] there were found n functionally independent integrals of the motion. The involution of the integrals was proved in the papers [4,5].
One can show that the Hamilton's equations are equivalent to the following matrix equation: where B is ... a n−2 0 −a n−1 0 0 ... a n−1 0 The equation (2) is the compatibility condition of the system: where Ψ ∈ SO(n, R) and Λ is the eigenvalue matrix of the Lax operator.
One can show that non-periodic Toda lattice can be treated as a dynamical system on the orbits of the coadjoint action of the Borel subgroup B + n of SL(n, R) (equal to the group of upper triangular matrices with determinant equal to 1) see [6,7,8,9]. It is also possible to give an alternative description of the phase space of the dynamical system, if we identify the algebra sl n with its dual using Killing form on sl n . In this way one obtains generalizations of the classical Toda lattice (tri-diagonal Toda chain). Namely, use the following identifications: sl n = so n ⊕ b + n , sl * n = (b + n ) * ⊕ (so n ) * ∼ = Symm n ⊕ n + n , (b + n ) * ∼ = (so n ) ⊥ = Symm n , (so n ) * ∼ = (b + n ) ⊥ = n + n , where b + n is the algebra of upper triangular matrix and n + n is the algebra of strictly upper triangular matrix. As one sees, these identifications map the space of symmetric matrices into the dual space of Lie algebra of Borel subgroup: Symm n ∼ = (b + n ) * and hence we can introduce a symplectic structure on Symm n pulling it back from (b + n ) * ; it is by restriction of this pullback, that one obtains the symplectic structure on the tridiagonal symmetric matrices, used in Toda system.
Based on this approach one can get further generalizations of the non-periodic Toda lattice: just by plugging in other Cartan pairs in this construction. In particular, so one obtains the Full Symmetric sl n Toda lattice (elsewhere FS Toda lattice). In contrast to the Lax matrix of non-periodic Toda lattice the Lax matrix of FS Toda lattice is not a tri-diagonal symmetric matrix but the full symmetric matrix: The decomposition of the Lax operator has the following form: It turns out, that FS Toda lattice is also integrable (refer to [11,6,13,12,17] for details). Moser in [15] showed that at t → −∞ the Lax operator of the usual tri-diagonal Toda lattice converges to the diagonal matrix with eigenvalues put in the increasing order, and when t → +∞ it converges to the diagonal matrix with decreasing eigenvalues. This property has been further studied in [10], [13], [16], [19], [20]. The present paper deals with the FS Toda lattice in arbitrary dimension n. It is a natural continuation of the previous one, [19], in which we study the behavior of the Lax operator L at t → ±∞; this question is again in the center of our attention here. The difference is that in paper [19] we treated Lax matrices L with n distinct eigenvalues and now we allow the eigenvalues of L coincide. In the previous case the Bruhat order on the symmetric group S n played a crucial rôle. On the other hand, as one knows there exists a Bruhat order on the permutations of multisets, induced in a natural way from the Bruhat order on the symmetric group (one can regard this phenomenon as a combinatoric manifestation of the fact that the Schubert cells in the full flag space project into Schubert cells in partial flag manifolds). So it is natural to ask, if this (induced) order has something to do with the limit behavior of the Lax matrix with coinciding eigenvalues.
It turns out that this is precisely the case. More specifically, we show that one can regard the restriction of FS Toda lattice on the space of symmetric matrices with non-distinct eigenvalues as a gradient system on a partial flag space so that the set of singular points of the gradient vector field is naturally identified with the permutations of multisets (see section 3), and we show, that the gradient system at hand verifies the same properties as before. That is we prove that when t → ±∞ the trajectories of the FS Toda lattice converge to a finite set of singular points in the partial flag space indexed by the Schubert cells, so that any two points of this set are connected by a trajectory iff the corresponding cells are adjoint (see theorem 4.1).
The paper is organized as follows: in Section 2 we give a list of facts from the geometry of partial flag spaces, describe the Bruhat order and introduce Schubert cells; all this is used in the rest of the paper. In Section 3 we consider the FS Toda lattice and describe how it induces a gradient flow on partial flag manifolds, so that some elementary facts from the Morse theory are applicable. Finally, in section 4 we give two simple explicit examples that illustrate our theorem and prove the main results of this paper (see section 4.3).

Notation and assumptions
In what follows, unless otherwise stated, all manifolds will be assumed smooth and compact (without boundary), vector spaces are assumed to be real and finite-dimensional.
We shall also consider the full symmetric Toda system in a generic dimension n, so we set L denote the real symmetric n × n Lax matrix of the system, and Λ the diagonal matrix of eigenvalues of L. As L is symmetric its eigenvalues are real, so we assume that they are ordered naturally in Λ. We shall use the notation O(n, R) (respectively SO(n, R)) for the group of n-dimensional orthogonal matrices (respectively, the group of orthogonal matrices with positive determinant), and so(n) will denote its Lie algebra, the space of real antisymmetric n × n-matrices. Similarly, SL(n, R) will denote the n-dimensional special linear group over real numbers, and B + n ⊂ SL(n, R) (respectively B − n ) the Borel group of upper (respectively lower) triangular matrices with unit determinant.

Bruhat order and Schubert cells
The notion of Bruhat order, Schubert cells and their generalizations have long been crucial instruments in the research of geometry of Lie groups and homogenous spaces. In what follows we give a brief introduction to the subject. In most part of this section we draw on the exposition from classical books, see [14] and references therein.

Flag spaces, Grassmanians and their generalizations
First we recall some definitions: . . , i k ) be a set of positive integers, i 1 + i 2 + · · · + i k = n. Then the real partial flag space F l i 1 ,i 2 ,...,i k (R) is the set of all sequences of vector subspaces Its topology is induced from the angles between various spaces (or from the identification with quotient spaces, see below).
Let us make few remarks before we proceed. First of all, similar definitions work literally for vector spaces over arbitrary field k, giving us the flag spaces F l i 1 ,...,i k (k). Second, an important particular case of this construction is k = n, i 1 = i 2 = · · · = i n = 1, which is called the full flag manifold and denoted F l n (R) (or in a general case F l n (k)). Another important particular case of the flag space is the case k = 1, i.e. when there is only one subspace chosen insideR n . In this case the flag manifold is called the Grassman space or just Grassmannian and is denoted by Gr d,n (R), where d = i 1 is the dimension of the subspaces we choose.
An important property of flag spaces is that they are homogeneous spaces, being homeomorphic to the quotient spaces of the groups of linear transformations. The isomorphism is induced by the choice of compatible bases in the sub spaces V i . In particular, F l n (R) = SL(n, R)/B + n , Gr d,n (R) = SL(n, R)/P, for a parabolic subgroup P ⊂ SL(n, R). Similarly, general partial flag spaces are isomorphic to the quotient spaces of SL(n, R) by a suitable parabolic subgroup. On the other hand, choosing orthogonal bases of the corresponding subspaces so that the orientation on R n would match with the given one, we obtain homeomorphisms with the quotients of special orthogonal group. For instance, Here we have denoted by T n the group of diagonal matrices with eigenavalues equal to ±1, and T + n the intersection SO(n, R) ∩ T n . In both cases we see, that the groups SL(n, R) and SO(n, R) act transitively on the flag spaces.
The important rôle of the flag spaces in our investigation follows from the next observation: Proposition 2.1. The space of real symmetric n × n matrices with a fixed set of eigenvalues λ 1 , . . . , λ k with multiplicities i 1 , . . . , i k can be identified with the partial flag manifold F l i 1 ,...,i k .
Before we prove this in full generality, let us consider the simplest case: n = 3. Let α < β be real numbers. Then the set of all symmetric 3 × 3 matrices with eigenvalues α, α, β is equal to the orbit of the diagonal matrix diag(α, α, β) under the conjugations by the elements of SO(3, R). This action is not free, the stabilizer of diag(α, α, β) being equal to the subgroup SO(2, R) ⊂ SO(3, R) of matrices Ψ that have one of the following forms: So the orbit is equal to the quotient space of the sphere SO(3, R)/SO(2, R) = S 2 modulo the antipodal action of Z/2Z; to see this observe that the action is induced from the conjugation by the matrix In fact the subgroup SO(2, R) ⊂ SO(3, R) is generated by A and the subgroup of rotations around the Oz axis. Since the homeomorphism SO(3, R)/SO(2, R) = S 2 is given by the image of the point (0, 0, 1) under the action of SO(3, R), we see that the action of A on S 2 is by the antipodal map. So the quotient space that we need is More generally assume that the diagonal matrix Λ has k < n coinciding eigenvalues, and all the other eigenvalues of Λ are distinct. Without the loss of generality we can think that λ 1 = λ 2 = · · · = λ k . Then reasoning just as before, we can identify the set of symmetric matrices with such eigenvalues can with the quotient space of SO(n, R) modulo the subgroup SO(k, R) generated by SO(k, R) (orthogonal transformations of the subspace R k ⊂ R n spanned by the first k-axes) and the subgroup of diagonal matrices in SO(n, R). This subgroup is equal to the intersection of SO(n, R) and the cartesian product n−k times and the quotient space SO(n, R)/ SO(k, R) is equal to the partial flag space F l k,1,...,1 (with n − k units): Finally consider the most general case. Assume that the eigenvalues of Λ are divided into several "clusters": λ 1 = · · · = λ i 1 , λ i 1 +1 = · · · = λ i 1 +i 2 , . . . , λ i 1 +···+i k−1 = · · · = λ i 1 +···+i k , where i 1 + i 2 + · · · + i k = n. Then the orbit ΨΛΨ −1 will be equal to the partial flag space F l i 1 ,...,i k : where dim V l = i 1 + i 2 + · · · + i l . This can be proved either by the considerations of the symmetry (i.e. by finding the subgroup of matrices commuting with Λ) as before or one can use the following observation: every symmetric matrix L ∈ Symm n with eigenvalues λ 1 , . . . , λ k of multiplicities i 1 , . . . , i k is uniquely determined by the collection of eigenspaces: Clearly, dimL j = i j . So one naturally identifies such matrix with a point in F l i 1 ,...,i k by putting An important property of the partial flag spaces is the existence of surjective projections: given by "forgetting" the unnecessary subspaces. Alternatively these projections can be regarded as the additional factorization: It follows from this description that π is a locally trivial fibre bundle with the fibre equal to ).
Or using the notation we introduced earlier:
This permutation can be abbreviated to (ω(1), . . . , ω(n)). One defines the length of the permutation ω as the total number of involutions in the sequence (ω(1), . . . , ω(n)), that is: Let the numbers r ω [p, q] be equal to the "number of involutions with respect to p, q": There are many equivalent definitions of the Bruhat order on permutations; we give the following (see [18]): Definition 2.2. The Bruhat order on S n is the partial order, determined by the following relation: for any two permutations u and v in S n , one says that u precedes v (u ≺ v) iff A simple way to compute r ω [p, q] is to consider the matrix A ω representing the permutation ω: It is clear that for all p, q the number r ω [p, q] is equal to the rank of the submatrix A pq ω = ((A ω ) ij ) i≤q, j≤p, of A ω (informally one can say that A pq ω is the submatrix in the upper left corner of A ω determined by the element a pq ). Since there is only one non-zero entry in every row and column of A ω , it is enough to count the number of such nonzero elements in A pq ω . E.g., for ω = (4213) we obtain r ω [3, 2] = 2 from the following table (we replace 1 by •): The following lemma is proved in [14]: The relation between this order and the geometry of the full flag space is determined by the structure of Schubert cell decomposition of this space. There are many ways to define Schubert cells, we shall use the following one (more definitions can be found in the literature, see [14,18,22] and c.f. 2.3). Definition 2.3. Embed the group S n into SL(n, R) as it is explained above (one only should replace 1 in the definition of A ω to ±1 so as to make sure, that the determinant is equal to 1 and not −1). Using the projection p : SL(n, R) → F l n (R) = SL(n, R)/B + n , we obtain points p(A ω ) ∈ F l n (R), which we shall denote by [A ω ]. The subgroup B + n of SL(n, R) acts on F l n (R) and the the Schubert cell X ω ⊂ F l n (R) for ω ∈ S n is the orbit of [A ω ] with respect to this action: One can show (c.f. [14] and section2.3) that X ω is indeed a cell. In effect, The closure X ω of X ω in the flag space is a singular algebraic variety. It is called the Schubert variety. The following statement explains the relation of Schubert cells and the Bruhat order (see [14]): 2. An element w ∈ S n precedes v ∈ S n with respect to Bruhat order, w ≺ v, iff This proposition is sometimes used to give an alternative definition of Bruhat order: Bruhat order is the partial order, induced by the contiguity of the cells in the Schubert cell decomposition of flag manifold.
There is another important observation relating Schubert cells and Bruhat order. Namely, consider the B − n -orbits of the same points [A ω ]: They are called the dual Bruhat cells. These sets are as well homeomorphic to Euclidean spaces, their closures are called the dual Schubert varieties. Then the following is true (see again [14]): In the latter case, the intersection of cells is transversal.
This property of the Schubert cells was crucial in our description of asymptotic behavior of the FS Toda lattice, see [19]. Below we shall make use of analogous properties of the partial flag spaces.

Bruhat order on multiset permutations
In this section we give a description of Schubert cell decomposition of partial flag space and the Bruhat order, associated with it. Since we were not able to find a combinatoric description of this order in literature, we do not confine ourselves here to mere definitions and statement of results, and give proofs of few facts (in particular see proposition 2.4). Besides this one can refer to [22] and the book [18] for more details.
Let n > k be two natural numbers and let I = (i 1 , i 2 , . . . , i k ) be a partition of n into k parts, i.e. for each j = 1, . . . , k, we let i j be a positive integer so that Then we give the following definition: Definition 2.4. An I-permutation of multiset (or permutation with multiplicities, or permutation with repetitions) is any epimorphic map such, that |τ −1 (j)| = i j for all j = 1, . . . , k. We shall denote the set of all such permutations by S I n .
One can regard an element τ ∈ S I n as a string of elements of the form where 1 ≤ τ (j) ≤ k and every element j, 1 ≤ j ≤ k appears exactly i j times (which explains the name of these objects that we use).
Below we shall make an extensive use of the following map τ * 1 : S n → S I n : take the fixed multiset permutation τ 1 ∈ S I n , Then for every permutation w ∈ S n we define the multiset permutation by the formula Or in "linear form": The map τ * 1 is evidently an epimorphism and For all other elements τ ∈ S I n we see that (τ * 1 ) −1 (τ ) is a right coset of S n by the subgroup S i 1 ×S i 2 ×· · ·×S i k , i.e. the set of all permutations of the form x τ ·w for any w ∈ S i 1 ×S i 2 ×· · ·×S i k and some x τ ∈ S n such that τ * 1 (x τ ) = τ . We shall sometimes call these cosets clusters in S n , corresponding to τ ∈ S I n ; clearly every such cluster contains N = i 1 !i 2 !...i k ! elements. One can describe elements in S I n in the terms of clusters; for instance the string (1, 2, 2, 1) ∈ S (2,2) 4 corresponds to the cluster {(1, 3, 4, 2), (2, 3, 4, 1), (1, 4, 3, 2), (2, 4, 3, 1)}. Now one use these constructions to introduce Bruhat order on S I n . Loosely speaking it is obtained by "pulling back" from S n . However this procedure is not always well defined, so we give some details.
First of all, just as in section 2.2 we introduce the notion of length of a permutation with repeating elements: let τ {1, 2, . . . , n} → {1, 2, . . . , k} be an element in S I n , then i.e. the number of involutions in the sequence (τ (1), . . . , τ (n)). Observe, that we do not count the pairs j 1 < j 2 for which τ (j 1 ) = τ (j 2 ). Second, similarly to the definition 2.2 we define the Bruhat order on S I n with the help of ranks of some matrices: consider the n × k matrices A τ corresponding to the elements of S I n . Just as before we define the numbers r τ [p, q] for all 1 ≤ p ≤ n and 1 ≤ q ≤ k. Then: for all p, q. Observe that we use ≤ rather the strict inequality < here. Of course at least one inequality must be strict, if τ = υ The following example illustrates this definition very well: let I = (2, 2) as before and τ = (2, 1, 1, 2). Then A τ has this shape: Similarly, lengths of the elements of S (2,2) 4 are equal to These data are enough to draw the Hasse diagram of the Bruhat order in this case (c.f. figure  2).
The following propositions (propsitions 2.4 and 2.5, and lemma 2.2) explain, in what sense one can "pull back" the order from S n to S I n .
Proof. Let us denote τ * 1 (u) = U and τ * 1 (v) = V ; we will show that U ≺ V . First of all we observe that it is enough to show this in a particular case v = u · (i, j), l(v) = l(u) + 1, (here (i, j) denotes the transposition of i and j for i < j). Indeed, it follows directly from lemma 2.1 that every pair of Bruhat-comparable elements in S n can be connected by a "path" of intermediate elements verifying this condition at every stage.
So let u and v be as above.
Recall now, that we assumed that v = u(i, j), i < j and l(v) = l(u) + 1. This means, in particular, that u(i) < u(j) (otherwise the number of inversions would decrease from multiplication by (i, j)).
Counting the number of inversions in all three cases, we see that only the first two options are possible (in the third case the number of inversions will change at least by 3, when we swap i and j). In terms of the rectangular matrices associated with the elements of S n we can say, that the rectangle P abij cut from A u by the a-th and b-th rows and i-th and j-th columns contains no nonzero elements, see (14).
Clearly, no nonzero elements will appear in this rectangle when we swap i and j. When we pass from u and v to U and V , this matrix "shrinks" vertically and the interior of the corresponding rectangle P τ 1 (a)τ 1 (b)ij in the matrices A U and A V does not contain nonzero elements: new non-zero elements can appear only in the top and the bottom rows of the rectangle (i.e. in the rows number τ 1 (a) and τ 1 (b)) and this can happen only simultaneously for U and V . Hence, the matrices A U and A V differ only by positions of 1's in the corners of P kak b ij : in A U they stand at the entries (k a , i) and (k b , j), while in A V they stand at (k a , j) and (k b , i). Now a simple inspection shows that r U [p, q] < r V [p, q] for all p and q, see (15).
The following statement is similar to lemma 2.1; it gives a procedure by which we can move between comparable elements of S I n .
The proof is by direct inspection of the definitions and we omit it. Finally, we use this procedure to move "in the opposite direction": from S I n to S n . Proposition 2.5. Every coset (τ * 1 ) −1 (υ) ⊂ S n , υ ∈ S I n contains a unique minimal element u ∈ S n (with respect to Bruhat order in S n ) so that, if ξ ≺ ζ in S I n , then similar inequality holds for the corresponding minimal elements in S n : x ≺ z.
Recall, that minimal element in a partial order is the element, which is less than all other; in particular, it is comparable with all others. It is clear that it is unique, if it exists.
Proof. Observe, that every coset (τ * 1 ) −1 (υ) contains a unique element s υ , such that s −1 υ is a I-shuffle, i.e. a permutation which preserves the order of the elements inside the "blocks" of the partition I. We shall call such s υ an I-deshuffle; clearly, this deshuffle is the minimal element that we need. This follows easily from the fact that for any permutation u ∈ (τ * 1 ) −1 (υ), and any i < k < j, such that υ(i) = υ(j) = υ(k), we shall have either u(k) < u(i), u(k) < u(j), or u(i) < u(k), u(j) < u(k) (i.e. u(k) cannot lie between u(i) and u(j)). Hence, the order in (τ * 1 ) −1 (υ) (induced from S n ) depends only on the permutation of the elements inside separate blocks of the partition, and not on υ. In particular, the shuffle corresponds to the "unit" with respect to the permutations inside the blocks, hence it is minimal.
Consider now arbitrary U, V ∈ S I n and let s U , s V be the corresponding deshuffles. By lemma 2.2 we can assume that V is obtained from U by swapping two elements chosen as it is explained in its condition: V = U (l, m). Then as one can see , and the fact that it coincides with s V follows from the choice of l and m, see lemma 2.2. Similarly, from the same choice it follows that all the elements in the string corresponding to s U , that lie between l and m, are either greater than both s U (l) and s U (m), or less than both. Hence, Observe, that we have in fact established, that Bruhat order on S n , when restricted to clusters (τ * 1 ) −1 (υ), υ ∈ S I n , coincides with the lexicographic order on the direct product S i 1 ×S i 2 ×· · ·×S i k where each factor is equipped with the usual Bruhat order on S l . In particular, there is a unique maximal element in it too.

Schubert cells in partial flags and Bruhat order
The definition of Schubert cells given in section 2.2, can be extended to the more general situation of partial flag spaces, in particular, to Grassmannians, so that the canonical projection (8) maps Schubert cells in F l n (R) to the cells in the partial flags. We shall use this observation and the properties of Bruhat order to describe some of the structure of the Schubert cells, their duals and their intersections in partial flag spaces. Main references for this section are [21], [22] and [14].
We begin with the general definition of the Schubert cells in an important particular case: k = 1, i.e. the flag space is equal to a Grassmannian Gr d,n (R), the space of d-planes in R n .
Let {e 1 , ..., e n } be a basis of R n . Any d-dimensional hyperplane inside R n is determined by a collection of d linearly-independent vectors in R n , which is the same (when the base of R n is fixed) as a d × n, d < n matrix with maximal rank. It is clear, that two matrices Λ, Λ ′ determine the same d-plane, iff there exists g ∈ GL(d, R) such that: We shall use the same symbol Λ to denote the d-plane, corresponding to the matrix Λ. Observe that the natural GL(n, R)-action on the Grassmannian translates into the right multiplication of Λ by the matrices B ∈ GL(n, R). Also observe that in our notation the left and right actions exchange their roles: in section 2.3 we define Gr d,n as the quotient of SL(n, R) by the right action of a parabolic subgroup so that GL(n, R) acts on it from the left, and when we draw rectangular matrices, the action and factorization change directions. This can be amended by the use of n × d matrices instead of d × n matrices. Let for any multi-index A = (a 1 , . . . , a d ), 1 ≤ a 1 < a 2 < · · · < a d ≤ n symbol V A denote the subspace in R n , spanned by {e a 1 , e a 2 , . . . , e a d }. Then Definition 2.6. The Schubert cell X A is the B + n -orbit of the point V A in Grassmannian Gr d,n (R).
One can give a more geometric definition of the Schubert cells. To this end consider the subspaces V i , i = 1, . . . , n, V i ⊆ R n , spanned for each i by the vectors {e 1 , ..., e i } and consider for an arbitrary d-plane Λ ∈ Gr d,n (R) the intersections: Then for the same multi-index A = (a 1 , . . . , a d ) we have: Here 1 stand in the intersections of the i-th rows and the (n − a d−i+1 + 1)-th columns, and are the only nonzero elements of the column; asterisks are used to denote arbitrary real numbers. This flip of indices is caused by the fact that we have substitute the left action of B + n for the right.
As one can see, the sets X A are indeed cells: it follows from the shape of the matrices we use here that Here dn is the dimension of the space of d × n matrices and we subtract the number of fixed elements in a matrix. One can also introduce the "length" of the sequence A by putting : Then it is easy to see that l(A) = dn − (n + i − a i ), so It turns out, that A X A is a cell decomposition of Gr d,n (R). Moreover, one can show (see [22]) that a cell X B is adjacent to X A , iff b k ≤ a k for all k. In this case we shall write B ≤ A; this is a partial order on the set of all sequences.
It is easy to see, that the Bruhat order on S  4) and (3, 4). The corresponding Schubert cells are spanned by the matrices of the following shapes (we use the equality signs to denote this): In the general case of an arbitrary partial flag space F l i 1 ,...,i k (R) (which we shall often abbreviate to just F l I (R)) we can define the Schubert cells decomposition similarly by choosing a base {e 1 , ..., e n } of R n ; then for any partial flag of vector subspaces in R n and any multiset permutation ω ∈ S I n (where I is a k-partition of n) we say that E • is in the cell X ω , iff for all 1 ≤ p ≤ n, 1 ≤ q ≤ k. It is easy to see that these sets are indeed cells, that the cells, defined for different ω ∈ S I n , do not intersect and that F l I (R) is equal to their disjoint union. Closures of these cells will be denoted X ω ; they are called the Schuber varieties in F l I (R). One can describe them by replacing the equalities in (20) by the constrict inequality ≥; they are indeed singular algebraic subvarieties in F l I (R).
As before the same sets can be described as orbits of the group of upper-triangular matrices B + n ; if we view the partial flag space as the quotient space of SL(n, R) (or SO(n, R)) by some parabolic subgroup P , we can define the Schubert cells as the orbits of certain elements within F l I (R) with respect to the action of B + n (or some block-diagonal matrices, if we speak about SO(n, R)): take the matrix A w , corresponding to the minimal representative w in S n of an element ω ∈ S I n (see section 2.3), then X ω = B + n (A w )P/P ⊂ SL(n; R)/P.
It clearly follows from this definition that Schubert cells in partial flag spaces are equal to the projections of Schubert cells in the full flag space under the projection π, see (8). Moreover, one can show (see [22]) that they are in effect homeomorphic to the cell of the minimal element w in F l n (R). Now by comparing this description with the results of the previous section (see propositions 2.4 and 2.5 and lemma 2.2), we obtain the following statement: Proposition 2.6. For any two permutations with repetitions ψ, ω ∈ S I n , we have ψ ≺ ω in Bruhat order, iff the corresponding cells are adjacent: X ψ ⊂ X ω .
Finally, similarly to the case of the full flag space F l n (R) in F l I (R) one can also define dual Schubert cells X ∧ ω , ω ∈ S I n (and the corresponding Schubert varieties). This can be done either by a modification of the rank (in)equalities (20), or as orbits of the corresponding elements in F l I (R) with respect to the action of B − n , or finally simply as projections of the dual cells in F l n (R). As before, a quick comparison of the definitions proves the following proposition: Proposition 2.7. A Schubert cell X ψ ⊂ F l I (R) intersects with the dual Schubert cell X ∧ ω ⊂ F l I (R), off ψ ≺ ω in Bruhat order in S I n , and in the latter case the intersection is transversal.
The proof follows from the observation that X ∧ ω corresponds to the cell, determined by the maximal element w ′ in the cosets (τ * 1 ) −1 (ω) just as X ω corresponds to the minimal one: i.e. X ∧ ω is a homeomorphic image of the dual cell, corresponding to w ′ in F l n (R) under the projection π. Since the statement of the proposition 2.7 holds in F l n (R), it should hold in F l I (R) as well (we should also use the fact that π is submersion).

Fibre bundles and Bott-Morse functions
In our study of the Toda system on symmetric matrices with coinciding eigenvalues we shall need few statements from the differential geometry of homogeneous spaces of Lie groups. We prove them here for the sake of completeness. Let M be a smooth closed manifold with a smooth right transitive action of a compact Lie groupG, g ∈G, x ∈ M, x → x g . Let G ⊂G be a compact subgroup ofG We assume that π : M → M/G is a locally trivial bundle (in which the fibre containing x is homeomorphic to the stabilizer of x, G x ⊂ G); in particular, we assume that X = M/G is a smooth manifold. Let g be a G-invariant Riemannian metric on M and f be a G-invariant smooth function. In this case the gradient vector field ξ = grad g f is G-invariant as well. Since g and f are G-invariant they induce a Riemannian structure g X and a smooth function f X on X; similarly, the vector field ξ induces a field ξ X on X; we just put ξ X (π(m)) to be equal to the projection dπ m (ξ(m)) for arbitrary point m in M .
We shall need the following statement: Proposition 3.1. The field ξ X is equal to the gradient of the function f X with respect to the metric g X : Proof. The statement follows almost immediately from the definitions of a gradient vector field and induced metric g X on M/G: recall that the field grad g X f X is characterized by the equality g X (η X , grad g X f X ) = η X (f X ); and that g X (η X , ζ X ) for two vectors η X , ζ X ∈ T x X (here X = M/G) is defined as g X (η X , ζ X ) = g(η, ζ), for any η, ζ ∈ T m M (m ∈ π −1 (x) is arbitrary point), such that dπ(η) = η X , dπ(ζ) = ζ X and they both are perpendicular to the "vertical" subspace T m (mG) of T m M . Now since the function f is constant along the fibres of π, the field grad g f is orthogonal to the vertical subspaces; so where the second equality follows from the definition of grad g f , and the third one from the definition of f X (it is enough to consider the local structure of cartesian product in M ).
Below we shall need to know that f X is a Morse function. Of course, they cannot follow directly from the fact that f is Morse. Moreover, f cannot be a Morse function unless the group G is discrete (and finite): because of the G-invariance of f every singular point of f X in X = M/G will correspond to a G-orbit of singular points of f in M . So in order to fill this gap we give a simple criterion for f X to be Morse: Proof. First, recall that the gradient of a function is always perpendicular to its level sets. So in the case we consider it is perpendicular to the G-orbits. It follows that it is not in the kernel of the projection dπ : T m M → T π(m) M/G. Hence the singular points of f X correspond to the "singular orbits" of f in M , i.e. to the G-orbits consisting of the singular points of f .
Second, Hessian of f X at a singular point x 0 ∈ X is the symmetric quadratic form on tangent space given by the formula where ξ, η are two vector fields in a neighborhood of x 0 : it is easy to show, that the right hand side of this formula is indeed symmetric 2-form on vectors in T x 0 X if x 0 is singular (in particular, it depends only on the values of ξ and η at the point x 0 ). Now the proposition follows from the fact that every vector field ξ on M/G can be locally lifted to a G-invariant vector fieldξ in a neighborhood of π −1 (x 0 ) ⊂ M , orthogonal to the fibre π −1 (x 0 ): to see this it is enough to use the local trivialization π −1 (U ) = U × G x 0 for some open neighborhood U of x 0 . So we have the following formula wherex 0 is a point in the fibre π −1 (x 0 ); the right hand side of this formula is G-equivariant, so the statement of the proposition follows.
As a matter of fact, the condition of proposition 3.2 is a variation of the Morse-Bott condition.
Recall that a function f ∈ C ∞ (X) (X is a smooth manifold) is called Morse-Bott function, if its singular set is a smooth (not connected) submanifold in X and the Hessian is nondegenerate in the normal direction to this sub manifold. In our case the singular set is equal to the orbit through singular points of f , hence it is smooth (by assumption on the nature of the action) and normal direction is the direction orthogonal to vertical vectors.

The FS Toda lattice on partial flags
As we have explained in the introduction, the FS Toda lattice in dimension n induces a gradient flow on the orthogonal group SO(n, R), which we here shall call the Toda flow,or just Toda system on SO(n, R). This flow is determined by the vector field where Λ is the diagonal matrix of eigenvalues of the symmetric Lax matrix L; in addition we identify the tangent space T Ψ SO(n, R) with the right translation of the Lie algebra so n (R) by Ψ. In fact, this vector field is equal to the gradient of a function with respect to an invariant metric on SO(n, R): for a fixed eigenvalues matrix Λ one can take The SO(n, R)-invariant Riemannian structure is determined by its values on so n , where it is given by the formula for any antisymmetric matrices A and B and a linear isomorphism J : so n → so n . Then see [19] and [23] for details. As a matter of fact, this property does not depend on whether the eigenvalues are all distinct, or not. Since all the objects here are T + n -invariant (see section 2.1) the same formulas (21), (22) and (23) induce a vector field, a function and a Riemannian structure on the full flag space F l n (R) = SO(n; R)/T + n .
It follows from the discussion of section 3.1 that the field M on F l n (R) is equal to the gradient of the corresponding function. It can be shown (see [19] and [23]) that the function F is Morse function both on SO(n, R) and on the flag space F l n (R).
In a similar way, one can use the propositions of section 3.1 to construct gradient flows on the partial flag spaces. Suppose, there are coinciding eigenvalues of Λ; for definiteness we can assume that they are where i 1 + i 2 + · · · + i k = n; that is we assume there are k < n distinct eigenvalues of Λ, and the multiplicity of the j-th eigenvalue is i j . In this case the vector field M (Ψ) on SO(n; R) is invariant with respect to O(i 1 , R) × · · · × O(i k , R), i.e. R)). So we obtain a vector field M on the partial flag space. In effect all the conditions of proposition 3.1 hold (i.e. the function F and the Riemannian structure , J are also invariant with respect tot the group action) hence the field M is equal to the gradient of the function F with respect to the induced Riemannian structure. Now we are going to show, that the function F on F l i,...,i k (R) is Morse. To this end (see proposition 3.2) we should show that the restriction of the Hessian of F on the directions normal to the "vertical" vectors is nondegenerate. This Hessian of F is given by the formula (39) in [19]: let s w ∈ SO(n; R) be the positive permutation matrix (i.e. the matrix in SO(n, R), which permutes the vectors ±e i , i = 1, . . . , n so that the permutation w ∈ S n emerges, when the signs are dropped) then s w is a singular point in SO(n; R) and the Hessian of F at s w is Here θ ij denote the local coordinates on SO(n, R) at the point s w ∈ SO(n, R), obtained from the standard coordinates on so n by right translation. Put Recall that we assume the eigenvalues of Λ to be partitioned into k "blocks", see (24), so that everything is G-invariant. In this case the "vertical" directions at s w are equal to the tangent space of the corresponding orbit The Lie algebra g of G is equal to the space of all antisymmetric matrices, spanned by the following set of elementary antisymmetric matrices: where we have put i 0 = 1 and use symbols e ij to denote the matrix units. Then the vertical directions at s w are equal to the linear span of (e w(i)w(j) − e w(j)w(i) )s w ∈ T sw SO(n; R) for the same set of indices i, j, as above. This follows either from the direct computations, or from the fact that the conjugate action Ad sw of s w on so n amounts to the permutation of indices. Now the non degeneracy of d 2 sw on the linear complement of T v sw SO(n, R) ⊂ T sw SO(n, R) can be seen from the comparison of the formula (25) with this description of the vertical subspace. And because of the G-invariance of the constructions, the same is true for an arbitrary point in the orbit through s w .
We conclude this section by an important observation: consider the natural projection (see section 2.1) π : F l n (R) → F l i 1 ,...,i k (R).
As we have shown above, when the eigenvalues of Λ are given by (24) the spaces on both sides of this diagram can be equipped with a Morse-Bott function and Riemannian metric, which give rise to the gradient vector fields M on them. In both cases, these structures are pulled to the flag spaces (full or partial) from the group SO(n, R). On the other hand we can use this projection π to pull the structures we need from F l n (R) to the partial flag space. It is clear that in this way we shall obtain the same result. In other words, when the eigenvalues of Λ are not distinct, the function F on SO(n, R) is a Morse-Bott function, which induces a Morse function on the base F l i 1 ,...,i k (R).

The asymptotic behavior
In this section we prove the main theorem of the paper: the asymptotic behavior of the trajectories of vector field, induced on F l i 1 ,...,i k (R) by the FS Toda lattice on the set of symmetric matrices with multiple eigenvalues (with multiplicities i 1 , . . . , i k ) is completely determined by the Bruhat order on S I n , see theorem 4.1. We begin with two particular cases in which the structure of the trajectories is easy to perceive: the case of real projective spaces and the case of Grassmannian Gr 2,4 (R). The general case is treated in the end.

Example 1: projective spaces
Let L be the Lax matrix of the FS Toda lattice, Λ is the diagonal matrix of its eigenvalues so that L = ΨΛΨ −1 , Ψ ∈ SO(n, R).
As we have explained this relation allows one introduce an analogue of FS Toda lattice on the partial flag space F l i 1 ,...,i k (R), where i 1 , i 2 , . . . , i k are the multiplicities of the eigenvalues of L. The geometry of the flag space (and hence the structure of Toda trajectories) depends in a great measure on the partition I = (i 1 , i 2 , . . . , i k ). In this section we consider the simplest possible case; namely, we assume that I = (1, n−1), i.e. we assume that there are only two different eigenvalues λ < µ with multiplicities 1 and n − 1 respectively (we also can assume that λ + (n − 1)µ = 0). The flag spaces here are equal to the projective spaces RP n−1 . We begin with the smallest possible dimension n = 3 and eigenvalues λ 1 = λ < λ 2 = λ 3 = µ. In this case we consider the system on projective plane RP 2 ; one can see that there are exactly 3 singular points of the vector field, induced by the Toda system on RP 2 , corresponding to the permutations 0 = (λ, µ, µ), 1 = (µ, λ, µ) and 2 = (µ, µ, λ); direct calculations in the local coordinates at these points (it is enough to transfer the coordinates from the unit of SO(3, R) and take the directions, complementary to the vertical) show, that Morse function takes in these points distinct values and their Morse indices are 2, 1 and 0 respectively (see formula (25)). Recall that index of a singular point of a Morse function is equal to the dimension of the submanifold, spanned by the trajectories exiting this point. Taking this into consideration, we see that there is a unique way one can connect 0, 1 and 2 by trajectories: It is easy to describe the structure of the vector field here: its pullback from RP 2 to S 2 has 6 singular points, which can be identified with the intersections of the sphere with the coordinate axes; indices of these points are 0, 1 and 2 so that the opposite points have same index. Besides this the vector field is invariant with respect to the natural involution of the sphere (exchange of the opposite points). We shall call this vector field Toda field on RP 2 , and use the same name for the similar fields on arbitrary projective spaces.
In the generic case n > 3 we shall have an analogous picture: just remark, that the diagonal embedding of SO(n, R) into SO(n + 1, R), given by corresponds on one hand to the hyperplane embedding of projective spaces: RP n−1 → RP n , and on the other hand it induces the evident embedding of symmetric n × n matrices into the space of symmetric (n + 1) × (n + 1) matrices: This embedding sends the symmetric matrices with spectrum λ < µ = µ = · · · = µ n−1 times to symmetric matrices with spectrum Also one sees that this inclusion intertwines the Toda systems on bigger matrices and smaller matrices. Thus the Toda vector field on RP n−1 coincides with the restriction of Toda field from RP n , and the latter has one more singular point on the complement of RP n−1 in RP n ; this new point has maximal possible index. So we see that in the case of a generic projective space RP n the phase diagram of singular points and trajectories between them is as follows: It is also possible to rephrase the results of this section in terms of the asymptotic behavior of the symmetric Lax matrix L: as one sees, when t → ±∞ it tends to a diagonal matrix with eigenvalues λ and µ of multiplicities 1 and n − 1 respectively, in which λ stands on an arbitrary position (depending on the corresponding multiset permutation.

Example 2: Gr 2,4 (R)
The next simplest case is, when there are only two distinct eigenvalues of L, but the dimensions of the corresponding eigenspaces are greater, than 1. In this case the phase space of the system can be identified (see section 2.1) with the Grassmann space of all dimension d > 1 hyperplanes in an n > d + 1 dimensional Eucledian space R n . This case is already quite complicated, so we restrict our discussion to the least-dimensional case: n = 4, d = 2; we assume that the eigenvalues of the 4 × 4 symmetric Lax matrix L are λ 1 = λ 2 = λ, λ 3 = λ 4 = µ and consider the induced gradient system on Gr 2,4 (R).
We begin with a detailed description of the Grassmannian. This is the manifold, parametrizing all 2-dimensional subspaces in R 4 . As we mentioned in section 2.1 one has a homeomorphism of spaces The identification is given by choosing an orthogonal basis in R 4 : Then we can regard the 2-dimensional plane, spanned by e 1 , e 2 as the "origin" x of Gr 2,4 (R) so that all the other planes in R 4 are equal to translations of x by appropriate elements of SO(4, R).
The stabilizer of x in SO(4, R) is the subgroup B x of SO(4, R), comprised of the elements of the form:  (27) As one easily sees, the plane x is preserved by the action of these elements: Now consider the generic point y ∈ Gr 2,4 (R): as we know y = g y x for some g y ∈ SO(4, R). Then the stabilizer of y is equal to the conjugation of B x by g y : This simple observation allows one find a suitable description of the tangent space of Gr 2,4 (R) at y: T y Gr 2,4 (R) ∼ = so(4)/Ad g y (g), where g is the Lie algebra of the group B x and Ad g , g ∈ SO(4, R) denotes the adjoint action of the group SO(4, R) on its Lie algebra so(4) (by conjugation of matrices). We choose the standard base in so(4) (the space of all 4 × 4 anti-symmetric matrices) so that the generic element X of so(4) takes the form The functions θ 1 , . . . , θ 6 are coordinates in so(4). The exponential map allows one to pull these coordinates to an open neighbourhood of the unit matrix in SO(4, R), and the right translations by the group elements then give coordinate systems in open neighbourhoods of points in SO(4, R) and on tangent spaces at these points. Below we shall use these coordinates extensively without explanation, preserving (by a slight abuse of language) their original names.
This notation is well-suited for the description of the tangent spaces of Gr 2,4 (R) at the singular points of the FS Toda lattice: recall that these singular points are equal to the projections to Gr 2,4 (R) of the singular fibres of the FS Toda lattice in SO(4, R) (or in F l 4 (R)). These singular fibres are determined by the singular points of the FS Toda lattice on the matrices with distinct eigenvalues, which they contain. As one knows, the singular points in SO(4, R) of FS Toda lattice corresponding to the Lax matrices with distinct eigenvalues are given by the matrices, which permute the basis vectors and their opposites (see [19]). Conjugation by such matrices induces permutations of the coordinates θ 1 , . . . , θ 6 .
With the help of these observations one can draw the following table, describing the tangent spaces at the singular points, values of the Morse function F 2,4 , positive and the negative directions of corresponding Hessians at these singular points of Toda system on Gr 2,4 (R). We index the singular points on Gr 2,4 (R) by the corresponding permutations of the eigenvalues' set (λ, λ, µ, µ), λ < µ.

Point
Morse index Value of F 2,4 + − Minors (µ, µ, λ, λ) 4 5λ + µ 0 θ 2 , θ 3 , θ 4 , θ 5 ψ 11 , ψ 12 , ψ 43 , ψ 44 (29) In the fourth and the fifth columns we use the coordinates θ 1 , . . . , θ 6 on SO(4, R) introduced earlier and project them to Gr 2,4 (R): on the level of tangent spaces projection π is a linear epimorphism and we identify T π (x)Gr 2,4 (R) with a linear subspace, which maps isomorphically onto it. The last column gives the list of certain invariant surfaces of the Toda field in SO(4, R), which contain the corresponding "singular fibre" of the Toda field (i.e. the fibre of the projection π : SO(4, R) → Gr 2,4 (R), on which Toda field vanishes identically); as one knows (see [19]) such surfaces can be given by equations ψ ij = 0; there are more general surfaces of this sort, in which the equations are given by the condition that certain minor of the matrix Ψ = (ψ ij ) vanish (hence the name "minor surfaces" that we use in this and previous paper) but we restrict our attention to the simplest set of invariants listed in this table.
We illustrate these constructions by figure 1, in which the case of the singular point in Gr 2 (4, R), corresponding to the permutation (λ, µ, λ, µ), is considered (below we shall often identify such points with the corresponding permutations of eigenvalues): In this figure the little circle in the bottom represents the point in Gr 2,4 (R) that corresponds to the permutation (λ, µλ, µ) of the eigenvalues. Small squares above correspond to the permutation matrices in SO(4, R) that project into the chosen point. Arrows represent the coordinate directions in the tangent space of the group at the chosen points. The red arrows correspond to the directions inside the fibre, and black arrows to all the rest. As one knows (see the paper [19]) the coordinates θ i are (up to infinitesimal correction terms) the canonical coordinates, in which the Morse function takes the form of the difference of sums of squares. So the arrows are directed towards, or from the points, depending on whether the corresponding tangent directions are in positive or negative subspaces of the Hessian (red arrows in fact correspond to the directions in which the Hessian has kernel, but we preserve them for the sake of simplicity; their directions are determined under the assumption that all the eigenvalues are distinct and are ordered in a natural way, otherwise one can take arbitrary vectors in the fibre direction).
Finally we consider the minor surfaces, that pass through the fibre: it is clear, that if a surface is invariant under Toda flow on SO(4, R), its projection to the Grassmann space is an invariant set of the generalized flow. Using this simple observation we come up with the following diagram, see figure 2, representing the flows in Gr 2,4 (R); dotted lines represent the 1-parameter families of trajectories that connect points, whose Morse indices differ by 2; there are also 2-parameter families between points with Morse indices, differing by 3 and one 3-parametric family between the lowest and the highest points, which we have omitted to make the diagram more readable.
As one can see, this diagram coincides with the Hasse diagram of Bruhat order on multiset permutations. Below (see theorem 4.1), we shall show that it is always the case.
Let us give a brief explanation of how the diagram has been obtained: consider two singular points in Gr 2,4 (R) which correspond to the permutations (λ, λ, µ, µ) and (λ, µ, λ, µ) of the eigenvalues. Let us show, that there is a finite number of trajectories of the Toda flow between these points.
Observe that the surface is invariant with respect to the O(2, R) × O(2, R)-action and also is preserved by the Toda flow.
Thus it projects to an invariant surface Σ in Gr 2,4 (R), which contains only the above mentioned singular points on the base. Of course, Σ and Σ are a singular varieties in SO(4, R) and Gr 2,4 (R), so for our purposes it is enough to compute their dimensions in generic point. A generic matrix Ψ from Σ has the following form: Moreover, Ψ being an orthogonal matrix, the dimension of the surface Σ in a generic point is equal to 3: it is enough to compute the dimension of the tangent space to it at the unit matrix; the tangent space of SO(4, R) at the unit consists of antisymmetric matrices, so summing the conditions on the matrix elements we come to the above mentioned conclusion about the dimension of Σ. Thus the dimension Σ (in a generic point) is equal to 1. On the other hand Σ is invariant with respect to the Toda field on Gr 2,4 (R), so being 1-dimensional means it should consist of a finite number of trajectories connecting singular points in it. Thus the points (λ, λ, µ, µ) and (λ, µ, λ, µ) must be connected by a discrete set of trajectories.

The general case
Finally, let us consider the most general distribution of the eigenvalues of L; let us fix the multiindex I = (i 1 , i 2 , . . . , i k ) such that 0 < i j , i 1 + · · · + i k = n. As we have observed earlier, the set of all real symmetric matrices L with the given set Λ of eigenvalues λ 1 < λ 2 < · · · < λ k , such that λ j has multiplicity i j , is homeomrphic to the flag space F l I (R) (see section 2.1). We use the same symbol Λ for the diagonal matrix Λ with the set of eigenvalues equal to Λ.
As we have shown earlier (see section 3.2) FS Toda lattice on the space of symmetric matrices, conjugate with Λ, is induced by a gradient vector field ξ I on the partial flag space F l I (R): the image of the usual Toda vector field ξ on the full flag variety F l n (R). Moreover, the potential function, generating this field, is a Morse function; its singular points correspond to those fibres of the natural projection F l n (R) → F l I (R) on which the Toda vector field vanishes identically (singular fibres with respect to the Toda potential function on F l n (R)). As one knows, in this case the trajectories connect the singular points of the field, when t → ±∞. Our goal is to describe the order, in which these points are connected: we shall show, that this order is the same as the (strong) Bruhat order.
To this end recall (see section 3.2 and [19]), that the singular points of Toda vector field on the full flag space F l n (R) with Lax matrix all whose eigenvalues are distinct, correspond to the permutation matrices in SO(n, R), i.e. matrices, which permute the base vectors in R n and, if necessary, multiply them by −1 (to make sure the determinant is positive). In order to understand the structure of the singularities in F l I (R), it is convenient to look rather at the bundle SO(n, R) → F l I (R) with the fibres, isomorphic to G = SO(n, R) (O(i 1 , R) × · · · × O(i k , R)). The vector field ξ can be raised to a vector field on the orthogonal group with singular points at the matrices A w . The fibre through a permutation matrix A w , which is equal to the conjugation of G by A w , will contain all permutations of the form wuw −1 , where u ∈ S i 1 × · · · × S i k . Comparing this description with section 2.3, we conclude that the following proposition holds: Now it is our purpose to describe the trajectories, connecting different singular points in F l I (R). We shall prove the following statement: Theorem 4.1. Let ψ, ω ∈ S I n be two multiset permutations. Then the corresponding singular points of the FS Toda lattice in F l I (R) will be connected by a trajectory, if and only if ψ ≺ ω in the Bruhat order on S I n (see section 2.3). Moreover, the dimension of subset swept by the trajectories, connecting these points is equal to the length of the path in the Hasse diagram of S I n .
Proof. First of all observe, that the projection F l n (R) → F l I (R) maps the Toda field ξ to the field ξ I ; hence it sends invariant sub varieties in the full flag space to invariant subsets in F l I (R).
In particular, it means that the Schubert cells in F l I (R) are preserved by the generalized Toda flow since the Schubert cells in the full flags are. Moreover, since Schubert cells in F l n (R) coincide with the unstable subspace of ξ (i.e. the possibly singular subspace in M , swept by the trajectories, tending to the given singular point of the gradient system), we conclude that their images in partial flags coincide with the stable sub space of ξ I : this follows for example from the fact, that the cells in F l I (R) are homeomorphic images of minimal cells in F l n (R), see the end of section 2.4 (also compare the formula (25)). Similarly, the stable submanifolds of ξ I (i.e. the subsets, spanned by the outgoing trajectories of ξ I ) in F l I (R) coincide with the dual Schubert cells in this space, since this is true for the dual cells in F l n (R) (this was proved in [19]). But we know (see section 2.4 again), that Schubert cell and dual Schubert cells in the flag space intersect if and only if the corresponding multiset permutations are comparable in Bruhat order. The statement about the dimensions follow from the transversality of these intersections.
We conclude by the simple observation, concerning the picture that emerges on the level of the Lax matrices:

Further observations
As we have shown above (see section 3), the FS Toda lattice on symmetric matrices with nondistinct eigenvalues can be regarded as a dynamical system on partial flag manifolds. It is interesting that although the system on these spaces can be described as the image of the FS Toda lattice, the usual invariants of the FS Toda system do not descend easily to the flag spaces: direct computations show that they can become constants or functionally-dependent. So the question is, whether one can still find new invariants to prove Liuoville integrability of such systems (of course, one should first make up explicit definitions of the Poisson structures used there).
One can begin with studying few low-dimensional cases, first of all those,which correspond to the projective spaces (see section 4.1. Already in the simplest case n = 3 (and when two eigenvalues coincide), that is the system on RP 2 , we obtain a new integral of motion: Here λ 1 = λ 2 = λ and λ 3 = µ, λ < µ are the eigenvalues. In terms of the matrix entries a ij of the Lax matrix L, this function can be rewritten in the following form: I (RP 2 ,λ 1 =λ 2 ) (a ij ) = a 12 a 23 a 3
On the other hand, one can show that in this case the chopping procedure (see [11], for example) does not give any integrals, different from the traces of the powers of the Lax matrix. Thus this invariant is a new phenomenon, which makes the whole picture quite intriguing. Similar integrals can be found in the case of projective spaces in higher dimensions. These questions will be the subject of our further papers.