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

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

Yury B. CHERNYAKOV ${}^{{\dagger}^{1}{\dagger}^{2}}$, Georgy I. SHARYGIN
${}^{{\dagger}^{1}{\dagger}^{2}{\dagger}^{3}}$ and Alexander S. SORIN
${}^{{\dagger}^{2}{\dagger}^{4}{\dagger}^{5}}$

Yu.B. Chernyakov, G.I. Sharygin and A.S. Sorin

${}^{{\dagger}^{1}}$ Institute for Theoretical and Experimental Physics, 25
Bolshaya Cheremushkinskaya,
${}^{{\dagger}^{1}}$ 117218, Moscow, Russia<EMAIL_ADDRESS><EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Joint Institute for Nuclear Research, Bogoliubov
Laboratory of Theoretical Physics,
${}^{{\dagger}^{2}}$ 141980, Dubna, Moscow region, Russia<EMAIL_ADDRESS>
${}^{{\dagger}^{3}}$ Lomonosov Moscow State University, Faculty of Mechanics
and Mathematics,
${}^{{\dagger}^{3}}$ GSP-1, 1 Leninskiye Gory, Main Building, 119991, Moscow,
Russia

${}^{{\dagger}^{4}}$ National Research Nuclear University MEPhI (Moscow
Engineering Physics Institute),
${}^{{\dagger}^{4}}$ 31 Kashirskoye Shosse, 115409 Moscow, Russia

${}^{{\dagger}^{5}}$ Dubna International University, 141980, Dubna, Moscow
region, Russia

Received February 15, 2016, in final form August 10, 2016; Published online
August 20, 2016

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.

full symmetric Toda lattice; Bruhat order; integrals and semi-invariants;
partial flag space; Morse function; multiset permutation

06A06; 37D15; 37J35

##  1 Introduction

###  1.1 Toda system

The present paper is devoted to the study of the full symmetric
$\mathfrak{sl}_{n}$ Toda lattice which can be considered as a straightforward
generalization of the non-periodic Toda lattice. Let us briefly remind that
the 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 [22, 23]; in paper [16] $n$ functionally independent
integrals of the motion were found. The involution of the integrals was proved
in [11, 12].

The non-periodic Toda lattice has the Lax representation and the Lax operator
matrix has the following form in Flaschka’s variables:

$\displaystyle L=\left(\begin{matrix}b_{1}&a_{1}&0&\cdots&0\\\
a_{1}&b_{2}&a_{2}&\cdots&0\\\ 0&\cdots&\cdots&\cdots&0\\\
0&\cdots&a_{n-2}&b_{n-1}&a_{n-1}\\\
0&0&\cdots&a_{n-1}&b_{n}\end{matrix}\right)$

One can show that the Hamilton equations are equivalent to the following
matrix equation

$\displaystyle\dot{L}=[B,L],$ (1.1)

where $B$ is

$\displaystyle B=\left(\begin{matrix}0&-a_{1}&0&\cdots&0\\\
a_{1}&0&-a_{2}&\cdots&0\\\ 0&\cdots&\cdots&\cdots&0\\\
0&\cdots&a_{n-2}&0&-a_{n-1}\\\ 0&0&\cdots&a_{n-1}&0\end{matrix}\right).$

Equation (1.1) is the compatibility condition of the system

$\displaystyle L\Psi=\Psi\Lambda,\qquad\frac{\partial}{\partial t}\Psi=B\Psi,$

where $\Psi\in{\rm SO}(n,\mathbb{R})$ and $\Lambda$ is the eigenvalue matrix
of the Lax operator.

One can show that the 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 ${\rm SL}(n,\mathbb{R})$ (equal to the group of upper triangular matrices
with the determinant equal to $1$) see [1, 2, 19, 21]. It is also possible to
give an alternative description of the phase space of this dynamical system if
we identify the algebra $\mathfrak{sl}_{n}$ with its dual using the Killing
form on $\mathfrak{sl}_{n}$. In this way one obtains generalizations of the
classical Toda lattice (tri-diagonal Toda chain). Namely, use the following
identifications:

$\displaystyle\mathfrak{sl}_{n}=\mathfrak{so}_{n}\oplus\mathfrak{b}^{+}_{n},\qquad\mathfrak{sl}^{\ast}_{n}=(\mathfrak{b}^{+}_{n})^{\ast}\oplus(\mathfrak{so}_{n})^{\ast}\cong\operatorname{Symm}_{n}\oplus\mathfrak{n}_{n}^{+},$
$\displaystyle(\mathfrak{b}^{+}_{n})^{\ast}\cong(\mathfrak{so}_{n})^{\perp}=\operatorname{Symm}_{n},\qquad(\mathfrak{so}_{n})^{\ast}\cong(\mathfrak{b}^{+}_{n})^{\perp}=\mathfrak{n}_{n}^{+},$

where $\mathfrak{b}^{+}_{n}$ is the algebra of the upper triangular matrices
and $\mathfrak{n}_{n}^{+}$ is the algebra of the strictly upper triangular
matrices. As one can see this identification maps the space of symmetric
matrices into the dual space of the Lie algebra of the Borel subgroup:
$\operatorname{Symm}_{n}\cong(\mathfrak{b}_{n}^{+})^{*}$ and hence we can
introduce Kirillov–Kostant symplectic structure on $\operatorname{Symm}_{n}$
pulling it back from $(\mathfrak{b}_{n}^{+})^{*}$; it is by restriction of
this pullback that one obtains the symplectic structure on the tridiagonal
symmetric matrices used in the 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 in this way one obtains the full symmetric $\mathfrak{sl}_{n}$ Toda
lattice (elsewhere FS Toda lattice). In contrast to the Lax matrix of the non-
periodic Toda lattice the Lax matrix of the FS Toda lattice is not a tri-
diagonal symmetric matrix but an arbitrary symmetric matrix:

$\displaystyle L=\left(\begin{matrix}a_{11}&a_{12}&a_{13}&\cdots&a_{1n}\\\
a_{12}&a_{22}&a_{23}&\cdots&a_{2n}\\\ \cdots&\cdots&\cdots&\cdots&\cdots\\\
a_{1n}&a_{2n}&a_{3n}&\cdots&a_{nn}\end{matrix}\right).$

As one knows every symmetric matrix can be diagonalized in an orthogonal
basis, i.e., the Lax matrix $L$ can be decomposed as follows

$\displaystyle L=\Psi\Lambda\Psi^{-1},$

where $\Psi$ is an orthogonal matrix. It turns out, that the FS Toda lattice
is also integrable (we refer the reader to [2, 6, 8, 10, 13] for details).
Moser in [20] showed that as $t\rightarrow-\infty$ the Lax operator of the
usual tri-diagonal Toda lattice converges to the diagonal matrix with
eigenvalues put in increasing order, and when $t\rightarrow+\infty$ it
converges to the diagonal matrix with decreasing eigenvalues. This property
was further studied in [5, 9, 13, 17, 18].

###  1.2 Outline of the paper

The present paper deals with the FS Toda lattice in an arbitrary dimension
$n$. It is a natural continuation of the previous one [5] in which we study
the behavior of the Lax operator $L$ as $t\rightarrow\pm\infty$; this question
is again in the center of our attention here. The difference is that in [5] we
treated the Lax matrices $L$ with $n$ distinct eigenvalues and now we let 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 the Schubert cells in the partial flag
manifolds). So it is natural to ask if this (induced) order has something to
do with the asymptotic behavior of the Lax matrix with coinciding eigenvalues.

We show that one can regard the restriction of the FS Toda lattice on the
space of symmetric matrices with multiple 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, Proposition 3.3) and we show that the gradient system at hand verifies the
same properties as the system with distinct eigenvalues. That is we prove that
for $t\to\pm\infty$ 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 if and only if the corresponding cells are adjacent (see Theorem
4.2).

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 4.1 we give two simple explicit examples that
illustrate our theorem and prove the main results of this paper (see Section
4.3).

###  1.3 Notation and assumptions

In what follows, unless otherwise stated, all manifolds will be assumed smooth
and compact (without boundary), all 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 let $L$ denote the real symmetric $n\times n$ Lax matrix of the
system and $\Lambda$ the diagonal matrix of eigenvalues of $L$. As $L$ is
symmetric, its eigenvalues are real; so we assume that they are ordered
naturally in $\Lambda$.

We shall use the notation ${\rm O}(n,\mathbb{R})$ (respectively ${\rm
SO}(n,\mathbb{R})$) for the group of $n$-dimensional orthogonal matrices
(respectively the group of orthogonal matrices with positive determinant), and
$\mathfrak{so}(n)$ will denote its Lie algebra that is the space of real
antisymmetric $n\times n$-matrices. Similarly ${\rm SL}(n,\mathbb{R})$ will
denote the $n$-dimensional special linear group over real numbers and
$B_{n}^{+}\subset{\rm SL}(n,\mathbb{R})$ (respectively $B_{n}^{-}$) the Borel
group of upper (respectively lower) triangular matrices with unit determinant.

##  2 Bruhat order and Schubert cells

The notion of the 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.

###  2.1 Flag spaces, Grassmanians and their generalizations

First we recall some definitions:

######  Definition 2.1.

Let $I=(i_{1},i_{2},\dots,i_{k})$ be a set of positive integers,
$i_{1}+i_{2}+\dots+i_{k}=n$. Then the real partial flag space ${\rm
Fl}_{i_{1},i_{2},\dots,i_{k}}(\mathbb{R})$ is the set of all sequences of
vector subspaces

$\displaystyle\big{\\{}0=V_{0}\subset V_{1}\subset V_{2}\subset
V_{3}\subset\dots\subset V_{k}=\mathbb{R}^{n}\big{\\}}$

such that $\dim V_{l}=i_{1}+i_{2}+\dots+i_{l}$.

Let us make a few remarks before we proceed. First of all, similar definitions
work literally for vector spaces over arbitrary field $\Bbbk$ giving us the
flag spaces ${\rm Fl}_{i_{1},\dots,i_{k}}(\Bbbk)$. These spaces are easy to
define on the level of sets; however, their topology depends a lot on $\Bbbk$.
Below we shall work only with $\Bbbk=\mathbb{R}$ or $\Bbbk=\mathbb{C}$ with a
usual topology. Second, an important particular case of this construction is
$k=n$, $i_{1}=i_{2}=\dots=i_{n}=1$, which is called the full flag manifold and
is denoted by ${\rm Fl}_{n}(\mathbb{R})$ (or in a general case ${\rm
Fl}_{n}(\Bbbk)$). Another important particular case of the flag space is the
case $k=1$, i.e., when there is only one subspace chosen inside
$\mathbb{R}^{n}$. In this case the flag manifold is called the Grassman space
or just Grassmannian and is denoted by ${\rm Gr}_{d,n}(\mathbb{R})$, where
$d=i_{1}$ is the dimension of the subspaces we choose.

An important property of flag spaces is that they are homogeneous spaces, the
quotient spaces of the groups of linear transformations of $\mathbb{R}^{n}$ by
their subgroups. The isomorphism is induced by the choice of a compatible
basis in the subspaces $V_{i}$. In particular,

$\displaystyle{\rm Fl}_{n}(\mathbb{R})={\rm
SL}(n,\mathbb{R})/B_{n}^{+},\qquad{\rm Gr}_{d,n}(\mathbb{R})={\rm
SL}(n,\mathbb{R})/P,$

for a maximal parabolic subgroup $P\subset{\rm SL}(n,\mathbb{R})$. Similarly,
the general partial flag spaces are isomorphic to the quotient spaces of ${\rm
SL}(n,\mathbb{R})$ by a suitable parabolic subgroup. On the other hand, if we
choose an orthogonal basis of the corresponding subspaces so that the
orientation on $\mathbb{R}^{n}$ would match with the given one, we obtain
homeomorphisms with the quotients of a special orthogonal group. For instance

$\displaystyle{\rm Fl}_{n}(\mathbb{R})={\rm O}(n;\mathbb{R})/T_{n}={\rm
SO}(n,\mathbb{R})/T^{+}_{n},$ $\displaystyle{\rm Gr}_{d,n}(\mathbb{R})={\rm
O}(n;\mathbb{R})/{\rm O}(d;\mathbb{R})\times{\rm O}(n-d;\mathbb{R})$
$\displaystyle\hphantom{{\rm Gr}_{d,n}(\mathbb{R})}{}={\rm
SO}(n;\mathbb{R})/{\rm SO}(n,\mathbb{R})\bigcap\left({\rm
O}(d;\mathbb{R})\times{\rm O}(n-d;\mathbb{R})\right).$

Here we have denoted by $T_{n}$ the group of diagonal matrices with
eigenavalues equal to $\pm 1$, and $T^{+}_{n}$ the intersection ${\rm
SO}(n,\mathbb{R})\cap T_{n}$. In both cases we see that the groups ${\rm
SL}(n,\mathbb{R})$ and ${\rm SO}(n,\mathbb{R})$ act transitively on the flag
spaces.

The important rôle of the flag spaces in our investigation follows from the
next observation:

######  Proposition 2.2.

The space of real symmetric $n\times n$ matrices with a fixed set of
eigenvalues $\lambda_{1},\dots,\lambda_{k}$ with multiplicities
$i_{1},\dots,i_{k}$ can be identified with the partial flag manifold ${\rm
Fl}_{i_{1},\dots,i_{k}}$.

Before we prove this in full generality, let us consider the simplest case:
$n=3$. Let $\alpha<\beta$ be real numbers. Then the set of all symmetric
$3\times 3$ matrices with eigenvalues $\alpha$, $\alpha$, $\beta$ is equal to
the orbit of the diagonal matrix $\operatorname{diag}(\alpha,\alpha,\beta)$
under conjugations by the elements of ${\rm SO}(3,\mathbb{R})$. This action is
not free, the stabilizer of $\operatorname{diag}(\alpha,\alpha,\beta)$ being
equal to the subgroup $\widetilde{{\rm SO}}(2,\mathbb{R})\subset{\rm
SO}(3,\mathbb{R})$ of matrices $\Psi$ that have one of the following forms:

$\displaystyle\Psi=\begin{pmatrix}\cos{t}&\sin{t}&0\\\ -\sin{t}&\cos{t}&0\\\
0&0&1\end{pmatrix}\qquad\text{or}\qquad\Psi=\begin{pmatrix}\cos t&-\sin t&0\\\
-\sin t&-\cos t&0\\\ 0&0&-1\end{pmatrix},$

So the orbit is equal to the quotient space of the sphere ${\rm
SO}(3,\mathbb{R})/{\rm SO}(2,\mathbb{R})=S^{2}$ by the antipodal action of
$\mathbb{Z}/2\mathbb{Z}$; to see this observe that the action is induced from
the conjugation by the matrix

$\displaystyle A=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&-1\end{pmatrix}.$

In fact, the subgroup $\widetilde{{\rm SO}}(2,\mathbb{R})\subset{\rm
SO}(3,\mathbb{R})$ is generated by $A$ and the subgroup of rotations around
the $Oz$ axis. Since the homeomorphism ${\rm SO}(3,\mathbb{R})/{\rm
SO}(2,\mathbb{R})=S^{2}$ is given by the image of the point $(0,0,1)$ under
the action of ${\rm SO}(3,\mathbb{R})$, we see that the action of $A$ on
$S^{2}$ is given by the antipodal map. So the quotient space that we need is
$S^{2}/(\mathbb{Z}/2\mathbb{Z})=\mathbb{R}P^{2}$. As one can see this
construction amounts to sending each matrix $L$ from this set to the straight
line spanned by the eigenvectors corresponding to $\beta$. Alternatively, one
can consider $\mathbb{R}P^{2}$ in the dual sense as the space of all
2-dimensional subspaces in $\mathbb{R}^{3}$, and identify $L$ with the
eigenspace corresponding to $\alpha$.

More generally, assume that the diagonal matrix $\Lambda$ has $k<n$ coinciding
eigenvalues, and all the other eigenvalues of $\Lambda$ are distinct. Without
loss of generality we can think that
$\lambda_{1}=\lambda_{2}=\dots=\lambda_{k}$. Then reasoning just as before we
can identify the set of symmetric matrices with such eigenvalues with the
quotient space of ${\rm SO}(n,\mathbb{R})$ modulo the subgroup
$\widetilde{{\rm SO}}(k,\mathbb{R})$ generated by ${\rm SO}(k,\mathbb{R})$
(orthogonal transformations of the subspace
$\mathbb{R}^{k}\subset\mathbb{R}^{n}$ spanned by the first $k$-axes) and the
subgroup of diagonal matrices in ${\rm SO}(n,\mathbb{R})$. This subgroup is
equal to the intersection of ${\rm SO}(n,\mathbb{R})$ and the cartesian
product

$\displaystyle{\rm O}(k)\times\underbrace{{\rm O}(1)\times\dots\times{\rm
O}(1)}_{n-k\ \text{times}},$

and the quotient space ${\rm SO}(n,\mathbb{R})/\widetilde{{\rm
SO}}(k,\mathbb{R})$ is equal to the partial flag space ${\rm
Fl}_{k,1,\dots,1}$ (with $n-k$ units):

$\displaystyle{\rm Fl}_{k,1,\dots,1}=\big{\\{}0\subset W\subset V_{1}\subset
V_{2}\subset\dots\subset V_{n-k}=\mathbb{R}^{n}\big{\\}},$

where $\dim W=k$, $\dim V_{i}=k+i$.

Finally, consider the most general case. Assume that the eigenvalues of
$\Lambda$ are divided into several “clusters”:
$\lambda_{1}=\dots=\lambda_{i_{1}},\lambda_{i_{1}+1}=\dots=\lambda_{i_{1}+i_{2}},\dots,\lambda_{i_{1}+\dots+i_{k-1}}=\dots=\lambda_{i_{1}+\dots+i_{k}}$,
where $i_{1}+i_{2}+\dots+i_{k}=n$. Then the orbit $\Psi\Lambda\Psi^{-1}$ will
be equal to the partial flag space ${\rm Fl}_{i_{1},\dots,i_{k}}$:

$\displaystyle{\rm Fl}_{i_{1},i_{2},\dots,i_{k}}=\big{\\{}0=V_{0}\subset
V_{1}\subset V_{2}\subset V_{3}\subset\dots\subset
V_{k}=\mathbb{R}^{n}\big{\\}},$

where $\dim V_{l}=i_{1}+i_{2}+\dots+i_{l}$. This can be proved either by the
considerations of the symmetry (i.e., by finding the subgroup of matrices
commuting with $\Lambda$) as before or one can use the following observation:
every symmetric matrix $L\in\operatorname{Symm}_{n}$ with eigenvalues
$\lambda_{1},\dots,\lambda_{k}$ of multiplicities $i_{1},\dots,i_{k}$ is
uniquely determined by the collection of eigenspaces:

$\displaystyle L_{j}=\big{\\{}0\neq
v\in\mathbb{R}^{n}\,|\,Lv=\lambda_{j}v\big{\\}},\qquad j=1,\dots,k.$

Clearly $\dim L_{j}=i_{j}$. So one naturally identifies this matrix with a
point in ${\rm Fl}_{i_{1},\dots,i_{k}}$ by putting $V_{l}=L_{1}\oplus
L_{2}\oplus\dots\oplus L_{l}$.

An important property of the partial flag spaces is the existence of
surjective projections

$\displaystyle\pi\colon\ {\rm Fl}_{n}(\mathbb{R})\to{\rm
Fl}_{i_{1},i_{2},\dots,i_{n}}(\mathbb{R})$ (2.1)

given by “forgetting” the unnecessary subspaces. Alternatively, these
projections can be regarded as the additional factorization

$\displaystyle{\rm Fl}_{n}(\mathbb{R})={\rm SO}(n,\mathbb{R})/{\rm
SO}(n,\mathbb{R})\bigcap(\underbrace{{\rm O}(1)\times\dots\times{\rm
O}(1)}_{n\ \text{times}}),$ $\displaystyle{\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})={\rm SO}(n,\mathbb{R})/{\rm
SO}(n,\mathbb{R})\bigcap({\rm O}_{i_{1}}(\mathbb{R})\times\dots\times{\rm
O}_{i_{k}}(\mathbb{R})).$

It follows from this description that $\pi$ is a locally trivial fibre bundle
with the fibre equal to

$\displaystyle X={\rm SO}(n,\mathbb{R})\bigcap({\rm
O}_{i_{1}}(\mathbb{R})\times\dots\times{\rm O}_{i_{k}}(\mathbb{R}))/{\rm
SO}(n,\mathbb{R})\bigcap(\underbrace{{\rm O}(1)\times\dots\times{\rm
O}(1)}_{n\ \text{times}}).$

Or using the notation we introduced earlier

$\displaystyle X={\rm SO}(n,\mathbb{R})\bigcap({\rm
O}_{i_{1}}(\mathbb{R})\times\dots\times{\rm
O}_{i_{k}}(\mathbb{R}))/T_{n}^{+}.$

###  2.2 Bruhat order in $\boldsymbol{S_{n}}$ and full flag spaces

In this paragraph we will closely follow the exposition in [3, 14], with only
a few notations changed. Let $\omega$ be a permutation,

$\displaystyle\omega\in S_{n},\qquad\omega\colon\
\\{1,2,\dots,n\\}\to\\{1,2,\dots,n\\}.$

This permutation can be abbreviated to $(\omega(1),\dots,\omega(n))$. One
defines the length of the permutation $\omega$ as the total number of
involutions in the sequence $(\omega(1),\dots,\omega(n))$; that is,

$\displaystyle
l_{\omega}=\\#\\{j_{1}<j_{2}\,|\,\omega(j_{2})<\omega(j_{1})\\}.$

Let the numbers $r_{\omega}[p,q]$ be equal to the “number of involutions with
respect to $p$, $q$”:

$\displaystyle r_{\omega}[p,q]=\\#\\{j\leq p\,|\,\omega(j)\geq q\\},\qquad
1\leq p,q\leq n.$ (2.2)

There are many equivalent definitions of the Bruhat order on permutations; we
give the following (see [3]):

######  Definition 2.3.

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 $v$ ($u\prec v$) if and only if

$\displaystyle r_{u}[p,q]\leq r_{v}[p,q]\qquad\text{for all}\ p,\;q.$

A simple way to compute $r_{\omega}[p,q]$ is to consider the matrix
$A_{\omega}$ representing the permutation $\omega$:

$\displaystyle(A_{\omega})_{ij}=\begin{cases}1,&\omega(j)=n-i+1,\\\
0,&\text{otherwise}.\end{cases}$

It is clear that for all $p$, $q$ the number $r_{\omega}[p,q]$ is equal to the
rank of the submatrix $A^{pq}_{\omega}=\left((A_{\omega})_{ij}\right)_{i\leq
q,\,j\leq p,}$ of $A_{\omega}$ (informally one can say that $A^{pq}_{\omega}$
is the submatrix in the upper left corner of $A_{\omega}$ determined by the
element $a_{pq}$). Since there is only one nonzero entry in every row and
column of $A_{\omega}$, it is enough to count the number of such nonzero
elements in $A^{pq}_{\omega}$. E.g., for $\omega=(4213)$ we obtain
$r_{\omega}[3,2]=2$ from the following table (we replace $1$ by $\bullet$):

$4$ | $\bullet$ |  |  |
---|---|---|---|---
$3$ |  |  |  | $\bullet$
$2$ |  | $\bullet$ | $\leftarrow\uparrow$ |
$1$ |  |  | $\bullet$ |
| $1$ | $2$ | $3$ | $4$

The following lemma is proved in [14]:

######  Lemma 2.4.

Let $u\prec v$, $u\neq v$. Let $j$, $1\leq j\leq n$ be the smallest integer,
for which $u(j)\neq v(j)$ $($and hence $u(j)<v(j))$. Let $n\geq k>j$ be the
least integer for which $u(j)\leq v(k)<v(j)$ and let $v^{\prime}=v\cdot(j,k)$
denote the composition of $v$ with the swap of $j$ and $k$. Then $u\prec
v^{\prime}\prec v$.

The relation between this order and the geometry of the full flag space is
determined by the structure of the Schubert cell decomposition of this space.
There are many ways to define the Schubert cells, we shall use the following
one (more definitions can be found in the literature, see [3, 4, 14] and cf.
Section 2.3).

######  Definition 2.5.

Embed the group $S_{n}$ into ${\rm SL}(n,\mathbb{R})$ as it is explained above
(one should only replace $1$ in the definition of $A_{\omega}$ to $\pm 1$ so
as to make sure that the determinant is equal to $1$ and not $-1$). Using the
projection

$\displaystyle p\colon\ {\rm SL}(n,\mathbb{R})\to{\rm Fl}_{n}(\mathbb{R})={\rm
SL}(n,\mathbb{R})/B_{n}^{+},$

we obtain the points $p(A_{\omega})\in{\rm Fl}_{n}(\mathbb{R})$, which we
shall denote by $[A_{\omega}]$. The subgroup $B_{n}^{+}$ of ${\rm
SL}(n,\mathbb{R})$ acts on ${\rm Fl}_{n}(\mathbb{R})$ and the Schubert cell
$X_{\omega}\subset{\rm Fl}_{n}(\mathbb{R})$ for $\omega\in S_{n}$ is the orbit
of $[A_{\omega}]$ with respect to this action:

$\displaystyle X_{\omega}=B_{n}^{+}\cdot[A_{\omega}]\subseteq{\rm
Fl}_{n}(\mathbb{R}).$

One can show (cf. [14] and Section 2.3) that $X_{\omega}$ is indeed a cell. In
effect,

$\displaystyle X_{\omega}\cong\mathbb{R}^{l(\omega)}.$

The closure $\overline{X}_{\omega}$ of $X_{\omega}$ in the flag space is a
singular algebraic variety. It is called the Schubert variety. The following
statement explains the relation of the Schubert cells and the Bruhat order
(see [14, Proposition 7, p. 175]):

######  Proposition 2.6.

An element $w\in S_{n}$ precedes $v\in S_{n}$ with respect to the Bruhat
order, $w\prec v$, if and only if $\overline{X}_{w}\subseteq\overline{X}_{v}$.

This proposition is sometimes used to give an alternative definition of the
Bruhat order: the Bruhat order is the partial order induced by the contiguity
of the cells in the Schubert cell decomposition of the flag manifold.

There is another important observation relating the Schubert cells and Bruhat
order. Namely, consider the $B_{n}^{-}$-orbits of the same points
$[A_{\omega}]$:

$\displaystyle X^{\vee}_{\omega}=B^{-}_{n}\cdot[A_{\omega}].$

They are called the dual Bruhat cells. These sets are as well homeomorphic to
the Euclidean spaces, their closures are called the dual Schubert varieties.
Then the following is true (see again [14]):

######  Proposition 2.7.

Let $v,w\in S_{n}$ be two elements. Then $X_{v}\bigcap
X^{\vee}_{w}\neq\varnothing$ if and only if $v\prec w$ in Bruhat order. In the
latter case the intersection of cells is transversal.

This property of the Schubert cells was crucial in our description of the
asymptotic behavior of the FS Toda lattice, see [5]. Below we shall make use
of analogous properties of the partial flag spaces.

###  2.3 Bruhat order on multiset permutations

In this section we give a description of the Schubert cell decomposition of
the 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 a few facts (in particular, see Proposition 2.10). Besides this
one can refer to [4] and book [3] for more details.

Let $n>k$ be two natural numbers and let $I=(i_{1},i_{2},\dots,i_{k})$ be a
partition of $n$ into $k$ parts, i.e., for each $j=1,\dots,k$ we let $i_{j}$
be a positive integer so that

$\displaystyle i_{1}+i_{2}+\dots+i_{k}=n.$

Then we give the following definition:

######  Definition 2.8.

An $I$-permutation of multiset (or permutation with multiplicities, or
permutation with repetitions) is any surjective map

$\displaystyle\tau\colon\ \\{1,2,\dots,n\\}\to\\{1,2,\dots,k\\},$

such that $|\tau^{-1}(j)|=i_{j}$ for all $j=1,\dots,k$. We shall denote the
set of all such permutations by $S_{n}^{I}$.

One can regard an element $\tau\in S_{n}^{I}$ as a string of elements of the
form

$\displaystyle\tau=(\tau(1),\tau(2),\dots,\tau(n)),$

where $1\leq\tau(j)\leq k$ and every element $j$, $1\leq j\leq k$ 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 $\tau_{1}^{*}\colon
S_{n}\to S_{n}^{I}$, where $\tau_{1}\in S_{n}^{I}$ is the following multiset
permutation which we regard as a map

$\displaystyle\tau_{1}\colon\ \\{1,2,\dots,n\\}\to\\{1,2,\dots,k\\},$
$\displaystyle\tau_{1}(l)=\begin{cases}1,&1\leq l\leq i_{1},\\\ 2,&i_{1}+1\leq
l\leq i_{1}+i_{2},\\\ &\dots\dots\dots\dots\dots\dots\dots\\\
j,&i_{1}+\dots+i_{j-1}+1\leq l\leq i_{1}+\dots+i_{j-1}+i_{j},\\\
&\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\\\
k,&i_{1}+\dots+i_{k-1}+1\leq l\leq n.\end{cases}$

Then for every permutation $w\in S_{n}$ we define the multiset permutation
$\tau^{\ast}_{1}(w)=\tau_{w}$ by the formula

$\displaystyle\tau_{w}=\tau_{1}\circ w,$

or in the “linear form”:

$\displaystyle\tau_{w}=(\tau_{1}(w(1)),\tau_{1}(w(2)),\dots,\tau_{1}(w(n))).$

The map $\tau_{1}^{*}$ is evidently an epimorphism and

$\displaystyle(\tau_{1}^{*})^{-1}(\tau_{1})=S_{i_{1}}\times
S_{i_{2}}\times\dots\times S_{i_{k}}\subseteq S_{n},$

where we embed $S_{i_{j}}$ into $S_{n}$ by letting it permute the elements
$l$, $i_{1}+\dots+i_{j-1}<l\leq i_{1}+\dots+i_{j}$. For all other elements
$\tau\in S_{n}^{I}$ we see that $(\tau_{1}^{*})^{-1}(\tau)$ is a right coset
of $S_{n}$ by the subgroup $S_{i_{1}}\times S_{i_{2}}\times\dots\times
S_{i_{k}}$, i.e., the set of all permutations of the form $x_{\tau}\cdot w$
for any $w\in S_{i_{1}}\times S_{i_{2}}\times\dots\times S_{i_{k}}$ and some
$x_{\tau}\in S_{n}$ such that $\tau_{1}^{*}(x_{\tau})=\tau$. We shall
sometimes call these cosets clusters in $S_{n}$, corresponding to $\tau\in
S_{n}^{I}$; clearly every such cluster contains $N=i_{1}!i_{2}!\cdots i_{k}!$
elements. One can describe elements in $S_{n}^{I}$ in terms of clusters; for
instance, the string $(1,2,2,1)\in S_{4}^{(2,2)}$ corresponds to the cluster
$\\{(1,3,4,2),\,(2,3,4,1),\,(1,4,3,2),\,(2,4,3,1)\\}$.

Now one uses these constructions to introduce the Bruhat order on $S_{n}^{I}$.
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
$\tau\colon\\{1,2,\dots,n\\}\to\\{1,2,\dots,k\\}$ be an element in
$S_{n}^{I}$, then

$\displaystyle l_{\tau}=\\#\\{j_{1}<j_{2}\,|\,\tau(j_{2})>\tau(j_{1})\\},$

i.e., $l_{\tau}$ is the number of involutions in the sequence
$(\tau(1),\dots,\tau(n))$. Observe that we do not count the pairs
$j_{1}<j_{2}$ for which $\tau(j_{1})=\tau(j_{2})$.

Second, similarly to the Definition 2.3 we define the Bruhat order on
$S_{n}^{I}$ with the help of ranks of some matrices: consider the $n\times k$
matrices $A_{\tau}$ corresponding to the elements of $S_{n}^{I}$. Just as
before we define the numbers $r_{\tau}[p,q]$ (see formula (2.2)) for all
$1\leq p\leq n$ and $1\leq q\leq k$. Then:

######  Definition 2.9.

We say that $\tau\prec\upsilon$ (where both $\tau,\upsilon\in S_{n}^{I}$) if
and only if $r_{\tau}[p,q]\leq r_{\upsilon}[p,q]$ for all $p$, $q$. Observe
that we use $\leq$ rather than the strict inequality $<$ here. Of course at
least one inequality must be strict if $\tau\neq\upsilon$

The following example illustrates this definition very well: let $I=(2,2)$ as
before and $\tau=(2,1,1,2)$. Then $A_{\tau}$ has this shape:

$2$ | $\bullet$ |  |  | $\bullet$
---|---|---|---|---
$1$ |  | $\bullet$ | $\bullet$ |
| $1$ | $2$ | 3 | $4$

Similarly, the lengths of the elements of $S_{4}^{(2,2)}$ are equal to

$\displaystyle(1,1,2,2),\quad l_{(1,1,2,2)}=0,\qquad(1,2,1,2),\quad
l_{(1,2,1,2)}=1,\qquad(1,2,2,1),\quad l_{(1,2,2,1)}=2,$
$\displaystyle(2,1,1,2),\quad l_{(2,1,1,2)}=2,\qquad(2,1,2,1),\quad
l_{(2,1,2,1)}=3,\qquad(2,2,1,1),\quad l_{(2,2,1,1)}=4.$

These data are sufficient to draw the Hasse diagram of the Bruhat order in
this case (cf. Fig. 2)111Recall that the Hasse diagram of a partially ordered
set is the oriented graph whose vertices are given by the elements of the set
and edges connect elements $a$ and $b$ if and only if $a\prec b$ and there is
no $c$ such that $a\prec c\prec b$..

The following propositions (Propositions 2.10 and 2.12, and Lemma 2.11)
explain, in what sense one can “pull back” the order from $S_{n}$ to
$S_{n}^{I}$.

######  Proposition 2.10.

Let $u,v\in S_{n}$ be two permutations such that $u\prec v$ in the Bruhat
order on $S_{n}$ and $\tau_{1}^{*}(u)\neq\tau_{1}^{*}(v)$. Then
$\tau_{1}^{*}(u)\prec\tau_{1}^{*}(v)$ in the Bruhat order in $S_{n}^{I}$.

###### Proof.

Let us denote $\tau_{1}^{*}(u)=U$ and $\tau_{1}^{*}(v)=V$; we will show that
$U\prec V$. First of all we observe that it is enough to show this in a
particular case

$\displaystyle v=u\cdot(i,j),\qquad l(v)=l(u)+1$

(here $(i,j)$ denotes the transposition of $i$ and $j$ for $i<j$). Indeed, it
follows directly from Lemma 2.4 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. Since $u\prec v$ we have $r_{u}[p,q]\leq
r_{v}[p,q]$. We shall show that $r_{U}[p,q]\leq r_{V}[p,q]$ for all $1\leq
p\leq n$, $1\leq q\leq k$. It is impossible that all the numbers $r_{U}[p,q]$
and $r_{V}[p,q]$ are equal (we assumed that $U\neq V$) and so we have that
$U\prec V$. 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 $(i,j)$). Let
$u(i)=a$, $u(j)=b$ then $v(i)=b$, $v(j)=a$; let $i<l<j$ and for $c=u(l)$ there
are three possibilities: $c<a<b$, $a<b<c$ or $a<c<b$. 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
(2.14).

$n$ |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---
$\dots$ |  |  |  |  |  |  |  |  |  |  |
$b$ |  |  |  |  |  |  |  |  | $1$ |  |
|  |  |  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |  |  |
$a$ |  |  |  | $1$ |  |  |  |  |  |  |
$\dots$ |  |  |  |  |  |  |  |  |  |  |
$2$ |  |  |  |  |  |  |  |  |  |  |
$1$ |  |  |  |  |  |  |  |  |  |  |
| $1$ | $2$ | $\dots$ | $i$ |  |  |  |  | $j$ | $\dots$ | $n$
(2.14)

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 $P_{\tau_{1}(a)\tau_{1}(b)ij}$
in the matrices $A_{U}$ and $A_{V}$ does not contain nonzero elements: new
nonzero elements can appear only in the top and bottom rows of the rectangle
(i.e., in the rows number $\tau_{1}(a)$ and $\tau_{1}(b)$) 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 $P_{k_{a}k_{b}ij}$: 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 (2.23).

$\displaystyle
A_{U}=\begin{tabular}[]{c|c|c|c|c|c|c|c|}\hline\cr\cline{1-0}\cr$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\
\hline\cr\cline{1-0}\cr$k_{b}$&$$&$$&$$&$\dots$&$$&$1$&$$\\\
\hline\cr\cline{1-0}\cr$$&$$&$$&$$&$\dots$&$$&$$&$$\\\
\hline\cr\cline{1-0}\cr$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\
\hline\cr\cline{1-0}\cr$$&$$&$$&$$&$\dots$&$$&$$&$$\\\
\hline\cr\cline{1-0}\cr$k_{a}$&$$&$1$&$$&$\dots$&$$&$$&$$\\\
\hline\cr\cline{1-0}\cr$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\
\hline\cr\cline{1-0}\cr$$&$\dots$&$i$&$$&$\dots$&$$&$j$&$\dots$\end{tabular}$
(2.23) $\displaystyle
A_{V}=\begin{tabular}[]{c|c|c|c|c|c|c|c|}\hline\cr\cline{1-0}\cr$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\
\hline\cr\cline{1-0}\cr$k_{b}$&$$&$1$&$$&$\dots$&$$&$$&$$\\\
\hline\cr\cline{1-0}\cr$$&$$&$$&$$&$\dots$&$$&$$&$$\\\
\hline\cr\cline{1-0}\cr$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\
\hline\cr\cline{1-0}\cr$$&$$&$$&$$&$\dots$&$$&$$&$$\\\
\hline\cr\cline{1-0}\cr$k_{a}$&$$&$$&$$&$\dots$&$$&$1$&$$\\\
\hline\cr\cline{1-0}\cr$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$&$\dots$\\\
\hline\cr\cline{1-0}\cr$$&$\dots$&$i$&$$&$\dots$&$$&$j$&$\dots$\end{tabular}$
∎

The following statement is similar to Lemma 2.4; it gives a procedure by which
we can move between comparable elements of $S_{n}^{I}$.

######  Lemma 2.11.

Let $U,V\in S_{n}^{I}$, $U\prec V$, $U\neq V$. Let $j_{0}$ be the least $j$,
$1\leq j\leq n$, for which $U(j_{0})\neq V(j_{0})$; then $U(j_{0})<V(j_{0})$.
Let $l\geq j_{0}$ be the maximal number for which
$V(j_{0})=V(j_{0}+1)=\dots=V(l)$ so that $U(j_{0})<V(l)$. Let $m$ denote the
least integer $l\leq m\leq n$ verifying the inequality $U(j)\leq V(m)<V(l)$.
Put $V^{\prime}=V(l,m)$, i.e., in terms of the corresponding strings
$V^{\prime}$ is $V$ in which the values of $V(l)$ and $V(m)$ are swapped. Then
$U\prec V^{\prime}\prec V$.

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 $S_{n}^{I}$ to
$S_{n}$.

######  Proposition 2.12.

Every coset $(\tau_{1}^{*})^{-1}(\upsilon)\subset S_{n}$, $\upsilon\in
S_{n}^{I}$ contains the unique least element $u\in S_{n}$ (with respect to the
Bruhat order in $S_{n}$) so that if $\xi\prec\zeta$ in $S_{n}^{I}$, then a
similar inequality holds for the corresponding least elements in $S_{n}$:
$x\prec z$.

Recall that the least element in a partial order is the element, which is less
than all others, in particular, it is comparable with all others. It is clear
that it is unique if it exists.

###### Proof.

Observe that every coset $(\tau_{1}^{*})^{-1}(\upsilon)$ contains a unique
element $s_{\upsilon}$ such that $s_{\upsilon}^{-1}$ 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 $s_{\upsilon}$ the $I$-deshuffle;
clearly, this deshuffle is the least element that we need. This follows easily
from the fact that for any permutation $u\in(\tau_{1}^{*})^{-1}(\upsilon)$ and
any $i<k<j$ such that $\upsilon(i)=\upsilon(j)\neq\upsilon(k)$ 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
$(\tau_{1}^{*})^{-1}(\upsilon)$ (induced from $S_{n}$) depends only on the
permutation of the elements inside separate blocks of the partition, and not
on $\upsilon$. In particular, the shuffle corresponds to the “unit” with
respect to the permutations inside the blocks; hence, it is least.

Consider now arbitrary $U,V\in S_{n}^{I}$ and let $s_{U}$, $s_{V}$ be the
corresponding deshuffles. By Lemma 2.11 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 $s_{V}=s_{U}(l,m)$: clearly
$s_{U}(l,m)\in(\tau_{1}^{*})^{-1}(V)$ and the fact that it coincides with
$s_{V}$ follows from the choice of $l$ and $m$, see Lemma 2.11. 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,

$\displaystyle l_{s_{V}}=l_{s_{U}(l,m)}=l_{s_{U}}+1.$ ∎

Observe that we have in fact established that the Bruhat order on $S_{n}$,
when restricted to clusters $(\tau_{1}^{*})^{-1}(\upsilon)$, $\upsilon\in
S_{n}^{I}$, coincides with the lexicographic order on the direct product
$S_{i_{1}}\times S_{i_{2}}\times\dots\times S_{i_{k}}$ where each factor is
equipped with the usual Bruhat order on $S_{l}$. In particular, there is a
unique greatest element in it too.

###  2.4 Schubert cells in partial flags and Bruhat order

The definition of the Schubert cells given in Section 2.2 can be extended to a
more general situation of the partial flag spaces, in particular to
Grassmannians, so that the canonical projection (2.1) maps the Schubert cells
in ${\rm Fl}_{n}(\mathbb{R})$ to the cells in the partial flags. We shall use
this observation and the properties of the Bruhat order to describe some of
the structures of the Schubert cells, their duals and their intersections in
the partial flag spaces. The main references for this section are [4, 14, 15].

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 ${\rm
Gr}_{d,n}(\mathbb{R})$, the space of $d$-planes in $\mathbb{R}^{n}$.

Let $\\{e_{1},\dots,e_{n}\\}$ be a basis of $\mathbb{R}^{n}$. Any
$d$-dimensional hyperplane inside $\mathbb{R}^{n}$ is determined by a
collection of $d$ linearly-independent vectors in $\mathbb{R}^{n}$, which is
the same (when the basis of $\mathbb{R}^{n}$ is fixed) as a $d\times n$, $d<n$
matrix with a maximal rank. It is clear that two matrices $\Lambda$,
$\Lambda^{\prime}$ determine the same $d$-plane if and only if there exists
$g\in{\rm GL}(d,\mathbb{R})$ such that:

$\displaystyle\Lambda^{\prime}=g\cdot\Lambda.$

We shall use the same symbol $\Lambda$ to denote the $d$-plane corresponding
to the matrix $\Lambda$. Observe that the natural ${\rm
GL}(n,\mathbb{R})$-action on the Grassmannian translates into the right
multiplication of $\Lambda$ by the matrices $B\in{\rm GL}(n,\mathbb{R})$. Also
observe that in our notation the left and right actions exchange their roles:
in Section 2.3 we define ${\rm Gr}_{d,n}$ as the quotient of ${\rm
SL}(n,\mathbb{R})$ by the right action of a parabolic subgroup so that ${\rm
GL}(n,\mathbb{R})$ 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 the $n\times d$ matrices instead of the $d\times n$ matrices.

Let for any multi-index $A=(a_{1},\dots,a_{d})$, $1\leq
a_{1}<a_{2}<\dots<a_{d}\leq n$ symbol $V_{A}$ denote the subspace in
$\mathbb{R}^{n}$ spanned by $\\{e_{a_{1}},e_{a_{2}},\dots,e_{a_{d}}\\}$. Then

######  Definition 2.13.

The Schubert cell $X_{A}$ is the $B_{n}^{+}$-orbit of the point $V_{A}$ in
Grassmannian ${\rm Gr}_{d,n}(\mathbb{R})$.

One can give a more geometric definition of the Schubert cells. To this end
consider the subspaces $V_{i}$, $i=1,\dots,n$, $V_{i}\subseteq\mathbb{R}^{n}$
spanned for each $i$ by the vectors $\\{e_{1},\dots,e_{i}\\}$ and consider for
an arbitrary $d$-plane $\Lambda\in{\rm Gr}_{d,n}(\mathbb{R})$ the
intersections:

$\displaystyle 0\subset\Lambda\cap V_{1}\subset\Lambda\cap
V_{2}\subset\cdots\subset\Lambda\cap V_{n-1}\subset\Lambda\cap V_{n}=\Lambda.$

Then for the same multi-index $A=(a_{1},\dots,a_{d})$ we have

######  Definition 2.14.

The Schubert cell $X_{A}$ is the set of such planes $\Lambda\in{\rm
Gr}_{d,n}(\mathbb{R})$ that

$\displaystyle\dim(\Lambda\cap V_{j})=\\#\\{k\,|\,1\leq k\leq d,\
a_{k}<j\\},\qquad\text{for all}\ j=1,\dots,n.$

The rectangular matrices $\Lambda$ corresponding to the planes in $X_{A}$ can
be chosen in a special form:

$\displaystyle\left(\begin{array}[]{@{}cccccccccccc@{}}0&\cdots&0&1&{*}&\cdots&0&{*}&0&\cdots&\cdots&{*}\\\
0&\cdots&0&0&\cdots&0&1&{*}&0&\cdots&\cdots&{*}\\\
\cdots&\cdots&\cdots&0&\cdots&\cdots&0&\cdots&0&\cdots&\cdots&{*}\\\
0&\cdots&\cdots&0&\cdots&\cdots&0&0&1&{*}&\cdots&{*}\end{array}\right).$

Here $1$’s stand in the intersections of the $i$-th rows and
$(n-a_{d-i+1}+1)$-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 substituted 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

$\displaystyle X_{A}\cong R^{dn-\sum(n+i-a_{i})}.$

Here $dn$ is the dimension of the space of the $d\times n$ matrices and we
subtract the number of fixed elements in a matrix. One can also introduce the
“length” of the sequence $A$ by putting

$\displaystyle l(A)=\sum_{k=1}^{d}a_{k}-k.$

Then it is easy to see that $l(A)=dn-\sum(n+i-a_{i})$ and so
$X_{A}\cong\mathbb{R}^{l(A)}$. It turns out that $\bigcup_{A}X_{A}$ is a cell
decomposition of ${\rm Gr}_{d,n}(\mathbb{R})$. Moreover, one can show (see
[4]) that a cell $X_{B}$ is adjacent to $X_{A}$ if and only if $b_{k}\leq
a_{k}$ for all $k$. In this case we shall write $B\leq A$; this is a partial
order on the set of all sequences.

Finally, we associate a sequence $A$ with the multiset permutation $\omega\in
S_{n}^{(d,n-d)}$ given by

$\displaystyle\omega(a_{1})=\dots=\omega(a_{d})=1,\qquad\omega(j)=2,\qquad
j\not\in\\{a_{1},\dots,a_{d}\\}.$

It is easy to see that the Bruhat order on $S_{n}^{(d,n-d)}$ corresponds to
the partial order given by the inequalities $B\leq A$.

######  Example 2.15.

Consider the Grassmannian ${\rm Gr}_{2,4}(\mathbb{R})$, the space of all real
$2$-planes in $\mathbb{R}^{4}$. In this case there are $6$ cells corresponding
to the sequences $(1,2)$, $(1,3)$, $(1,4)$, $(2,3)$, $(2,4)$ and $(3,4)$. The
corresponding Schubert cells are spanned by the matrices of the following
shapes (we use the equality signs to denote this):

$\displaystyle C_{(1,2)}=\left(\begin{matrix}0&0&1&0\\\
0&0&0&1\end{matrix}\right),\qquad C_{(1,3)}=\left(\begin{matrix}0&1&*&0\\\
0&0&0&1\end{matrix}\right),\qquad C_{(1,4)}=\left(\begin{matrix}1&*&*&0\\\
0&0&0&1\end{matrix}\right),$ $\displaystyle
C_{(2,3)}=\left(\begin{matrix}0&1&0&*\\\ 0&0&1&*\end{matrix}\right),\qquad
C_{(2,4)}=\left(\begin{matrix}1&*&0&*\\\ 0&0&1&*\end{matrix}\right),\qquad
C_{(3,4)}=\left(\begin{matrix}1&0&*&*\\\ 0&1&*&*\end{matrix}\right).$

In the general case of an arbitrary partial flag space ${\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$ (which we shall often abbreviate to just
${\rm Fl}_{I}(\mathbb{R})$) we can define the Schubert cells decomposition
similarly by choosing a basis $\\{e_{1},\dots,e_{n}\\}$ of $\mathbb{R}^{n}$;
then for any partial flag

$\displaystyle E_{\bullet}=(E_{1}\subset E_{2}\subset\dots\subset E_{k})$

of vector subspaces in $\mathbb{R}^{n}$ and any multiset permutation
$\omega\in S_{n}^{I}$ (where $I$ is a $k$-partition of $n$) we say that
$E_{\bullet}$ is in the cell $X_{\omega}$ if and only if

$\displaystyle\dim(E_{p}\cap E_{q})=\\#\\{i\leq p\,|\,\omega(i)\leq q\\}$
(2.32)

for all $1\leq p\leq n$, $1\leq q\leq k$. It is easy to see that these sets
are indeed cells, that the cells defined for different $\omega\in S_{n}^{I}$
do not intersect and that ${\rm Fl}_{I}(\mathbb{R})$ is equal to their union.
The closures of these cells will be denoted by $\overline{X}_{\omega}$; they
are called the Schubert varieties in ${\rm Fl}_{I}(\mathbb{R})$. One can
describe them by replacing the equalities in (2.32) by the non-strict
inequality $\geq$; they are indeed singular algebraic subvarieties in ${\rm
Fl}_{I}(\mathbb{R})$.

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 ${\rm SL}(n,\mathbb{R})$ (or ${\rm SO}(n,\mathbb{R})$) by
some parabolic subgroup $P$, we can define the Schubert cells as the orbits of
certain elements within ${\rm Fl}_{I}(\mathbb{R})$ with respect to the action
of $B^{+}_{n}$ (or some block-diagonal matrices, if we speak about ${\rm
SO}(n,\mathbb{R})$): take the matrix $A_{w}$ corresponding to the minimal
representative $w$ in $S_{n}$ of an element $\omega\in S_{n}^{I}$ (see Section
2.3), then

$\displaystyle X_{\omega}=B^{+}_{n}(A_{w})P/P\subset{\rm SL}(n;\mathbb{R})/P.$

It clearly follows from this definition that the Schubert cells in the partial
flag spaces are equal to the projections of the Schubert cells in the full
flag space under the projection $\pi$, see (2.1). Moreover, one can show (see
[4]) that they are in effect homeomorphic to the cell of the least element $w$
in ${\rm Fl}_{n}(\mathbb{R})$.

Now by comparing this description with the results of the previous section
(see Propositions 2.10 and 2.12 and Lemma 2.11) we obtain the following
statement:

######  Proposition 2.16.

For any two permutations with repetitions $\psi,\omega\in S_{n}^{I}$ we have
$\psi\prec\omega$ in the Bruhat order if and only if the corresponding cells
$X_{\psi}$ and $X_{\omega}$ are adjacent; on the level of Schubert varieties
this implies the inclusion $\overline{X}_{\psi}\subset\overline{X}_{\omega}$.

Finally, similarly to the case of the full flag space ${\rm
Fl}_{n}(\mathbb{R})$ in ${\rm Fl}_{I}(\mathbb{R})$ one can also define the
dual Schubert cells $X^{\wedge}_{\omega}$, $\omega\in S_{n}^{I}$ (and the
corresponding Schubert varieties). This can be done either by a modification
of the rank (in)equalities (2.32), or as orbits of the corresponding elements
in ${\rm Fl}_{I}(\mathbb{R})$ with respect to the action of $B^{-}_{n}$, or
finally simply as projections of the dual cells in ${\rm Fl}_{n}(\mathbb{R})$.
As before a quick comparison of the definitions proves the following
proposition:

######  Proposition 2.17.

A Schubert cell $X_{\psi}\subset{\rm Fl}_{I}(\mathbb{R})$ intersects with the
dual Schubert cell $X^{\wedge}_{\omega}\subset{\rm Fl}_{I}(\mathbb{R})$ if and
only if $\psi\prec\omega$ in the Bruhat order in $S_{n}^{I}$ and in the latter
case the intersection is transversal.

The proof follows from the observation that $X^{\wedge}_{\omega}$ corresponds
to the cell determined by the greatest element $w^{\prime}$ in the cosets
$(\tau_{1}^{*})^{-1}(\omega)$ just as $X_{\omega}$ corresponds to the least
one: i.e., $X_{\omega}^{\wedge}$ is a homeomorphic image of the dual cell
corresponding to $w^{\prime}$ in ${\rm Fl}_{n}(\mathbb{R})$ under the
projection $\pi$. Since the statement of the Proposition 2.17 holds in ${\rm
Fl}_{n}(\mathbb{R})$, it must hold in ${\rm Fl}_{I}(\mathbb{R})$ as well (we
must also use the fact that $\pi$ is submersion).

##  3 The FS Toda lattice on partial flag spaces

###  3.1 Fibre bundles and Bott–Morse functions

In our study of the Toda system on symmetric matrices with coinciding
eigenvalues we shall need a 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 group $\tilde{G}$, $h\in\tilde{G}$, $x\in M$, $x\mapsto x^{h}$.
Let $G\subset\tilde{G}$ be a compact subgroup of $\tilde{G}$. We assume that
$\pi\colon M\to M/G$ is a locally trivial bundle (in which the fibre
containing $x$ is homeomorphic to the stabilizer of $x$, $G^{x}\subset G$); 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
$\xi=\operatorname{grad}_{g}f$ 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 $\xi$ induces a field $\xi_{X}$ on
$X$; we just put $\xi_{X}(\pi(m))$ to be equal to the projection
$d\pi_{m}(\xi(m))$ for an arbitrary point $m$ in $M$.

We shall need the following statement:

######  Proposition 3.1.

The field $\xi_{X}$ is equal to the gradient of the function $f_{X}$ with
respect to the metric $g_{X}$:

$\displaystyle\xi_{X}=\operatorname{grad}_{g_{X}}f_{X}.$

###### 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
$\operatorname{grad}_{g_{X}}f_{X}$ is characterized by the equality

$\displaystyle
g_{X}(\eta_{X},\operatorname{grad}_{g_{X}}f_{X})=\eta_{X}(f_{X});$

and that $g_{X}(\eta_{X},\,\zeta_{X})$ for two vectors $\eta_{X},\zeta_{X}\in
T_{x}X$ (here $X=M/G$) is defined as

$\displaystyle g_{X}(\eta_{X},\zeta_{X})=g(\eta,\zeta),$

for any $\eta,\zeta\in T_{m}M$ ($m\in\pi^{-1}(x)$ is an arbitrary point), such
that $d\pi_{m}(\eta)=\eta_{X}$, $d\pi_{m}(\zeta)=\zeta_{X}$ and they are both
perpendicular to the “vertical” subspace $T_{m}(mG)$ of $T_{m}M$.

Now since the function $f$ is constant along the fibres of $\pi$, the field
$\operatorname{grad}_{g}f$ is orthogonal to the vertical subspaces; so

$\displaystyle
g_{X}(\eta_{X},d\pi(\operatorname{grad}_{g}f))=g(\eta,\operatorname{grad}_{g}f)=\eta(f)=\eta_{X}(f_{X}),$

where the second equality follows from the definition of
$\operatorname{grad}_{g}f$, and the third one from the definition of $f_{X}$
(it is enough to consider the local structure of a cartesian product in $M$).
∎

Below we shall need to know what conditions should be imposed on $f$ to
provide that $f_{X}$ is a Morse function. Observe that in the general case $f$
cannot be a Morse function itself unless the group $G$ is discrete (and
finite): because of the $G$-invariance of $f$ every critical point of $f_{X}$
in $X=M/G$ will correspond to a $G$-orbit consisting of critical points of $f$
in $M$. So let us give a simple criterion for $f_{X}$ to be Morse:

######  Proposition 3.2.

Let $f$ be a $G$-invariant function and $g$ be a $G$-invariant metric on $M$.
The function $f_{X}\in C^{\infty}(M/G)$, induced from $f$, is a Morse function
if and only if at every critical point $m$ of $f$ the Hessian $H(f)$ is
nondegenerate when restricted to the orthogonal complement to the space of
“vertical” vectors $($i.e., vectors tangent to the orbit$)$.

###### Proof.

First, recall that the gradient of a function is always perpendicular to its
level sets. So in the case we consider it to be perpendicular to the
$G$-orbits. It follows that it is not in the kernel of the projection
$d\pi_{m}\colon T_{m}M\to T_{\pi(m)}M/G$ (as we observed earlier, the gradient
of $f_{X}$ is equal to the projection of the gradient of $f$). Hence the
critical points of $f_{X}$ correspond to the “critical orbits” of $f$ in $M$,
i.e., to the $G$-orbits consisting of the critical points of $f$.

Second, Hessian of $f_{X}$ at a critical point $x_{0}\in X$ is the symmetric
quadratic form on tangent space given by the formula

$\displaystyle
H(f_{X})_{x_{0}}(\xi(x_{0}),\eta(x_{0}))=\xi(\eta(f_{X}))(x_{0}),$

where $\xi$, $\eta$ are two vector fields in the 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 $\xi$ and $\eta$ at the point $x_{0}$).

Now the proposition follows from the fact that every vector field $\xi$ on
$M/G$ can be locally lifted to a $G$-invariant vector field $\tilde{\xi}$ in
the neighborhood of $\pi^{-1}(x_{0})\subset M$ orthogonal to the fibre
$\pi^{-1}(x_{0})$: to see this it is enough to use the local trivialization
$\pi^{-1}(U)=U\times G^{x_{0}}$ for some open neighborhood $U$ of $x_{0}$. So
we have the following formula:

$\displaystyle
H(f_{X})_{x_{0}}(\xi(x_{0}),\eta(x_{0}))=\tilde{\xi}(\tilde{\eta}(f))(\tilde{x}_{0})=H(f)_{\tilde{x}_{0}}(\tilde{\xi}(\tilde{x}_{0}),\tilde{\eta}(\tilde{x}_{0})),$

where $\tilde{x}_{0}$ is a point in the fibre $\pi^{-1}(x_{0})$; 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 $f\in C^{\infty}(X)$ ($X$ is a
smooth manifold) is called the Morse–Bott function if its critical set is a
smooth (not connected) submanifold in $X$ and the Hessian is nondegenerate in
the normal direction to this submanifold. In our case the set of critical
points is equal to the $G^{x}$-orbits through critical points $x$ of $f$;
hence, it is smooth (by assumption on the nature of the action) and normal
direction is the direction orthogonal to the vertical vectors.

###  3.2 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 ${\rm SO}(n,\mathbb{R})$,
which we here shall call the Toda flow or just Toda system on ${\rm
SO}(n,\mathbb{R})$. This flow is determined by the vector field

$\displaystyle
M(\Psi)=\big{(}\big{(}\Psi\Lambda\Psi^{-1}\big{)}_{+}-\big{(}\Psi\Lambda\Psi^{-1}\big{)}_{-}\big{)}\Psi,$
(3.1)

where $\Lambda$ is the diagonal matrix of eigenvalues of the symmetric Lax
matrix $L$; in addition we identify the tangent space $T_{\Psi}{\rm
SO}(n,\mathbb{R})$ with the right translation of the Lie algebra
$\mathfrak{so}_{n}(\mathbb{R})$ by $\Psi$. In fact, this vector field is equal
to the gradient of a function with respect to an invariant metric on ${\rm
SO}(n,\mathbb{R})$: for the fixed eigenvalues matrix $\Lambda$ one can take

$\displaystyle
F(\Psi)=\operatorname{Tr}\big{(}\Psi\Lambda\Psi^{T}N\big{)},\qquad\text{where}\quad
N=\operatorname{diag}(0,1,\dots,n-1).$ (3.2)

The ${\rm SO}(n,\mathbb{R})$-invariant Riemannian structure is determined by
its values on $\mathfrak{so}_{n}$, where it is given by the formula

$\displaystyle\langle
A,B\rangle_{J}=-\operatorname{Tr}\big{(}AJ^{-1}(B)\big{)},$ (3.3)

for any antisymmetric matrices $A$ and $B$ and a linear isomorphism
$J\colon\mathfrak{so}_{n}\to\mathfrak{so}_{n}$. Then

$\displaystyle M(\Psi)=\operatorname{grad}_{\langle\,,\,\rangle_{J}}\,F,$

see [5, 7] for details. This property does not depend on the eigenvalues of
$L$, in particular it holds for both distinct and coinciding eigenvalues.

Since all the objects here are $T^{+}_{n}$-invariant (see Section 2.1) the
same formulas (3.1), (3.2) and (3.3) induce a vector field, a function and a
Riemannian structure on the full flag space

$\displaystyle{\rm Fl}_{n}(\mathbb{R})={\rm SO}(n;\mathbb{R})/T_{n}^{+}.$

It follows from the discussion of Section 3.1 that the field $M$ on ${\rm
Fl}_{n}(\mathbb{R})$ is equal to the gradient of the corresponding function.
It can be shown (see [5, 7]) that the function $F$ is a Morse function on both
${\rm SO}(n,\mathbb{R})$ and the flag space ${\rm Fl}_{n}(\mathbb{R})$.

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 $\Lambda$; for definiteness we can assume that they are

$\displaystyle\lambda_{1}=\lambda_{2}=\dots=\lambda_{i_{1}}<\lambda_{i_{1}+1}=\lambda_{i_{1}+2}=\dots=\lambda_{i_{1}+i_{2}}<\cdots$
$\displaystyle\hphantom{\lambda_{1}}{}<\lambda_{i_{1}+\dots+i_{k-1}+1}=\lambda_{i_{1}+\dots+i_{k-1}+2}=\dots=\lambda_{i_{1}+\dots+i_{k-1}+i_{k}},$
(3.4)

where $i_{1}+i_{2}+\dots+i_{k}=n$; that is we assume there are $k<n$ distinct
eigenvalues of $\Lambda$ and the multiplicity of the $j$-th eigenvalue is
$i_{j}$. In this case the vector field $M(\Psi)$ on ${\rm SO}(n;\mathbb{R})$
is invariant with respect to ${\rm O}(i_{1},\mathbb{R})\times\dots\times{\rm
O}(i_{k},\mathbb{R})$, i.e., $M(\Psi g)=M(\Psi)g$ for all $g\in{\rm
SO}(n,\mathbb{R})\bigcap({\rm O}(i_{1},\mathbb{R})\times\dots\times{\rm
O}(i_{k},\mathbb{R}))$. So we obtain a vector field $\widetilde{M}$ on the
partial flag space. In effect, all the conditions of Proposition 3.1 hold
(i.e., the function $F$ and the Riemannian structure $\langle\,,\,\rangle_{J}$
are also invariant with respect to the group action); hence, the field
$\widetilde{M}$ is equal to the gradient of a function $\widetilde{F}$ with
respect to the induced Riemannian structure.

Now we are going to show that the function $\widetilde{F}$ on ${\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$ is Morse. To this end (see Proposition
3.2) we should show that the restriction of the Hessian of $F$ on the
directions orthogonal to the “vertical” vectors is nondegenerate. This Hessian
of $F$ is given by formula (39) in [5]: let $s_{w}\in{\rm SO}(n;\mathbb{R})$
be a positive permutation matrix corresponding to $w\in S_{n}$ (i.e., the
matrix in ${\rm SO}(n,\mathbb{R})$ which permutes the vectors $\pm e_{i}$,
$i=1,\dots,n$ so that the permutation $w\in S_{n}$ emerges when the signs are
dropped); then $s_{w}$ is a singular point in ${\rm SO}(n;\mathbb{R})$ and the
Hessian of $F$ at $s_{w}$ is

$\displaystyle
d^{2}_{s_{w}}F=\sum_{i<j}\theta_{ij}^{2}(\lambda_{w(i)}-\lambda_{w(j)})(j-i).$
(3.5)

Here $\theta_{ij}$ denote the local coordinates on ${\rm SO}(n,\mathbb{R})$ at
the point $s_{w}\in{\rm SO}(n,\mathbb{R})$ obtained from the standard
coordinates on $\mathfrak{so}_{n}$ by right translation. Put

$\displaystyle G={\rm SO}(n,\mathbb{R})\bigcap\big{(}{\rm
O}(i_{1},\mathbb{R})\times\dots\times{\rm O}(i_{k},\mathbb{R})\big{)}.$

Recall that we assume the eigenvalues of $\Lambda$ to be partitioned into $k$
“blocks”, see (3.4), 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

$\displaystyle T^{v}_{s_{w}}{\rm SO}(n;\mathbb{R})=T_{s_{w}}(s_{w}\cdot G).$

The Lie algebra $\mathfrak{g}$ of $G$ is equal to the space of all
antisymmetric matrices, spanned by the following set of elementary
antisymmetric matrices:

$\displaystyle e_{ij}-e_{ji},\qquad\text{such that}\quad\exists\,p,\quad 1\leq
p\leq k,\quad i_{0}+\dots+i_{p-1}<i<j\leq i_{0}+\dots+i_{p},$

where we have put $i_{0}=0$ and use symbols $e_{ij}$ to denote the matrix
units. Then the vertical directions at $s_{w}$ are equal to the linear span of

$\displaystyle(e_{w(i)w(j)}-e_{w(j)w(i)})s_{w}\in T_{s_{w}}{\rm
SO}(n;\mathbb{R})$

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
$\operatorname{Ad}_{s_{w}}$ of $s_{w}$ on $\mathfrak{so}_{n}$ amounts to the
permutation of indices. Now the non-degeneracy of $d^{2}_{s_{w}}$ on the
linear complement of $T^{v}_{s_{w}}{\rm SO}(n,\mathbb{R})\subset T_{s_{w}}{\rm
SO}(n,\mathbb{R})$ can be seen from the comparison of formula (3.5) with this
description of the vertical subspace. Due to the $G$-invariance of the
constructions, the same is true for an arbitrary point in the orbit through
$s_{w}$.

We sum up our observations in the following proposition:

######  Proposition 3.3.

The full symmetric Toda system on the matrices with non-distinct eigenvalues
induces a dynamical system on the partial flag manifold ${\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$ $($where $i_{1},\dots,i_{k}$ are the
multiplicities of eigenvalues$)$. This dynamical system is equal to the
gradient flow of a Morse function on ${\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$.

We conclude this section by an important observation: consider the natural
projection (see Section 2.1)

$\displaystyle\pi\colon\ {\rm Fl}_{n}(\mathbb{R})\to{\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R}).$

As we have shown above, when the eigenvalues of $\Lambda$ are given by (3.4),
the spaces on both sides of this diagram can be equipped with a Morse–Bott
function and a Riemannian metric, which give rise to the gradient vector
fields $\widetilde{M}$ on them. In both cases these structures are pulled to
the flag spaces (full or partial) from the group ${\rm SO}(n,\mathbb{R})$. On
the other hand we can use this projection $\pi$ to pull the structures we need
from ${\rm Fl}_{n}(\mathbb{R})$ to the partial flag space. It is clear that in
either way we shall obtain the same result. In other words, when the
eigenvalues of $\Lambda$ are not distinct, the function $F$ on ${\rm
SO}(n,\mathbb{R})$ is a Morse–Bott function, which induces a Morse function on
the base ${\rm Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$.

##  4 The asymptotic behavior

In this section we prove the main theorem of the paper: the asymptotic
behavior of the trajectories of the vector field induced on ${\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$ by the FS Toda lattice on the set of
symmetric matrices with multiple eigenvalues (with multiplicities
$i_{1},\dots,i_{k}$) is completely determined by the Bruhat order on
$S_{n}^{I}$, see Theorem 4.2. 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 ${\rm Gr}_{2,4}(\mathbb{R})$. The general
case is treated at the end.

###  4.1 Example 1: projective spaces

Let $L$ be the Lax matrix of the FS Toda lattice; $\Lambda$ is the diagonal
matrix of its eigenvalues so that

$\displaystyle L=\Psi\Lambda\Psi^{-1},\qquad\Psi\in{\rm SO}(n,\mathbb{R}).$

As we have explained this relation allows one to introduce an analogue of the
FS Toda lattice on the partial flag space ${\rm
Fl}_{i_{1},\dots,i_{k}}(\mathbb{R})$, where $i_{1},i_{2},\dots,i_{k}$ are the
multiplicities of the eigenvalues of $L$. The geometry of the flag space (and
hence the structure of Toda trajectories) depends to a great measure on the
partition $I=(i_{1},i_{2},\dots,i_{k})$. 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 $\lambda<\mu$ with
multiplicities $1$ and $n-1$, respectively (we can also assume that
$\lambda+(n-1)\mu=0$). The flag spaces here are equal to the projective spaces
$\mathbb{R}P^{n-1}$.

We begin with the smallest possible dimension $n=3$ and eigenvalues
$\lambda_{1}=\lambda<\lambda_{2}=\lambda_{3}=\mu$. In this case we consider
the system on the projective plane $\mathbb{R}P^{2}$. One can see that there
are exactly $3$ singular points of the vector field induced by the Toda system
on $\mathbb{R}P^{2}$ corresponding to the permutations $0=(\lambda,\mu,\mu)$,
$1=(\mu,\lambda,\mu)$ and $2=(\mu,\mu,\lambda)$; direct calculations in the
local coordinates at these points (it is enough to transfer the coordinates
from the unit of ${\rm SO}(3,\mathbb{R})$ and take the directions
complementary to the vertical) show that the Morse function takes distinct
values at these points so that their Morse indices are $2$, $1$ and $0$,
respectively (see formula (3.5)). Recall that the 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 of connecting $0$, $1$ and $2$ by trajectories:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2.}$

It is easy to describe the structure of the vector field here: its pullback
from $\mathbb{R}P^{2}$ to $S^{2}$ has 6 singular points, which can be
identified with the intersections of the sphere with the coordinate axes; the
indices of these points are $0,1$ and $2$ so that the opposite points have the
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 the Toda field on $\mathbb{R}P^{2}$, and use the same
name for similar fields on arbitrary projective spaces.

In the general case $n>3$ we shall have an analogous picture: just remark that
the diagonal embedding of ${\rm SO}(n,\mathbb{R})$ into ${\rm
SO}(n+1,\mathbb{R})$, given by

$\displaystyle\Psi\mapsto\begin{pmatrix}\Psi&0\\\ 0&1\end{pmatrix},$

corresponds on the one hand, to the hyperplane embedding of projective spaces
$\mathbb{R}P^{n-1}\to\mathbb{R}P^{n}$, and on the other hand, it is induced by
the evident embedding of symmetric $n\times n$ matrices into the space of the
symmetric $(n+1)\times(n+1)$ matrices:

$\displaystyle L\to\begin{pmatrix}L&0\\\ 0&\mu\end{pmatrix}.$

This embedding sends the symmetric matrices with the spectrum

$\displaystyle\lambda<\underbrace{\mu=\mu=\dots=\mu}_{n-1\ \text{times}}$

to the symmetric matrices with the spectrum

$\displaystyle\lambda<\underbrace{\mu=\mu=\dots=\mu}_{n\ \text{times}}.$

Also, one can see that this inclusion intertwines Toda systems on bigger
matrices and smaller matrices. Thus, the Toda vector field on
$\mathbb{R}P^{n-1}$ coincides with the restriction of the Toda field from
$\mathbb{R}P^{n}$, and the latter field has one more singular point in the
complement of $\mathbb{R}P^{n-1}$ in $\mathbb{R}P^{n}$. This new point has the
maximum possible index. So we see that in the case of a generic projective
space $\mathbb{R}P^{n}$ the phase diagram of singular points and trajectories
between them is as follows

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{n\,.}$

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 can see, when
$t\to\pm\infty$ the matrix $L$ tends to a diagonal matrix with eigenvalues
$\lambda$ and $\mu$ of multiplicities $1$ and $n-1$, respectively, in which
$\lambda$ stands on an arbitrary position (depending on the corresponding
multiset permutation).

###  4.2 Example 2: $\boldsymbol{{\rm Gr}_{2,4}(\mathbb{R})}$

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 Euclidian space $\mathbb{R}^{n}$. 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 $4\times 4$ symmetric Lax
matrix $L$ are $\lambda_{1}=\lambda_{2}=\lambda$,
$\lambda_{3}=\lambda_{4}=\mu$ and consider the induced gradient system on
${\rm Gr}_{2,4}(\mathbb{R})$.

We begin with a detailed description of the Grassmannian. This is the manifold
parameterizing all $2$-dimensional subspaces in $\mathbb{R}^{4}$. As we
mentioned in Section 2.1 one has a homeomorphism of spaces

$\displaystyle{\rm Gr}_{2,4}(\mathbb{R})={\rm SO}(4,\mathbb{R})/{\rm
SO}(4,\mathbb{R})\cap({\rm O}(2,\mathbb{R})\times{\rm O}(2,\mathbb{R})).$

The identification is given by choosing an orthogonal basis in
$\mathbb{R}^{4}$:

$\displaystyle e_{1}=\left(\begin{matrix}1\\\ 0\\\ 0\\\
0\end{matrix}\right),\qquad e_{2}=\left(\begin{matrix}0\\\ 1\\\ 0\\\
0\end{matrix}\right),\qquad e_{3}=\left(\begin{matrix}0\\\ 0\\\ 1\\\
0\end{matrix}\right),\qquad e_{4}=\left(\begin{matrix}0\\\ 0\\\ 0\\\
1\end{matrix}\right).$

We can regard the $2$-dimensional plane spanned by $e_{1}$, $e_{2}$ as the
“origin” $x$ of ${\rm Gr}_{2,4}(\mathbb{R})$ so that all the other planes in
$\mathbb{R}^{4}$ are equal to translations of $x$ by appropriate elements of
${\rm SO}(4,\mathbb{R})$. The stabilizer of $x$ in ${\rm SO}(4,\mathbb{R})$ is
the subgroup $B_{x}$ of ${\rm SO}(4,\mathbb{R})$ comprised of the elements of
the form

$\displaystyle\left(\begin{matrix}\cos(t_{a})&\sin(t_{a})&0&0\\\
-\sin(t_{a})&\cos(t_{a})&0&0\\\ 0&0&\cos(t_{b})&\sin(t_{b})\\\
0&0&-\sin(t_{b})&\cos(t_{b})\\\
\end{matrix}\right),\qquad\left(\begin{matrix}-\cos(t_{a})&\sin(t_{a})&0&0\\\
\sin(t_{a})&\cos(t_{a})&0&0\\\ 0&0&-\cos(t_{b})&\sin(t_{b})\\\
0&0&\sin(t_{b})&\cos(t_{b})\end{matrix}\right).$

As one can easily see, the plane $x$ is preserved by the action of these
elements:

$\displaystyle e_{1}B_{x}=e_{1}\cos(t_{a})+e_{2}(-\sin(t_{a})),\qquad
e_{2}B_{x}=e_{1}\sin(t_{a})+e_{2}\cos(t_{a}).$

Now consider the generic point $y\in{\rm Gr}_{2,4}(\mathbb{R})$: as we know
$y=g^{y}x$ for some $g^{y}\in{\rm SO}(4,\mathbb{R})$. Then the stabilizer of
$y$ is equal to the conjugation of $B_{x}$ by $g^{y}$:

$\displaystyle B_{y}=g^{y}B_{x}(g^{y})^{-1}.$

This simple observation allows one to find a suitable description of the
tangent space of ${\rm Gr}_{2,4}(\mathbb{R})$ at $y$:

$\displaystyle T_{y}{\rm
Gr}_{2,4}(\mathbb{R})\cong\mathfrak{so}(4)/\operatorname{Ad}_{g^{y}}(\mathfrak{g}),$

where $\mathfrak{g}$ is the Lie algebra of the group $B_{x}$ and
$\operatorname{Ad}_{g}$, $g\in{\rm SO}(4,\mathbb{R})$ denotes the adjoint
action of the group ${\rm SO}(4,\mathbb{R})$ on its Lie algebra
$\mathfrak{so}(4)$ (by conjugation of matrices).

We choose the standard basis in $\mathfrak{so}(4)$ (the space of all $4\times
4$ anti-symmetric matrices) so that the generic element $X$ of
$\mathfrak{so}(4)$ takes the form

$\displaystyle X=\begin{pmatrix}0&\theta_{1}&\theta_{2}&\theta_{3}\\\
-\theta_{1}&0&\theta_{4}&\theta_{5}\\\ -\theta_{2}&-\theta_{4}&0&\theta_{6}\\\
-\theta_{3}&-\theta_{5}&-\theta_{6}&0\end{pmatrix}.$

The functions $\theta_{1},\dots,\theta_{6}$ are coordinates in
$\mathfrak{so}(4)$. The exponential map allows one to pull these coordinates
to an open neighbourhood of the unit matrix in ${\rm SO}(4,\mathbb{R})$ and
the right translations by the group elements then give coordinate systems in
open neighbourhoods of the points in ${\rm SO}(4,\mathbb{R})$ and on tangent
spaces at these points. Below we shall use these coordinates extensively
without explanation, preserving (by a slight abuse of the language) their
original names.

This notation is well-suited for the description of the tangent spaces of
${\rm Gr}_{2,4}(\mathbb{R})$ at the singular points of the FS Toda lattice:
recall that these singular points are equal to the projections to ${\rm
Gr}_{2,4}(\mathbb{R})$ of the singular fibres of the FS Toda lattice in ${\rm
SO}(4,\mathbb{R})$ (or in ${\rm Fl}_{4}(\mathbb{R})$). 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 ${\rm SO}(4,\mathbb{R})$ of the 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 [5]). The conjugation by
such matrices induces permutations of the coordinates
$\theta_{1},\dots,\theta_{6}$.

With the help of these observations one can draw the following table,
describing the tangent spaces at the critical points, values of the Morse
function $F_{2,4}$, positive and negative directions of Hessians at these
critical points, i.e., of the singular points of the Toda vector field on
${\rm Gr}_{2,4}(\mathbb{R})$. We index the singular points on ${\rm
Gr}_{2,4}(\mathbb{R})$ by the corresponding permutations of the eigenvalues’
set $(\lambda,\lambda,\mu,\mu)$, $\lambda<\mu$.

Point | Morse index | Value of $F_{2,4}$ | $+$ | $-$ |  Minors2pt2pt
---|---|---|---|---|---
$(\lambda,\lambda,\mu,\mu)$ | 0 | $\lambda+5\mu$ |  $\theta_{2}$, $\theta_{3}$, $\theta_{4}$, $\theta_{5}$ | 0 |  $\psi_{13}$, $\psi_{14}$, $\psi_{41}$, $\psi_{42}$2pt2pt
$(\lambda,\mu,\lambda,\mu)$ | 1 | $2\lambda+4\mu$ |  $\theta_{1}$, $\theta_{4}$, $\theta_{6}$ | $\theta_{2}$ |  $\psi_{13}$, $\psi_{14}$, $\psi_{41}$, $\psi_{42}$2pt2pt
$(\lambda,\mu,\mu,\lambda)$ | 2 | $3\lambda+3\mu$ |  $\theta_{5}$, $\theta_{6}$ |  $\theta_{1}$, $\theta_{3}$ |  $\psi_{13}$, $\psi_{14}$, $\psi_{43}$, $\psi_{44}$2pt2pt
$(\mu,\lambda,\lambda,\mu)$ | 2 | $3\lambda+3\mu$ |  $\theta_{1}$, $\theta_{3}$ |  $\theta_{5}$, $\theta_{6}$ |  $\psi_{11}$, $\psi_{12}$, $\psi_{41}$, $\psi_{42}$2pt2pt
$(\mu,\lambda,\mu,\lambda)$ | 3 | $4\lambda+2\mu$ | $\theta_{2}$ |  $\theta_{1}$, $\theta_{4}$, $\theta_{6}$ |  $\psi_{11}$, $\psi_{12}$, $\psi_{43}$, $\psi_{44}$2pt2pt
$(\mu,\mu,\lambda,\lambda)$ | 4 | $5\lambda+\mu$ | 0 |  $\theta_{2}$, $\theta_{3}$, $\theta_{4}$, $\theta_{5}$ |  $\psi_{11}$, $\psi_{12}$, $\psi_{43}$, $\psi_{44}$2pt2pt

In the fourth and the fifth columns we use the coordinates
$\theta_{1},\dots,\theta_{6}$ on ${\rm SO}(4,\mathbb{R})$ introduced earlier
and project them to ${\rm Gr}_{2,4}(\mathbb{R})$: on the level of tangent
spaces the differential $d\pi$ of this projection is a linear epimorphism and
we identify $T_{\pi(x)}{\rm Gr}_{2,4}(\mathbb{R})$ with a linear subspace,
which maps isomorphically onto it. The last column gives the list of certain
invariant surfaces of the Toda field in ${\rm SO}(4,\mathbb{R})$, which
contain the corresponding “singular fibre” of the Toda field (i.e., the fibre
of the projection $\pi\colon{\rm SO}(4,\mathbb{R})\to{\rm
Gr}_{2,4}(\mathbb{R})$, on which the Toda field vanishes identically). As one
knows (see [5]) such surfaces can be given by the equations $\psi_{ij}=0$;
there are more general surfaces of this sort, in which the equations are given
by the condition that a certain minor of the matrix $\Psi=(\psi_{ij})$
vanishes (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 Fig. 1, in which the case of the singular
point in ${\rm Gr}_{2}(4,\mathbb{R})$, corresponding to the permutation
$(\lambda,\mu,\lambda,\mu)$, is considered (below we shall often identify such
points with the corresponding permutations of eigenvalues).

Figure 1: The bundle over $(\lambda,\mu,\lambda,\mu)\in{\rm
Gr}_{2,4}(\mathbb{R})$.

In this figure the little circle at the bottom represents the point in ${\rm
Gr}_{2,4}(\mathbb{R})$ that corresponds to the permutation
$(\lambda,\mu,\lambda,\mu)$ of the eigenvalues. The small squares above
correspond to the permutation matrices in ${\rm SO}(4,\mathbb{R})$ that
project into the chosen point. The arrows represent the coordinate directions
in the tangent space of the group at the chosen points. The red (bold) arrows
correspond to the directions inside the fibre, and the black (thin) arrows to
all the rest. As one knows (see [5]) the coordinates $\theta_{i}$ are (up to
infinitesimal correction terms) the canonical coordinates, in which the Morse
function takes the form of the sums of the 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 (the red
arrows in fact correspond to the directions which lie in the kernel of the
Hessian 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 the Toda flow on ${\rm
SO}(4,\mathbb{R})$, its projection to the Grassmann space is an invariant set
of the generalized flows. Using this simple observation we come up with the
following diagram, see Fig. 2, representing the flows in ${\rm
Gr}_{2,4}(\mathbb{R})$; the 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 the points with the Morse indices that
differ by $3$ and one $3$-parametric family between the lowest and the highest
points, which we have omitted to make the diagram more readable.

$\textstyle{{\mathrm{ind}=4}}$$\textstyle{{\mu\mu\lambda\lambda}}$$\textstyle{{\mathrm{ind}=3}}$$\textstyle{{\mu\lambda\mu\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{ind}=2}}$$\textstyle{{\mu\lambda\lambda\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\lambda\mu\mu\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{ind}=1}}$$\textstyle{{\lambda\mu\lambda\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{ind}=0}}$$\textstyle{{\lambda\lambda\mu\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 2: The generalized Toda flow on ${\rm Gr}_{2}(4,\mathbb{R})$.

As one can see, this diagram coincides with the Hasse diagram of the Bruhat
order on multiset permutations. Below (see Theorem 4.2), 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 ${\rm Gr}_{2,4}(\mathbb{R})$, which correspond to the
permutations $(\lambda,\lambda,\mu,\mu)$ and $(\lambda,\mu,\lambda,\mu)$ 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

$\displaystyle\Sigma=(\psi_{13}=0)\cap(\psi_{14}=0)\cap(\psi_{41}=0)\cap(\psi_{42}=0),\qquad\Sigma\subset{\rm
SO}(4,\mathbb{R})$

is invariant with respect to the ${\rm O}(2,\mathbb{R})\times{\rm
O}(2,\mathbb{R})$-action and also is preserved by the Toda flow. Thus, it
projects to an invariant surface $\widetilde{\Sigma}$ in ${\rm
Gr}_{2,4}(\mathbb{R})$, which contains only the above mentioned singular
points on the base. Of course, $\Sigma$ and $\widetilde{\Sigma}$ are singular
varieties in ${\rm SO}(4,\mathbb{R})$ and ${\rm Gr}_{2,4}(\mathbb{R})$, so for
our purposes it is enough to compute their dimensions at the generic point.
The generic matrix $\Psi$ from $\Sigma$ has the following form

$\displaystyle\Psi=\left(\begin{matrix}\psi_{11}&\psi_{12}&0&0\\\
\psi_{21}&\psi_{22}&\psi_{23}&\psi_{24}\\\
\psi_{31}&\psi_{32}&\psi_{33}&\psi_{34}\\\
0&0&\psi_{43}&\psi_{44}\end{matrix}\right).$

Moreover, $\Psi$ being an orthogonal matrix the dimension of the surface
$\Sigma$ at the 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
${\rm SO}(4,\mathbb{R})$ 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 $\Sigma$.

Thus, the dimension $\widetilde{\Sigma}$ (at the generic point) is equal to
$1$. On the other hand $\widetilde{\Sigma}$ is invariant with respect to the
Toda field on ${\rm Gr}_{2,4}(\mathbb{R})$; since it is $1$-dimensional, it
should consist of a finite number of trajectories connecting singular points
in it. Thus, the points $(\lambda,\lambda,\mu,\mu)$ and
$(\lambda,\mu,\lambda,\mu)$ must be connected by a discrete set of
trajectories.

###  4.3 The general case

Finally, let us consider the most general distribution of the eigenvalues of
$L$; let us fix the multi-index $I=(i_{1},i_{2},\dots,i_{k})$ such that
$0<i_{j}$, $i_{1}+\dots+i_{k}=n$. As we have observed earlier, the set of all
real symmetric matrices $L$ with the given set $\Lambda$ of eigenvalues
$\lambda_{1}<\lambda_{2}<\dots<\lambda_{k}$, such that $\lambda_{j}$ has
multiplicity $i_{j}$, is homeomorphic to the flag space ${\rm
Fl}_{I}(\mathbb{R})$ (see Section 2.1). We use the same symbol $\Lambda$ for a
diagonal matrix with the set of eigenvalues equal to $\Lambda$.

As we have shown earlier (see Section 3.2) the FS Toda lattice on the space of
symmetric matrices conjugate with $\Lambda$ is induced by a gradient vector
field $\xi_{I}$ on the partial flag space ${\rm Fl}_{I}(\mathbb{R})$: the
image of the usual Toda vector field $\xi$ on the full flag variety ${\rm
Fl}_{n}(\mathbb{R})$. Moreover, the potential function generating this field
is a Morse function; its singular points correspond to those fibres of the
natural projection ${\rm Fl}_{n}(\mathbb{R})\to{\rm Fl}_{I}(\mathbb{R})$ on
which the Toda vector field vanishes identically (singular fibres with respect
to the Toda potential function on ${\rm Fl}_{n}(\mathbb{R})$). As one knows,
in this case the trajectories connect the singular points of the field when
$t\to\pm\infty$. 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 [5]) that the singular points of the
Toda vector field on the full flag space ${\rm Fl}_{n}(\mathbb{R})$ with the
Lax matrix, whose eigenvalues are all distinct, correspond to the permutation
matrices in ${\rm SO}(n,\mathbb{R})$, i.e., matrices which permute the basis
vectors in $\mathbb{R}^{n}$ and if necessary, multiply them by $-1$ (to make
sure the determinant is positive). In order to understand the structure of the
singularities in ${\rm Fl}_{I}(\mathbb{R})$, it is convenient to look at the
bundle

$\displaystyle{\rm SO}(n,\mathbb{R})\to{\rm Fl}_{I}(\mathbb{R})$

with the fibres isomorphic to $G={\rm SO}(n,\mathbb{R})\bigcap({\rm
O}(i_{1},\mathbb{R})\times\dots\times{\rm O}(i_{k},\mathbb{R}))$. The vector
field $\xi$ 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 $u\in
S_{i_{1}}\times\dots\times S_{i_{k}}$. Comparing this description with Section
2.3 we conclude that the following proposition holds:

######  Proposition 4.1.

The singular points of the Morse field induced on ${\rm Fl}_{I}(\mathbb{R})$
by the FS Toda lattice with eigenvalues $\Lambda$ $($of multiplicities $I)$
are indexed by the multiset permutations $S_{n}^{I}$.

Now it is our purpose to describe the trajectories connecting different
singular points in ${\rm Fl}_{I}(\mathbb{R})$. We shall prove the following
statement:

######  Theorem 4.2.

Let $\psi,\omega\in S_{n}^{I}$ be two multiset permutations. Then the
corresponding singular points of the FS Toda lattice in ${\rm
Fl}_{I}(\mathbb{R})$ will be connected by a trajectory, if and only if
$\psi\prec\omega$ in the Bruhat order on $S_{n}^{I}$ $($see Section 2.3).
Moreover, the dimension of the subset swept by the trajectories connecting
these points is equal to the length of the path in the Hasse diagram of
$S_{n}^{I}$.

###### Proof.

First of all observe that the projection ${\rm Fl}_{n}(\mathbb{R})\to{\rm
Fl}_{I}(\mathbb{R})$ maps the Toda field $\xi$ to the field $\xi_{I}$; hence
it sends invariant subvarieties in the full flag space to invariant subsets in
${\rm Fl}_{I}(\mathbb{R})$. In particular, it means that the Schubert cells in
${\rm Fl}_{I}(\mathbb{R})$ are preserved by the generalized Toda flow since
the Schubert cells in the full flags are. Moreover, since the Schubert cells
in ${\rm Fl}_{n}(\mathbb{R})$ coincide with the unstable subspace of $\xi$
(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 the partial flags coincide with the stable subspaces of $\xi_{I}$:
this follows for example from the fact that the cells in ${\rm
Fl}_{I}(\mathbb{R})$ are the homeomorphic images of the minimal cells in ${\rm
Fl}_{n}(\mathbb{R})$, see the end of Section 2.4 (also compare formula (3.5)).

Similarly, the stable submanifolds of $\xi_{I}$ (i.e., the subsets spanned by
the outgoing trajectories of $\xi_{I}$) in ${\rm Fl}_{I}(\mathbb{R})$ coincide
with the dual Schubert cells in this space since this is true for the dual
cells in ${\rm Fl}_{n}(\mathbb{R})$ (this was proved in [5]). However, we know
(see Section 2.4 again) that the Schubert cell and dual Schubert cell in the
flag space intersect if and only if the corresponding multiset permutations
are comparable in the Bruhat order. The statement about the dimensions follows
from the transversality of these intersections. ∎

We conclude by the simple observation concerning the picture that emerges on
the level of the Lax matrices:

######  Corollary 4.3.

The Toda flow $t\to L(t)$ on symmetric matrices converges to a diagonal matrix
when $t\to\pm\infty$ $($the set of eigenvalues of these matrices is fixed$)$.
Two such matrices are connected by a trajectory if and only if the
corresponding permutations of the eigenvalues are comparable with respect to
the Bruhat order on permutations $($or permutations with repetitions if there
are multiple eigenvalues$)$.

##  5 Further observations

As we have shown above (see Section 3), the FS Toda lattice on symmetric
matrices with non-distinct 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 a 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 for the
system on $\mathbb{R}P^{2}$, we obtain a new integral of motion:

$\displaystyle
I_{(RP^{2},\lambda_{1}=\lambda_{2})}=\frac{1}{(\mu-\lambda)}\frac{\psi_{23}^{2}}{\psi_{13}^{2}\psi_{33}^{2}}.$

Here $\lambda_{1}=\lambda_{2}=\lambda$ and $\lambda_{3}=\mu$, $\lambda<\mu$
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:

$\displaystyle
I_{(RP^{2},\lambda_{1}=\lambda_{2})}(a_{ij})=\frac{a_{12}a_{23}}{a_{13}^{3}}.$

On the other hand, one can show that in this case the chopping procedure (see
[8] for example) does not give any integrals which are 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.

### Acknowledgments

The authors would like to thank G. Koshevoy for the fruitful discussion. We
also would like to thank the referees, whose remarks helped in a great measure
to improve the paper. The work of Yu.B. Chernyakov was supported by grant
RFBR-15-01-08462. The work of G.I. Sharygin was supported by grant
RFBR-15-01-05990. The work of A.S. Sorin was partially supported by RFBR
grants 15-52-05022-Arm-a and 16-52-12012-NNIO-a.

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