The Full Symmetric Toda Flow and Intersections of Bruhat Cells
In this short note, we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements 𝓌, 𝓌′ in the Weyl group 𝑊(𝖌), the corresponding real Bruhat cell 𝑋𝓌 intersects with the dual Bruhat cell 𝑌𝓌′ iff 𝓌 ≺ 𝓌′ in the Bruh...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
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Інститут математики НАН України
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| Цитувати: | The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Yuri B. Chernyakov, Georgy I. Sharygin, Alexander S. Sorin and Dmitry V. Talalaev. SIGMA 16 (2020), 115, 8 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859465075525419008 |
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| author | Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. |
| author_facet | Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. |
| citation_txt | The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Yuri B. Chernyakov, Georgy I. Sharygin, Alexander S. Sorin and Dmitry V. Talalaev. SIGMA 16 (2020), 115, 8 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this short note, we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements 𝓌, 𝓌′ in the Weyl group 𝑊(𝖌), the corresponding real Bruhat cell 𝑋𝓌 intersects with the dual Bruhat cell 𝑌𝓌′ iff 𝓌 ≺ 𝓌′ in the Bruhat order on 𝑊(𝖌). Here 𝖌 is a normal real form of a semisimple complex Lie algebra 𝖌ℂ. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.
|
| first_indexed | 2026-03-12T14:01:24Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 115, 8 pages
The Full Symmetric Toda Flow
and Intersections of Bruhat Cells
Yuri B. CHERNYAKOV †1†2†3, Georgy I. SHARYGIN †1†2†4, Alexander S. SORIN †2†5†6
and Dmitry V. TALALAEV †1†4†7
†1 Institute for Theoretical and Experimental Physics, Bolshaya Cheremushkinskaya 25,
117218 Moscow, Russia
E-mail: chernyakov@itep.ru, sharygin@itep.ru, talalaev@itep.ru
†2 Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics,
141980 Dubna, Moscow region, Russia
E-mail: sorin@theor.jinr.ru
†3 Institute for Information Transmission Problems, Bolshoy Karetny per.19, build. 1,
127994, Moscow, Russia
†4 Lomonosov Moscow State University, Faculty of Mechanics and Mathematics,
GSP-1, 1 Leninskiye Gory, Main Building, 119991 Moscow, Russia
†5 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),
Kashirskoye shosse 31, 115409 Moscow, Russia
†6 Dubna State University, 141980 Dubna, Moscow region, Russia
†7 Centre of integrable systems, P.G. Demidov Yaroslavl State University,
150003, 14 Sovetskaya Str., Yaroslavl, Russia
Received July 13, 2020, in final form November 02, 2020; Published online November 11, 2020
https://doi.org/10.3842/SIGMA.2020.115
Abstract. In this short note we show that the Bruhat cells in real normal forms of
semisimple Lie algebras enjoy the same property as their complex analogs: for any two
elements w, w′ in the Weyl group W (g), the corresponding real Bruhat cell Xw intersects
with the dual Bruhat cell Yw′ iff w ≺ w′ in the Bruhat order on W (g). Here g is a normal
real form of a semisimple complex Lie algebra gC. Our reasoning is based on the proper-
ties of the Toda flows rather than on the analysis of the Weyl group action and geometric
considerations.
Key words: Lie groups; Bruhat order; integrable systems; Toda flow
2020 Mathematics Subject Classification: 22E15; 70H06
1 Introduction and the statement of result
1.1 General remarks
Schubert cells in standard flag manifolds and their generalizations to arbitrary Lie groups –
Bruhat cells – play an important role in algebraic geometry and representation theory. In the
complex case their structure is well understood and has been extensively studied in the last 40
years.
It is natural to suppose that the intersection properties (described below) of real Bruhat
cells are similar to the ones in the complex case. Although this has been implicitly assumed by
many authors, the proof has not been accurately written down yet (or at least we couldn’t find
a proper reference). The modest purpose of the present note is to fill this regrettable gap, at
mailto:chernyakov@itep.ru
mailto:sharygin@itep.ru
mailto:talalaev@itep.ru
mailto:sorin@theor.jinr.ru
https://doi.org/10.3842/SIGMA.2020.115
2 Yu.B. Chernyakov, G.I. Sharygin, A.S. Sorin and D.V. Talalaev
least for the normal real forms, and to show the close relation of the full symmetric Toda system
with the geometry of Bruhat cells.
1.2 Preliminaries on Bruhat decomposition
The Bruhat cells (or Schubert cells) are open dense subsets in algebraic subvarieties in flag spaces
(Schubert varieties), enumerated by the elements of the corresponding Weyl group. Namely, the
Schubert cell XC
w , corresponding to w ∈W (gC), is equal to the positive Borel subgroup B+
C orbit
of the element [w] ∈ Fl(gC) = GC/B
+
C . Here [w] is the point in Fl(gC) represented by the element
w̃ in the normalizer N(h) of the Cartan subalgebra corresponding to w ∈W (gC). Similarly the
dual Schubert cell Y C
w corresponding to w is the orbit B−C · [w] of the negative Borel subgroup.
It turns out that the properties of the Bruhat cells are closely related with the combinatorics
of the corresponding Weyl group, in particular with the Bruhat order on it. Recall that the
Bruhat order in a Coxeter group W with the set S = {σ1, . . . , σN} of generators is the unique
partial order on W determined by the following rule: we say that w precedes w′ if a minimal
word s(w′) (in letters σk) representing w′ has the form σi1 · · ·σik · s(w) · σj1 · · ·σjm (where s(w)
is a minimal word for w). The lengths of these words are called lengths of the elements and
denoted l(w), l(w′) respectively.
Now using complex geometry and algebraic groups theory one can show that XC
w ⊆ X
C
w′ iff
w ≺ w′ in Bruhat order (here X denotes the closure of X). Besides this, one can also show
that Y C
w ∩ XC
w′ 6= ∅ iff w′ ≺ w in Bruhat order (see [3, 10]). Moreover, in the latter case the
intersection is always transversal and the (complex) dimension of this variety is equal to the
difference l(w)− l(w′). In particular, the complex dimensions of the cells are equal to l(w) and
dimC Fl(gC)− l(w′) respectively (see the cited papers).
1.3 Preliminaries on Cartan decomposition
Let g be a real Lie algebra whose complexification is equal to gC (i.e., g is a real form of gC).
Let G be the corresponding real Lie group. For any such real form of gC one has the following
Cartan decomposition:
g = k⊕ p,
where k is the Lie algebra of a maximal compact subgroup, and p is the algebraic complement
of k on which the Killing form of g is positive-definite.
Alternatively (and maybe more properly) one can define the Cartan decomposition with the
help of a Cartan involution. Namely let θ be a Lie algebra automorphism θ : g → g, such that
θ2 = 1 and the quadratic form B(X, θY ) where B is the Killing form, is positive definite. Then
k = {X ∈ g | θ(X) = −X}, p = {Y ∈ p | θ(Y ) = Y }.
The Cartan decomposition of a real form g is unique (up to an isomorphism) and one has
[k, k] ⊆ k, [p, p] ⊆ k, [k, p] ⊆ p.
Let a be a maximal commutative Lie subalgebra in p. Then the real form g is called normal or
non-split if a is a maximal commutative subalgebra in whole g.
In this case we shall denote this commutative subalgebra by h and call it Cartan subalgebra.
The normal real form enjoys most of the properties of the complex algebra: one can determine
the root system with respect to h and decompose g into root subspaces:
g = h⊕
∑
α∈Φ
gα,
The Full Symmetric Toda Flow and Intersections of Bruhat Cells 3
where α ∈ Φ ⊂ h∗ are the roots and gα are the root subspaces, i.e.,
gα = {X ∈ g | [H,X] = α(H)X, H ∈ h}.
In the case of normal real form these subspaces are 1-dimensional and fixing a basis in h so that
one can tell which roots are positive and which are negative, one has the following decompositions
of k and p in the corresponding Cartan decomposition:
k =
∑
α∈Φ+
R(eα + e−α), p = h⊕
∑
α∈Φ+
R(eα − e−α), (1)
where Φ+ is the set of positive roots and eα, e−α are basis vectors in gα, g−α such that
e−α = θ(eα). It is easy to see that in this case the Cartan decomposition of the correspond-
ing complex Lie algebra gC is given by the complexification of the corresponding elements of g.
The Weyl group of g, which coincides with the Weyl group of gC, can now be defined as
the quotient group W (g) = NK(h)/ZK(h), where NK(h) and ZK(h) are the normalizer and the
centralizer of h in the maximal compact subgroup K of G, corresponding to k. It follows from
the maximality of h that these groups are discrete and that ZK(h) is a normal subgroup in it.
1.4 Main result
One can define real Bruhat cells Xw and dual real Bruhat cells Yw in Fl(G) = G/B+ = K/ZK(h)
(where B+ is the real Borel subgroup) as the orbits of [w] with respect to B+ and B−. In what
follows below, we shall show that the following proposition is true.
Proposition 1. The intersection Xw ∩ Yw′ is nonempty if and only if w′ ≺ w in Bruhat order;
moreover, if it is not empty, its (real) dimension is equal to the difference l(w)− l(w′).
The explanation of this fact that we found in literature is based on the geometric description
of Schubert cells in terms of matrix ranks, see, e.g., [10]; it works well only in the case of
G = SLn(R). The main purpose of this article is to extend this result to other groups by
applying the connection of this theory with the full symmetric Toda flow. Namely, we are going
to derive the result we need from the general properties of the Toda flow, and not vice-versa: in
our previous work we used the properties of Schubert cells in the standard flag variety (i.e., the
one associated with SLn(R)) in order to show that the phase portrait of this dynamical system
is completely determined by Bruhat order [5].
Also observe that in our other papers [6, 11] we used explicit computations to show that sim-
ilar statements hold for all real groups of rank 2 and for the non-split real form, SO(2, 4). In the
latter case the role of Weyl group is played by the group W (g, k) = NK(a)/ZK(a) where NK , ZK
are the (non-discrete) normalizer and centralizer of a in K. One can show that W (g, k) is again
a Coxeter group, thus we can introduce Bruhat order on it. Flag variety is replaced by K/ZK(a)
and the (dual) Bruhat cells are defined as orbits of Borel subgroups in flag varieties similarly to
the usual case. However, in this case the dimensions of the root spaces (with respect to a) are
not necessarily equal to 1. This is manifested in the dimensions of the intersections of the cells.
Below we shall describe a possible generalization of our main result to this case.
2 Full symmetric Toda flow and its properies
In this section we recall few properties of the full symmetric Toda system on Lie algebras and
flag spaces. For the connection between non-periodic Toda lattice and the geometric aspects of
Lie groups see, for example [4]. Namely, the full symmetric Toda system is the dynamical system
on p, defined as follows (we follow the exposition from the papers by de Mari, Pedroni [7] and
4 Yu.B. Chernyakov, G.I. Sharygin, A.S. Sorin and D.V. Talalaev
Bloch, Gekhtman [2]): it is the Hamiltonian system with respect to a Poisson structure, pulled
to p from (b+)∗ via the Killing form, and with the Hamiltonian H(L) = Trg
(
ad2
g(L)
)
, L ∈ p.
However, for our purposes it is better to consider only the corresponding Hamiltonian vector
field τ on p; it can be written down in the following commutator form:
τ(L) = [L,M(L)].
Here for
L = X +
∑
α∈Φ+
aα(eα + e−α) ∈ p, X ∈ h, aα ∈ R
(see (1)) we put
M(L) =
∑
α∈Φ+
aα(eα − e−α) ∈ k.
Thus the Toda system can be written in the following Lax form:
L̇ = [L,M(L)]. (2)
Below we shall refer to the map M : p → k as to the Toda projector, and the system (2) as the
full symmetric Toda system on g, or by a slight abuse of terminology, just Toda system on g
since it is clear that it depends only on the real form g (conventionally, the term Toda system
is used to denote the dynamical system determined by (2) on the subspace in p spanned by h
and the simple root spaces).
It turns out that the dynamics of the Toda system (2) is uniquely determined by the following
construction: it is known that almost all (i.e., up to a measure 0 subset) orbits of K on p pass
through a unique (up to the action of N ∈ h) element of h; on the other hand, the vector field τ
is evidently tangent to these orbits (because it is determined by the adjoint action of k). So we
choose a generic element Λ ∈ h such that its centralizer coincides with the centralizer of h and
consider the vector field T at a point Ψ ∈ K, given by
TΛ(Ψ) = −M(AdΨ(Λ)) ·Ψ,
i.e., it is the right translation by Ψ of the element M(AdΨ(Λ)) ∈ k. Then this vector field
gives a dynamical system Ψ = Ψ(t) on K, such that the solution of Toda system on p through
a generic point L = L(0) is given by the formula
L(t) = AdΨ(t)(Λ),
where Λ ∈ h is the eigenmatrix of L (the one fixed above) and Ψ(t) is the trajectory of the
field T on K, such that AdΨ(0)(Λ) = L. It would probably be more accurate to consider instead
of the field T on K an analogous field T ′ on the flag space. But in that case the formula for L(t)
would involve a choice of section F (G)→ K.
2.1 Invariant surfaces of the Toda field
It is our purpose to study the behaviour of the vector field τ on p by studying the equivalent
fields TΛ on K. It has been studied by many authors in the last 25 years so here we shall just
give references for the well-known facts and give brief proofs for those which seem to be less
known.
The Full Symmetric Toda Flow and Intersections of Bruhat Cells 5
One can show (see [2] and [1]) that the vector field TΛ on K is gradient. The corresponding
potential function F = FΛ : K → R is given by
FΛ(Ψ) = Trg(adg(AdΨ(Λ)) adg(N)),
for a suitable element N ∈ h. It turns out that FΛ is a Morse function. The construction also
involves a proper choice of the Riemann structure 〈 , 〉 on K. We shall not use the exact form of
the metric and of the element N here. It is enough to say that the scalar product in question is
left-invariant and that in the basis eα−e−α of the Lie algebra k it coincides with the Killing form
up to scalar factors jα and N is uniquely defined up to a central element of g, look into [5, 7]
for details.
From the very definition of the field TΛ it follows that for a generic Λ the only singular
points of this field coincide with the group NK(h) or with the set of elements of W (g) on the
level of the flag space Fl(G). Further, one can show that there always exists a vast family of
invariant (with respect to TΛ) curves in K, i.e., of 1-dimensional invariant submanifolds of the
Toda flow. In fact, it is possible to give an explicit description of such curves. To this end we let
w ∈ NK(h) ⊂ K be a point and let us fix a root α ∈ Φ+. Consider the smooth path γw = γw,α
through the point w on K given by
γw(t) = exp (t(eα − e−α)) · w.
Proposition 2. The path γw is tangent to the vector field T on K.
Proof. Let us consider the path
γw,α(t) = exp (t(eα − e−α)) · w,
and the corresponding path in the Lie algebra
L(t) = Adγw,α(t)(Λ) = Adexp (t(eα−e−α))(Λw),
where the element Λw ∈ h is given by
Λw = Adw(Λ).
Since the action of h on e±α is diagonal, we can prove by induction that L(t) is an element in
the subspace in p generated by h and eα + e−α. Hence M(L(t)) is proportional to eα− e−α. Let
us introduce a scalar function k(t) by
M(L(t)) = k(t)(eα − e−α).
Then the Toda field T at the point γw(t) is equal to k(t)(eα − e−α) · γw(t) and is proportional
to the tangent field to the trajectory γw,α(t) with coefficient k(t). �
In addition to these invariant curves we shall need the following result, which one can find,
for instance in [7, 9]: Bruhat cells are invariant manifolds of this system. In fact, one can make
this statement more precise, bringing it to the following form.
For every Λ ∈ h we consider the ordering of the elements of W (g) by the values of FΛ; let
w ∈W (g) be the minimal point. Then for any w′ we consider the shifted Bruhat cell Xw
w′ and the
dual shifted Bruhat cell Y w
w′ , i.e., the orbits (B+)w·[w′] and (B−)w·[w′], where (B±)w = w·B±·w−1
and [w′] is the point in Fl(G) corresponding to w′. Then the shifted and dual shifted Bruhat cells
are invariant submanifolds of TΛ. Moreover Bruhat cells are unstable and dual Bruhat cells are
stable submanifolds of this system. See the cited papers for proof.
Before we proceed, let us make the following observation: Bruhat cells, dual Bruhat cells and
their intersections are invariant surfaces of vector fields TΛ for all Λ ∈ h. This follows directly
from the previously cited facts: just observe that if two vector fields are tangent to a surface,
then all their linear combinations also are and notice that the fields TΛ are linear in Λ.
6 Yu.B. Chernyakov, G.I. Sharygin, A.S. Sorin and D.V. Talalaev
3 Proof of the main result
We begin with the following simple observation: if the intersection Xw ∩ Yw′ is nonempty, then
the elements w and w′ are comparable in Bruhat order so that w′ ≺ w. In fact real Bruhat
cells lie inside complex ones and in the complex case intersection of cells is equivalent to the
comparison w′ ≺ w. Also observe, that the real dimension of the intersections of the real Bruhat
cells with dual real Bruhat cells does not exceed l(w)− l(w′) since this is the complex dimension
of the intersection of complex cells.
Further, suppose that Λ ∈ h is such that the minimal point of the Morse function FΛ is equal
to the unit element e ∈ W (g) (this is possible since FΛ(Ψ · w) = FAdw(Λ)(Ψ). So if we choose
Λ′ = Adw(Λ), where w is the current minimal element, then e will be minimum of FΛ′). Then
it follows from the properties of Bruhat cells and Toda system, listed at the end of the previous
section, that if there exists a trajectory k(t) of TΛ which tends to w when t → +∞ and to w′
when t → −∞ then w′ ≺ w. Indeed, in this case the corresponding real Bruhat and dual real
Bruhat cells intersect and hence complex cells intersect too.
Consider now the paths γw,α corresponding to the roots α1, . . . , αN . It is known from the
classical Lie groups theory that there exists t0 ∈ R such that exp(t0(eα − e−α)) = σα, where σα
is the reflection of h, corresponding to α. To verify this one can choose the eigenfunction
basis. Thus, the paths γw,αi connect w and w′ = σα · w. Since these paths coincide (up to
reparametrization) with the trajectories of the Toda system, which is gradient system with
stable points equal to W (g), we conclude from the definition of Bruhat order that there exist
trajectories of Toda system connecting any two elements of W (g) which differ by only one
reflection.
Let now w′ ∈ W (g) be any element equal to the product of exactly two reflections. Let us
prove that there exists a trajectory of the Toda system connecting the unit e ∈ W (g) and w′.
But it follows from the properties of Toda system, listed in the end of previous section, that the
dimension of unstable manifold of w′ is equal to the dimension of the corresponding Bruhat cell,
i.e., equals 2. On the other hand, there can be no trajectories entering w′ from the elements
whose length is greater or equal to 2 (since these points do not verify the condition w ≺ w′).
Similarly, the dimension of the space, spanned by the trajectories, connecting w′ with elements
of length 1, can not exceed 1 (it follows from the first observation in this section). Thus, this
space should consist of the trajectories coming from e (up to a measure zero subset, spanned
by the trajectories from “intermediate” points). It means that the set of such trajectories is
nonempty, i.e., the corresponding Bruhat cell and the dual Bruhat cell intersect. It also follows
that the dimension of this intersection is 2.
Next, if w ≺ w′ and l(w′) − l(w) = 2 we choose Λ′ so that w is minimal (this is done as
before by passing to the function FΛ(Ψ ·w) = FAdw(Λ)(Ψ) for suitable w. One can also say that
this amounts to the change of the set of positive roots or to the action of Weyl group on the
Weyl chambers of h). In this new setting w will play the role of unit in Weyl group and w′ will
correspond to an element of length 2. Then as before, we conclude that there exist trajectories
of TΛ′ from w to w′ and the space of these trajectories is 2-dimensional. Now by the remark we
made in the end of the previous section this surface is also invariant for TΛ, thus, since w is local
minimum and w′ is maximum of FΛ on this surface (in fact FΛ′ = w∗(FΛ) and the Riemannian
structure used to determine the Toda flow is (right) invariant) we conclude that there exists
a 2-dimensional space of trajectories of TΛ from w to w′. And hence the corresponding Bruhat
and dual Bruhat cell intersect along a 2-dimensional variety.
Finally, we proceed by induction: assuming that we know the existence of intersections of all
cells with l(w) − l(w′) ≤ k, we consider first w′ = e, l(w) = k + 1 and conclude (by dimension
counting) that the space of trajectories from e to w is nonempty and has dimension k+1. Then,
using the shift we reduce the general case to this one.
The Full Symmetric Toda Flow and Intersections of Bruhat Cells 7
Observe that in addition to the main proposition we have also proved that two stable points w
and w′ of Toda system on a normal real form are connected by a trajectory iff they are comparable
in Bruhat order.
Remark 3. It seems that the same intersection result can be obtained from the standard
reasoning, based on the combinatoric considerations and on the use of Bott–Samelson varieties
and their generalizations similarly to the Deodhar’s construction for algebraically-closed fields,
see [8].
3.1 Non-split case and further questions
In non-split case we have the problem of identifying the real Bruhat cells with the real part
of complex cells. Thus the very first stage of our inductive process fails. In fact, the Weyl
group W (g, k) in this case coincides with the Weyl group of the smaller root system, namely, of
the so-called Satake projection in which a plays the role of Cartan algebra. Besides this, some
of the root spaces gα can be multidimensional.
On the other hand, most part of the statements from the previous section remain true: the
points w ∈W (g, k) are invariant points of Toda system, their Bruhat cells and dual Bruhat cells
(or their shifted versions) are unstable and stable manifolds of the Toda system (see [9]) and
the paths γα,w are invariant with respect to Toda flow.
Using these facts, one can make the following conjecture, concerning the structure of trajec-
tories of Toda flow on non-split real forms, and the intersections of the Bruhat cells:
Conjecture 4. There exist trajectories connecting w and w′ in W (g, k) iff the corresponding
elements are comparable in Bruhat order, in which case the space of such trajectories coincides
with the intersection of Bruhat and dual Bruhat cell, and its dimension is equal to the sum:
k∑
i=1
dim gαi ,
where w = σαik · · ·σαi1 · w
′. In particular, this sum does not depend on the choice of the
reflections σαi in this formula.
Another interesting question is related to the fact that in Deodhar’s work [8] one obtains
further decomposition of the intersections of Bruhat cells into smaller algebraic varieties. One
can ask if these varieties can also be somehow characterized in terms of the Toda system.
Acknowledgments
The authors thank Sergey Loktev, Vladimir Rubtsov, Evgeniy Smirnov for useful discussions.
The authors would like to express their gratitude to the anonymous referee, whose remarks have
helped us to improve the presentation of this paper.
The work of Yu.B. Chernyakov was partly supported by the grant RFBR-18-02-01081. The
work of G.I. Sharygin was partly supported by the grant RFBR-18-01-00398. The work of
D.V. Talalaev was partly supported by the grant Leader(math) 20-7-1-21-1 of the foundation
for the advancement of theoretical physics and mathematics “BASIS” and within the frame-
work of a development program for the Regional Scientific and Educational Mathematical Cen-
ter of the Yaroslavl State University with financial support from the Ministry of Science and
Higher Education of the Russian Federation (Agreement No. 075-02-2020-1514/1 additional to
the agreement on provision of subsidies from the federal budget No. 075-02-2020-1514).
8 Yu.B. Chernyakov, G.I. Sharygin, A.S. Sorin and D.V. Talalaev
References
[1] Bloch A.M., Brockett R.W., Ratiu T.S., Completely integrable gradient flows, Comm. Math. Phys. 147
(1992), 57–74.
[2] Bloch A.M., Gekhtman M.I., Hamiltonian and gradient structures in the Toda flows, J. Geom. Phys. 27
(1998), 230–248.
[3] Brion M., Lakshmibai V., A geometric approach to standard monomial theory, Represent. Theory 7 (2003),
651–680, arXiv:math.AG/0111054.
[4] Casian L., Kodama Y., Toda lattice, cohomology of compact Lie groups and finite Chevalley groups, Invent.
Math. 165 (2006), 163–208, arXiv:math.AT/0504329.
[5] Chernyakov Yu.B., Sharygin G.I., Sorin A.S., Bruhat order in full symmetric Toda system, Comm. Math.
Phys. 330 (2014), 367–399, arXiv:1212.4803.
[6] Chernyakov Yu.B., Sharygin G.I., Sorin A.S., Bruhat order in the Toda system on so(2, 4): an example of
non-split real form, J. Geom. Phys. 136 (2019), 45–51, arXiv:1712.0913.
[7] De Mari F., Pedroni M., Toda flows and real Hessenberg manifolds, J. Geom. Anal. 9 (1999), 607–625.
[8] Deodhar V.V., On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells,
Invent. Math. 79 (1985), 499–511.
[9] Faybusovich L., Toda flows and isospectral manifolds, Proc. Amer. Math. Soc. 115 (1992), 837–847.
[10] Fulton W., Young tableaux with applications to representation theory and geometry, London Mathematical
Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
[11] Sorin A.S., Chernyakov Yu.B., Sharygin G.I., Phase portrait of the full symmetric Toda system on rank-2
groups, Theoret. and Math. Phys. 193 (2017), 1574–1592, arXiv:1512.05821.
https://doi.org/10.1007/BF02099528
https://doi.org/10.1016/S0393-0440(97)00081-8
https://doi.org/10.1090/S1088-4165-03-00211-5
https://arxiv.org/abs/math.AG/0111054
https://doi.org/10.1007/s00222-005-0492-6
https://doi.org/10.1007/s00222-005-0492-6
https://arxiv.org/abs/math.AT/0504329
https://doi.org/10.1007/s00220-014-2035-8
https://doi.org/10.1007/s00220-014-2035-8
https://arxiv.org/abs/1212.4803
https://doi.org/10.1016/j.geomphys.2018.10.015
https://arxiv.org/abs/1712.0913
https://doi.org/10.1007/BF02921975
https://doi.org/10.1007/BF01388520
https://doi.org/10.2307/2159235
https://doi.org/10.1134/S0040577917110022
https://arxiv.org/abs/1512.05821
1 Introduction and the statement of result
1.1 General remarks
1.2 Preliminaries on Bruhat decomposition
1.3 Preliminaries on Cartan decomposition
1.4 Main result
2 Full symmetric Toda flow and its properies
2.1 Invariant surfaces of the Toda field
3 Proof of the main result
3.1 Non-split case and further questions
References
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| id | nasplib_isofts_kiev_ua-123456789-211005 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:01:24Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. 2025-12-22T09:24:28Z 2020 The Full Symmetric Toda Flow and Intersections of Bruhat Cells. Yuri B. Chernyakov, Georgy I. Sharygin, Alexander S. Sorin and Dmitry V. Talalaev. SIGMA 16 (2020), 115, 8 pages 1815-0659 2020 Mathematics Subject Classification: 22E15; 70H06 arXiv:1810.09622 https://nasplib.isofts.kiev.ua/handle/123456789/211005 https://doi.org/10.3842/SIGMA.2020.115 In this short note, we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements 𝓌, 𝓌′ in the Weyl group 𝑊(𝖌), the corresponding real Bruhat cell 𝑋𝓌 intersects with the dual Bruhat cell 𝑌𝓌′ iff 𝓌 ≺ 𝓌′ in the Bruhat order on 𝑊(𝖌). Here 𝖌 is a normal real form of a semisimple complex Lie algebra 𝖌ℂ. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations. The authors thank Sergey Loktev, Vladimir Rubtsov, and Evgeniy Smirnov for useful discussions. The authors would like to express their gratitude to the anonymous referee, whose remarks have helped us to improve the presentation of this paper. The work of Yu.B. Chernyakov was partly supported by the grant RFBR-18-02-01081. The work of G.I. Sharygin was partly supported by the grant RFBR-18-01-00398. The work of D.V. Talalaev was partly supported by the grant Leader(math) 20-7-1-21-1 of the foundation for the advancement of theoretical physics and mathematics BASIS and within the framework of a development program for the Regional Scienti c and Educational Mathematical Center of the Yaroslavl State University with nancial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-02-2020-1514/1 additional to the agreement on provision of subsidies from the federal budget No. 075-02-2020-1514). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Full Symmetric Toda Flow and Intersections of Bruhat Cells Article published earlier |
| spellingShingle | The Full Symmetric Toda Flow and Intersections of Bruhat Cells Chernyakov, Yuri B. Sharygin, Georgy I. Sorin, Alexander S. Talalaev, Dmitry V. |
| title | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_full | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_fullStr | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_full_unstemmed | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_short | The Full Symmetric Toda Flow and Intersections of Bruhat Cells |
| title_sort | full symmetric toda flow and intersections of bruhat cells |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211005 |
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