On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class
The fundamental invariants for vector ODEs of order ≥ 3 considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant 𝒰, we give a local (point) classifica...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2023
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211971 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class. Johnson Allen Kessy and Dennis The. SIGMA 19 (2023), 058, 29 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859459401785540608 |
|---|---|
| author | Kessy, Johnson Allen The, Dennis |
| author_facet | Kessy, Johnson Allen The, Dennis |
| citation_txt | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class. Johnson Allen Kessy and Dennis The. SIGMA 19 (2023), 058, 29 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The fundamental invariants for vector ODEs of order ≥ 3 considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant 𝒰, we give a local (point) classification for all submaximally symmetric ODEs of C-class with 𝒰 ≢ 0 and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain.
|
| first_indexed | 2026-03-12T12:31:13Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 058, 29 pages
On Uniqueness of Submaximally Symmetric Vector
Ordinary Differential Equations of C-Class
Johnson Allen KESSY and Dennis THE
Department of Mathematics and Statistics, UiT The Arctic University of Norway,
9037 Tromsø, Norway
E-mail: johnson.a.kessy@uit.no, dennis.the@uit.no
Received April 07, 2023, in final form August 01, 2023; Published online August 10, 2023
https://doi.org/10.3842/SIGMA.2023.058
Abstract. The fundamental invariants for vector ODEs of order ≥ 3 considered up to
point transformations consist of generalized Wilczynski invariants and C-class invariants.
An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class
invariant U , we give a local (point) classification for all submaximally symmetric ODEs of
C-class with U ̸≡ 0 and all remaining C-class invariants vanishing identically. Our results
yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie.
Fundamental invariants correspond to the harmonic curvature of the associated Cartan
geometry. A key new ingredient underlying our classification results is an advance concerning
the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for
each irreducible C-class module, we provide an explicit identification of a lowest weight
vector as a harmonic 2-cochain.
Key words: submaximal symmetry; system of ODEs; C-class equations; Cartan geometry
2020 Mathematics Subject Classification: 35B06; 53A55; 17B66; 57M60
1 Introduction
Finite dimensionality of the contact symmetry algebra for scalar ODEs un+1 = f(t, u, u1, . . . , un)
of order n+1 ≥ 4 is a classical result due to Sophus Lie [21] (see also [24, Theorem 6.44]). (We
use jet notation uk instead of the more standard notation u(k) to denote the k-th derivative of u
with respect to t.) The maximal symmetry dimension and the submaximal (i.e., next largest
realizable) symmetry dimension are respectively
M := n+ 5 and S :=
{
M− 1 for n = 4 or 6,
M− 2 otherwise.
The former is realized locally uniquely by the trivial ODE un+1 = 0. For ODEs realizing S, we
have the following result (over C) due to Lie [20] (see also [24, pp. 205–206]): Any submaximally
symmetric scalar ODE of order n+ 1 ≥ 4 is locally contact-equivalent to
(a) a linear equation, or
(b) exactly one of 1
(i) n = 4: 9(u2)
2u5 − 45u2u3u4 + 40(u3)
3 = 0.
This paper is a contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of
Peter J. Olver. The full collection is available at https://www.emis.de/journals/SIGMA/Olver.html
1In [24, p. 206], the scalar ODE 3u2u4 − 5(u3)
2 = 0 is also listed, but this is in fact contact-equivalent to
nun−1un+1 − (n + 1)(un)
2 = 0 when n = 3. We have verified this using Cartan-geometric techniques – details
will be given elsewhere.
mailto:johnson.a.kessy@uit.no
mailto:dennis.the@uit.no
https://doi.org/10.3842/SIGMA.2023.058
https://www.emis.de/journals/SIGMA/Olver.html
2 J.A. Kessy and D. The
(ii) n = 6: 10(u3)
3u7 − 70(u3)
2u4u6 − 49(u3)
2(u5)
2 + 280u3(u4)
2u5 − 175(u4)
4 = 0.
(iii) n ̸= 4, 6: nun−1un+1 − (n+ 1)(un)
2 = 0.
The aim of our article is to establish analogous results for vector ODEs E of order n+1 ≥ 3:
un+1 = f(t,u,u1, . . . ,un), (1.1)
where u is an Rm-valued function of t (for m ≥ 2), and uk is its k-th derivative. More precisely,
we consider and completely resolve the classification problem (up to local contact equivalence)
for submaximally symmetric vector ODEs (1.1) of order ≥ 3 of C-class [3, 5] (see below for
motivation). Note that by the Lie–Bäcklund theorem, contact-equivalence agrees with point-
equivalence for vector ODEs.
For vector ODEs (1.1) of order n+1 ≥ 3, the maximal and submaximal symmetry dimensions
are
M = m2 + (n+ 1)m+ 3 and S = M− 2, (1.2)
with the latter established in our earlier work [16], along with numerous other symmetry gap
results. The trivial vector ODE un+1 = 0 is locally uniquely maximally symmetric – see for
example [16, Corollary 2.8]. Examples of some submaximally symmetric vector ODEs were given
in [16, Table 8], but no definitive classification lists for the submaximal strata were asserted.
This is a focus of our current article.
Following Cartan [5] (see also [1, 3, 15]), a class of vector ODE (1.1) of order ≥ 3 is said
to be a C-class if it is invariant under all contact transformations, and all (contact) differential
invariants of any ODE in this class are first integrals of that ODE. Hence, generic C-class
equations (having sufficiently many functionally independent first integrals) can be solved using
these invariants. In [3, Theorem 4.1 and 4.2], the C-class was characterized by the vanishing of
the (generalized) Wilczynski invariants. (This vanishing also leads to the existence of geometric
structures on ODE solution spaces, which has been an important recent theme [6, 12, 13, 14, 19].)
The Wilczynski invariants are a subset of the fundamental (relative) invariants (see Section 2.4),
which additionally consist of C-class invariants (in the terminology of [16]).
We note from [16, Tables 8 and 10] that a vector ODE realizing S given in (1.2) is either
a 3rd order ODE pair, i.e., (n,m) = (2, 2), of C-class or it is of Wilczynski type (i.e., an ODE
with all C-class invariants vanishing identically). We will prove the following generalization of
Lie’s result above for vector ODEs:
Theorem 1.1. Any submaximally symmetric vector ODE (1.1) of order n+ 1 ≥ 3 is either
(a) of Wilczynski type, or
(b) locally equivalent2 over R to exactly one of the three 3rd order ODE pairs in Table 1.
Over C, the two 3rd order ODE pairs in the second row of Table 1 are locally equivalent.
Lie obtained his result for submaximally symmetric scalar ODEs using his complete classifica-
tion of Lie algebras of contact vector fields on the (complex) plane and classified invariant ODEs
having sufficiently many symmetries. Certainly, this approach generalizes to vector ODEs, but
it is not feasible: complete classifications for Lie algebras of (point) vector fields on Cn or Rn
for n ≥ 3 are known to be very difficult to establish [7, 25]. So, different techniques are needed
to establish analogous results for submaximally symmetric vector ODEs.
Our approach to classifying all submaximally symmetric vector ODEs (1.1) of C-class of order
≥ 3 is motivated by that of [28, 29] in the setting of parabolic geometries [4], and is based on
2More precisely, “local equivalence” here is meant in a neighbourhood of a point in E where at least one of the
C-class invariants is non-zero.
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 3
an equivalent reformulation of vector ODEs (1.1) as (strongly) regular, normal Cartan geome-
tries (G → E , ω) of type (G,P ) for a certain Lie group G and closed subgroup P ⊂ G [3, 8, 10]
(see Section 2.3 below).
For such a (non-parabolic) Cartan geometry, the harmonic curvature κH , which corresponds
to the fundamental invariants, is valued in a certain P -module that is completely reducible
[3, Corollary 3.8], so only the action of the reductive part G0 ⊂ P is relevant. Via a known
algebraic Hodge theory associated with G0, the codomain of κH can be identified with a certain
G0-submodule E ⊊ H2(g−, g) of a Lie algebra cohomology group called the effective part (see
Definition 2.3). This has been already computed for ODEs (1.1) of order 3 in [22, 23] and of
order ≥ 4 in [11]. The aforementioned fundamental invariants are valued in corresponding G0-
irreducible submodules U ⊂ E; see [16, Table 6] for a summary. The irreducible C-class modules
are listed in Table 2.
We next formulate our second main result, which concerns the classification of vector ODEs
(1.1) of C-class realizing the so-called constrained submaximal symmetry dimensions SU iden-
tified in [16, Table 2]. Fix an irreducible C-class module U = B4,Atr
2 ,Atf
2 ⊂ E (see Section 2.4)
and its corresponding C-class invariant U = B4,Atr
2 ,Atf
2 (see Section 2.5). Let CU denote the
set of all ODEs (1.1) with U ̸≡ 0 and all remaining C-class invariants vanishing identically
(equivalently, 0 ̸≡ im(κH) ⊂ U), and let SU denote the largest realizable symmetry dimension
among ODEs in CU . We will prove the following classification result:
Theorem 1.2. Any vector ODE (1.1) E of C-class of order n+1 ≥ 3 in CU realizing SU, near
any point x ∈ E with U(x) ̸= 0, is locally (point) equivalent over R to exactly one of the ODEs
given in Table 1. Over C, the indicated 3rd order ODEs for U = B4 are locally equivalent.
n
Irreducible C-class
module U ⊂ E
SU
ODE of C-class with 0 ̸≡ im(κH) ⊂ U
with symmetry dimension realizing SU
2 B4 M−m ua3 =
3u12u
a
2
2u11
(1≤a≤m)
or
ua3 =
3u11u
1
2u
a
2
1 +
(
u11
)2
(1≤a≤m)
≥ 3 Atr
2 M−m− 1 uan+1 =
(n+ 1)u1nu
a
n
nu1n−1
(1≤a≤m)
≥ 2 Atf
2 M− 2m+ 1 + δn2
uan+1 =
(
u2n
)2
δa1
(1≤a≤m)
(Recall M = m2 + (n+ 1)m+ 3 from (1.2).)
Table 1. Classification over R of submaximally symmetric vector ODEs (1.1) of C-class of order n+1 ≥ 3.
Our method for proving Theorems 1.1 and 1.2 will rely on the Cartan-geometric viewpoint
for vector ODEs, and the associated computations will be efficiently done using representation
theory. This will require important refinements to the existing structural results for vector ODEs
of C-class stated in Table 2. Such refinements constitute our final main result, which we now
briefly describe. In our non-parabolic ODE setting, the aforementioned algebraic Hodge theory
establishes a G0-equivariant identification of H2(g−, g) with the subspace ker□ ⊂
∧2 g∗− ⊗ g of
harmonic 2-cochains (see Section 2.3). Analogous to Kostant’s theorem [17], which is funda-
mental in the study of parabolic geometries, we may seek harmonic realizations of lowest weight
vectors ΦU ∈ U for each irreducible C-class submodule U ⊂ E ⊊ H2(g−, g). Our Theorem 3.1
establishes such realizations (see Table 4). We anticipate that these structural results will be
important for future geometric studies of the C-class and vector ODEs in general.
4 J.A. Kessy and D. The
2 Cartan geometries and vector ODEs of C-class
We briefly review the Cartan-geometric reformulation for vector ODEs (1.1) of order ≥ 3 modulo
point transformations, and summarize all relevant facts about vector ODEs of C-class.
2.1 ODE geometry and symmetry
We begin by summarizing [16, Section 2.1], which is based on [8, 10, 11], and refer the reader
to these articles for more details. The (n + 1)-st order ODE (1.1) defines a submanifold
E = {un+1 = f} of co-dimension m ≥ 2 in the space of (n + 1)-jets of functions Jn+1(R,Rm)
that is transverse to the projection πn+1
n : Jn+1(R,Rm) → Jn(R,Rm). Let C denote the Car-
tan distribution on Jn+1(R,Rm) with standard local coordinates (t,u0,u1, . . . ,un+1), where
ur =
(
u1r , . . . , u
m
r
)
. Then C is given by
C = span{∂t + u1∂u0 + · · ·+ un+1∂un , ∂un+1},
where ui∂uj :=
∑m
a=1 u
a
i ∂ua
j
and ∂ur refers to ∂u1
r
, . . . , ∂um
r
. We also consider the restriction of C
to E and abuse notation by also referring to this distribution as C.
Contact transformations are diffeomorphisms Φ: Jn+1(R,Rm) → Jn+1(R,Rm) that pre-
serve C, i.e., dΦ(C) = C. By the Lie–Bäcklund theorem, since m ≥ 2, such transforma-
tions are the prolongations of diffeomorphisms on J0(R,Rm) ∼= R × Rm, i.e., all such contact
transformations are point transformations. Infinitesimally, a point vector field is a vector field
ξ ∈ X
(
Jn+1(R,Rm)
)
whose flow is a point transformation. Equivalently, LξC ⊂ C, where Lξ
is the Lie derivative with respect to ξ. A point symmetry of (1.1) is a point vector field that is
tangent to E .
We will consider ODEs (1.1) up to point transformations. The (point) geometry of E is
encoded by a pair (E, V ) of completely integrable sub-distributions of C on E :
E = span
{
d
dt
:= ∂t + u1∂u0 + · · ·+ un∂un−1 + f∂un
}
, V = span{∂un}. (2.1)
(Note that integral curves of E are lifts of solution curves to (1.1).) Moreover, the distribution
D := E⊕V ⊂ TE is bracket-generating, and its weak-derived flag defines the following filtration
on TE :
TE = D−n−1 ⊃ · · · ⊃ D−2 ⊃ D−1 := D,
where D−i−1 := D−i +
[
D−i, D−1
]
for i > 0. Since
[
Γ
(
Dj
)
,Γ
(
Dk
)]
⊂ Γ
(
Dj+k
)
, then the pair
(E , {Di}) forms a filtered manifold. As we will describe below, this leads to the formulation of
an ODE (1.1) as a filtered G0-structure [2, Section 2.1].
Letting T iE := Di ⊂ TE for −n − 1 ≤ i ≤ −1 and T 0E := 0, we define gr(TE) :=⊕−1
i=−n−1 gri(TE) where gri(TE) := T iE/T i+1E . Let gri(TxE) denote the fiber of gri(TE) at
x ∈ E , i.e., mi(x) := gri(TxE) = T i
xE/T i+1
x E . Then m(x) := gr(TxE) =
⊕−1
i=−n−1mi(x) is a nilpo-
tent graded Lie algebra (NGLA) under the (Levi) bracket induced by the Lie bracket of vector
fields. It is called the symbol algebra at x. Since the symbol algebras at all points are isomorphic,
then we let m denote a fixed NGLA with m ∼= m(x), ∀x ∈ E , and we say that (E , {Di}) is regular
of type m.
Let Autgr(m) ≤ GL(m) be the subgroup that preserves the grading of m. Since m is generated
by m−1, then we have Autgr(m) ↪→ GL(m−1). For x ∈ E , we let Fgr(x) denote the set of all
NGLA isomorphisms m → m(x). Then Fgr(E) :=
⋃
x∈E Fgr(x) defines a principal fiber bundle
Fgr(E) → E with structure group Autgr(m), cf. [2, Proposition 2.1]. The splitting of D implies
a splitting of m−1, and restricting to the subgroup G0 ≤ Autgr(m) that preserves the splitting
yields a principal subbundle G0 → E with reduced structure group G0
∼= R× ×GLm, i.e., this is
the filtered G0-structure associated to an ODE (1.1).
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 5
2.2 Structure underlying the trivial ODE
Let n,m ≥ 2. The trivial ODE un+1 = 0 has point symmetry Lie algebra g (see, for example,
[3, Section 2.2] for explicit symmetry vector fields) with abstract structure given by
g ∼= q⋉ V, q := sl2 × gl(W ), V := Vn ⊗W, W := Rm,
where Vn is the unique (up to isomorphism) sl2-irrep of dimension n+1, and W is the standard
rep of gl(W ). Here, V is taken to be an abelian subalgebra.
We now fix a basis for g. Let {wa}ma=1 be the standard basis for W , and let ea
b be the m×m
matrix such that ea
bwc = δc
bwa, so that
{
ea
b
}m
a,b=1
spans gl(W ). Letting {x, y} be the standard
basis for R2, we identify Vn
∼= SnR2. We obtain bases {Ei}ni=0 on Vn and {Ei,a : 0 ≤ i ≤ n, 1 ≤
a ≤ m} on V via
Ei :=
xiyn−i
(n− i)!
, Ei,a := Ei ⊗ wa.
(For convenience, we define Ei = 0 for i < 0 or i > n. We also caution that our Ei corresponds
to En−i in [16, Section 2.1.2].) We complete our bases of V and gl(W ) to a basis of g by
introducing the standard sl2-triple
X := x∂y, H := x∂x − y∂y, Y := y∂x.
Note that sl2 commutes with gl(W ), and the sl2-actions on Vn and V are naturally induced,
e.g.,
[X, Ei] = Ei+1, [H, Ei] = (2i− n)Ei, [Y, Ei] = i(n+ 1− i)Ei−1.
In particular, Ei and Ei,a are weight vectors for the sl2-action, i.e., eigenvectors with respect
to H.
Now endow g with a bi-grading as in [16, Section 3.1]. Letting idm :=
∑m
a=1 ea
a, define
Z1,Z2 ∈ g by
Z1 := −1
2
(H+ n idm), Z2 := − idm . (2.2)
Then g decomposes into the joint eigenspaces of adZ1 and adZ2 . We write
gs,t := {x ∈ g : Z1 · x = sx, Z2 · x = tx},
and refer to s and t as the Z1-degree and Z2-degree of x, respectively. The ordered pair (s, t) ∈
Z× Z is the bi-grade of x. It is helpful to picture g as in Figure 1.
· · ·
(−1, 0) (0, 0) (1, 0)
(0,−1)(−1,−1)(−n + 1,−1)(−n,−1)
X H, ea
b Y
En,a En−1,a E1,a E0,a
Figure 1. Bi-grading on g.
Defining the grading element Z ∈ z(g0,0), we similarly induce the structure of a Z-grading
on g via
Z := Z1 + Z2 = −1
2
(H+ (n+ 2) idm).
6 J.A. Kessy and D. The
(In a given representation, Z-eigenvalues will also be referred to as degrees.) Then we have the
decomposition g = g−n−1 ⊕ · · · ⊕ g1, where
g1 := g1,0 = RY,
g0 := g0,0 = RH⊕ glm,
g−1 := g−1,0 ⊕ g0,−1 = RX⊕ (RE0 ⊗W ),
g−i−1 := g−i,−1 = REi ⊗W, i = 1, . . . , n.
We note that g− := g−n−1 ⊕ · · · ⊕ g−1 ⊂ g is generated by g−1.
We also endow g with the canonical filtration gi :=
∑
j≥i gj , which turns g into a filtered Lie
algebra. Its associated graded gr(g) :=
⊕
k∈Z grk(g), where grk(g) = gk/gk+1, is isomorphic to g
as graded Lie algebras. Using the isomorphism, we let grk : g
k → gk denote the leading part.
Explicitly, if x ∈ gk with x = xk + xk+1 + · · · , where xj ∈ gj , then grk(x) := xk. The following
notations will be convenient:
p := g0 = g0 ⊕ g1, p+ := g1 = g1.
At the group level, let
G := (SL2×GLm)⋉ V, P := ST2×GLm, G0 := {g ∈ P : Adg(g0) ⊂ g0},
where ST2 ⊂ SL2 is the subgroup of lower triangular matrices. (Note that G0 is isomorphic to
that given in Section 2.1.) We also let P+ ⊂ P denote the connected Lie subgroup corresponding
to p+ ⊂ p. We remark that the canonical filtration on g is P -invariant.
2.3 Cartan geometries associated to ODE
Fix G, P and G0 as above. Recall also from Section 2.1 that all vector ODEs (1.1) can be
formulated as filtered G0-structures. Importantly, there is an equivalence of categories between
filtered G0-structures on E (which is a wider category than that arising from ODE – see below)
and regular, normal Cartan geometries (G → E , ω) of type (G,P ) [3, 10]. A Cartan geometry
consists of a (right) principal P -bundle G → E endowed with a Cartan connection ω, i.e.,
ω ∈ Ω1(G, g) is a g-valued 1-form on G such that
(a) For any u ∈ G, ωu : TuG → g is a linear isomorphism;
(b) R∗
gω = Adg−1 ◦ω for any g ∈ P , i.e., ω is P -equivariant;
(c) ω(ζA) = A, where A ∈ p, where ζA is the fundamental vertical vector field defined
by ζA(u) :=
d
dt
∣∣
t=0
u · exp(tA).
The curvature K ∈ Ω2(G, g) of the geometry is given by K(ξ, η) = dω(ξ, η) + [ω(ξ), ω(η)],
which is P -equivariant and horizontal, i.e., K(ζA, ·) = 0, ∀A ∈ p. Consequently, K is deter-
mined by the P -equivariant curvature function κ : G →
∧2(g/p)∗ ⊗ g, defined by κ(u)(A,B) =
K
(
ω−1(A), ω−1(B)
)
(u), ∀A,B ∈ g. Letting ωG be the Maurer–Cartan form on G, the Klein
geometry (G→ G/P , ωG) satisfies K ≡ 0 (Maurer–Cartan equation), and is the flat model for all
Cartan geometries of type (G,P ).
In terms of the canonical filtration
{
gi
}
on g from Section 2.2, ω is said to be regular if
κ
(
gi, gj
)
⊂ gi+j+1, ∀i, j. Importantly, it is known that for all filtered G0-structures arising from
ODE, the corresponding Cartan geometry has κ satisfying the strong regularity condition [3,
Remark 2.3]
κ
(
gi, gj
)
⊂ gi+j+1 ∩ gmin(i,j)−1, ∀i, j. (2.3)
To define normality, we first fix an inner product ⟨·, ·⟩ on g in terms of the basis introduced
in Section 2.2:
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 7
Definition 2.1. Let ⟨·, ·⟩ be an inner product on g such that
{
X,H,Y, ea
b, Ei,a
}
is an orthogonal
basis for g with squared lengths of basis elements given below
⟨X,X⟩ = ⟨Y,Y⟩ = 1, ⟨H,H⟩ = 2,
〈
ea
b, ea
b
〉
= 1, ⟨Ei,a, Ei,a⟩ =
i!
(n− i)!
.
Then ∀A,B ∈ q = sl2 × glm and ∀u, v ∈ V , we have ⟨A,B⟩ = tr
(
ATB
)
and ⟨Au, v⟩ =
〈
u,ATv
〉
.
Consider Ck(g, g) :=
∧k g∗⊗g equipped with the induced canonical filtration from g and let ∂g
be the standard differential of the complex for computing Lie algebra cohomology groupsHk(g,g).
Then, define the codifferential ∂∗ : Ck(g, g) → Ck−1(g, g) to be the adjoint of ∂g with respect
to the induced inner product from g, i.e., for each k we have ⟨∂gϕ, ψ⟩ = ⟨ϕ, ∂∗ψ⟩ for all ϕ ∈
Ck−1(g, g) and ψ ∈ Ck(g, g). By [3, Lemma 3.2], the codifferential descends to a P -equivariant
map ∂∗ :
∧k(g/p)∗⊗g →
∧k−1(g/p)∗⊗g. A Cartan connection ω has curvature function κ valued
in
∧2(g/p)∗ ⊗ g, and ω is said to be normal if ∂∗κ = 0. In this article, we will always work with
Cartan geometries of type (G,P ) that are normal and strongly regular.
Since (∂∗)2 = 0, then the (normal) curvature κ quotients to a P -equivariant function κH :
G → ker ∂∗
im ∂∗ called the harmonic curvature. By regularity, κH is valued in the filtrand of positive
degree of the P -module ker ∂∗
im ∂∗ , which by [3, Corollary 3.8] is completely reducible, i.e., P+ acts
on it trivially, and therefore only the G0-action is relevant. It is well known (see Theorem A.3
and references therein) that κH completely obstructs local flatness, i.e., κH ≡ 0 is equivalent
to κ ≡ 0.
Identify
∧k(g/p)∗⊗ g ∼=
∧k g∗−⊗ g as G0-modules, and recall from Section 2.2 that g ∼= q⋉V .
Given ϕ ∈ Ck(g−, g) :=
∧k g∗− ⊗ g, then we have ϕ = X∗ ∧ ϕ1 + ϕ2, for ϕ1 ∈ Ck−1(V, g)
and ϕ2 ∈ Ck(V, g), and where X∗ is dual to X. Denoting ϕ :=
(
ϕ1
ϕ2
)
, then ∂ϕ is given by [3,
Lemma 3.4]
∂
(
ϕ1
ϕ2
)
=
(
−∂V ϕ1 + X · ϕ2
∂V ϕ2
)
, (2.4)
where
∂V ϕ2(x0, . . . , xk) =
k∑
i=0
(−1)ixi · ϕ2
(
x0, . . . , x̂i, . . . , xk
)
for x0, . . . , xk ∈ V , and letting x̂i denote omission of xi. A direct consequence of (2.4) is:
Lemma 2.2. Let ϕ ∈
∧k V ∗ ⊗ g. Then ∂ϕ = 0 if and only if X · ϕ = 0 and ∂V ϕ = 0. Moreover,
if in fact ϕ ∈
∧k V ∗ ⊗ V , then ∂ϕ = 0 if and only if X · ϕ = 0.
Defining □ := ∂ ◦ ∂∗ + ∂∗ ◦ ∂ :
∧k g∗− ⊗ g →
∧k g∗− ⊗ g, we then have the following G0-
isomorphisms,
∧
kg∗− ⊗ g ∼=
ker ∂∗︷ ︸︸ ︷
im ∂∗ ⊕ ker□⊕ im ∂︸ ︷︷ ︸
ker ∂
, ker□ ∼=
ker ∂∗
im ∂∗
∼=
ker ∂
im ∂
=: Hk(g−, g). (2.5)
Consequently, for a regular, normal Cartan geometry, the codomain of κH can be identified
with the subspace H2
+(g−, g) ⊂ H2(g−, g) on which the grading element Z = Z1 + Z2 acts with
positive eigenvalues. However, it should be emphasized that only part of H2
+(g−, g) is in fact
realizable for geometries associated to ODE [11, 22]. Correspondingly, we define:
Definition 2.3. The effective part E ⊊ H2
+(g−, g) is the minimal G0-module in which κH is
valued, for any (strongly) regular, normal Cartan geometry of type (G,P ) associated to an
ODE (1.1) (for fixed (n,m)).
8 J.A. Kessy and D. The
2.4 Vector ODEs of C-class
We will focus on ODEs (1.1) of C-class, which have been characterized in [3] using curvatures κ
of corresponding canonical Cartan connections ω described above. We define [3, Definition 2.4]:
Definition 2.4. An ODE (1.1) is said to be of C-class if the curvature κ of the corresponding
strongly regular, normal Cartan geometry satisfies κ(X, ·) = 0, where X ∈ g−1 was defined in
Section 2.2.
Remark 2.5. Recall from Section 2.2 that g ∼= q⋉ V . We remark that for a Cartan geometry
corresponding to an ODE of C-class, we can identify κ ∈
∧2(g/q)∗ ⊗ g ∼=
∧2 V ∗ ⊗ g.
As shown in [3], the notion of C-class can be concretely reformulated in terms of fundamental
invariants for vector ODEs (1.1) of order ≥ 3 described below, which comprise the harmonic
curvature of the geometry. We then have the following characterization of the C-class given
in [3, Theorems 4.1 and 4.2]:
Theorem 2.6. A vector ODE (1.1) of order ≥ 3 is of C-class if and only if all of its generalized
Wilczynski invariants vanish.
For concreteness, we now explicitly describe the fundamental invariants for vector ODEs (1.1)
of order n + 1 ≥ 3, which consist of generalized Wilczynski invariants Wr [9] and C-class
invariants [11, 22, 23]:
� Consider a linear vector ODE of order n+ 1:
un+1 + Pn(t)un + · · ·+ P1(t)u1 + P0(t)u = 0, (2.6)
where Pj(t) is an End(Rm)-valued function. Using the invertible transformations (t,u) 7→
(f(t), h(t)u) where f : R → R× and h : R → GL(m), which preserve the form of equa-
tion (2.6), we may normalize to Pn = 0 and tr(Pn−1) = 0, i.e., Laguerre–Forsyth canonical
form. Then
Θr =
r−1∑
k=1
(−1)k+1 (2r − k − 1)!(n− r + k)!
(r − k)!(k − 1)!
P
(k−1)
n−r+k, r = 2, . . . , n+ 1,
are fundamental invariants found by Se-ashi [26], and r is the degree of the invariant.
For (1.1), the generalized Wilczynski invariants Wr (for r = 2, . . . , n + 1) are defined
as Θr above evaluated at its linearization along a solution u. Formally, Wr are obtained
from (2.6) by replacing Pr(t) by the matrices −
(
∂fa
∂ub
r
)
and the usual derivative by the
total derivative d
dt given in (2.1). Moreover, Wr do not depend on the choice of solution u,
and are therefore contact invariants.
� C-class invariants are the following:
n ≥ 2:
(
Atf
2
)a
bc
= tf
(
∂2fa
∂ubn ∂u
c
n
)
,
n ≥ 3:
(
Atr
2
)a
bc
= tr
(
∂2fa
∂ubn ∂u
c
n
)
,
n = 2: (B4)bc = −∂H
−1
c
∂ub1
+
∂
∂ub2
∂
∂uc2
Ht − ∂
∂uc2
d
dt
H−1
b
− ∂
∂uc2
(
m∑
a=1
H−1
a
∂fa
∂ub2
)
+ 2H−1
b H−1
c ,
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 9
where
H−1
b =
1
6(m+ 1)
m∑
a=1
∂2fa
∂ua2 ∂u
b
2
,
Ht = − 1
4m
m∑
a=1
(
∂fa
∂ua1
− d
dt
∂fa
∂ua2
+
1
3
m∑
c=1
∂fa
∂uc2
∂f c
∂ua2
)
.
2.5 C-class modules
The above fundamental invariants correspond to G0-irreducible submodules in the effective
part E ⊊ H2
+(g−, g) (see Definition 2.3), which we now describe. Recall that g0 ∼= span{Z1,Z2}⊕
sl(W ), and we have the induced action of Z1 and Z2 from (2.2) on H2
+(g−, g), and therefore on E.
Note that Z2 acts with degrees 0, 1, or 2. We define:
Definition 2.7. A G0-submodule U ⊂ E ⊊ H2
+(g−, g) on which Z2 acts with positive degree(s) is
called a C-class module, and we let EC ⊊ E denote the direct sum of all irreducible C-class mod-
ules. On the other hand, if Z2 acts on U with zero degree, we refer to U as a Wilczynski module.
Any g0-irrep U ⊂ E is determined by its bi-grade and its lowest weight λ with respect
to sl(W ) ∼= slm. Such λ can be expressed in terms of the fundamental weights λ1, . . . , λm−1
of slm with respect to the Cartan subalgebra h consisting of diagonal matrices in slm, and the
standard choice ofm−1 simple roots. Letting h = diag(h1, . . . , hm) ∈ h and ϵa : h → R the linear
functional ϵa(h) = ha, we then have ϵ1 + · · ·+ ϵm = 0 and λi = ϵ1 + · · ·+ ϵi for 1 ≤ i ≤ m− 1.
Table 2 contains a summary of results for EC ⊊ E for ODEs (1.1), due to Medvedev [22, 23]
for order 3, and Doubrov–Medvedev [11] for order ≥ 4. Using the G0-isomorphisms (2.5), we
identify each irreducible C-class module U ⊂ EC from Table 2 with the corresponding module
in ker□ ⊂ C2(g−, g) consisting of harmonic 2-cochains satisfying the strong regularity condi-
tion (2.3). (A further condition is formulated in Section 2.6 below.) Adopting the same notation
from [16, Table 6], we let A2 and B4 denote the C-class submodules with bi-grades (1, 1) and (2, 2)
respectively. From the respective Z2-degrees, and since κ ∈
∧2 V ∗⊗ g for C-class ODE, then we
deduce that we may identify
A2 ⊂
∧
2 V ∗ ⊗ V, B4 ⊂
∧
2 V ∗ ⊗ q. (2.7)
Since A2 is not irreducible, we decompose it into (irreducible) trace and trace-free parts: A2 =
Atr
2 ⊕Atf
2 . (The C-class invariants B4, Atr
2 , Atf
2 from Section 2.4 are valued in the corresponding
irreducible C-class modules B4, Atr
2 , Atf
2 respectively.)
n Irred. C-class moduleU Bi-grade sl(W )-module structure sl(W )-lowest weight λ
2 B4 (2, 2) S2W ∗ −2ϵ1
≥ 3 Atr
2 (1, 1) W ∗ −ϵ1
≥ 2 Atf
2 (1, 1)
(
S2W ∗ ⊗W
)
0
ϵm − 2ϵ1
Table 2. C-class modules in EC ⊊ E ⊊ H2
+(g−, g) for vector ODEs of order n+ 1 ≥ 3.
Since each U is a g0-irrep, then up to scale U contains a unique lowest weight vector ΦU. Since
g0 ∼= span{Z1,Z2} ⊕ sl(W ), then being “lowest” means that ΦU is annihilated by all lowering
operators, i.e., strictly lower triangular matrices, in sl(W ) ∼= slm. From Table 2, we can give an
explicit description of the annihilators ann(ΦU), which will be needed later. Namely, if p̃ ⊂ slm is
the parabolic subalgebra preserving ΦU up to scale, then ann(ΦU) ⊂ span{Z1,Z2}⊕ p̃. For a ̸= c,
if ea
c ∈ p̃, then ea
c ∈ ann(ΦU). It suffices to consider linear combinations of Z1, Z2, and diagonal
10 J.A. Kessy and D. The
elements h ⊂ p̃. If ΦU has slm-weight λ and Z2-degree t, then we conclude that ann(ΦU) is
spanned by
Z1 − Z2, h− λ(h)
t
Z2, h ∈ h, ea
c ∈ p̃, a ̸= c, (2.8)
where Z1 − Z2 ∈ ann(ΦU) because of the bi-grading of U. Applying (2.8) to (λ, t) from Table 2,
we obtain Table 3. Here, p̃1, p̃1,m−1 are the parabolic subalgebras in slm consisting of block
lower triangular matrices with diagonal blocks of sizes 1, m− 1 and 1, m− 2, 1 respectively.
n U dim ann(ΦU) Generators for ann(ΦU) ⊂ g0
2 B4
m2 −m+ 1
Z1 − Z2, ea
c ∈ p̃1, a ̸= c,
eb
b − eb+1
b+1 + δ1
bZ2, 1 ≤ b ≤ m− 1≥ 3 Atr
2
≥ 2 Atf
2 m2 − 2m+ 3
Z1 − Z2, ea
c ∈ p̃1,m−1, a ̸= c,
eb
b − eb+1
b+1 +
(
2δ1
b + δm−1
b
)
Z2, 1 ≤ b ≤ m− 1
Table 3. ann(ΦU) ⊂ g0 for irreducible C-class modules U ⊂ E.
2.6 The Doubrov–Medvedev condition
We will be able to precisely identify A2 with the help of an additional linear condition formulated
in [11, Section 3.1, Proposition 4], and which we now summarize. Consider the p-invariant
subspace F = span{E0, . . . , En−1}⊗W ⊂ V , and define δ : Hom(F,RX) → Hom
(∧2 F, V/F
)
by
(δB)(x, y) = (B(x) · y −B(y) · x) mod F, ∀B ∈ Hom(F,RX). (2.9)
We have the inclusion ιF : F → V , which induces V/F ∼= W (as p-modules) and natural quo-
tient πW : V → V/F . Also induced is the inclusion ι∧2 F :
∧2 F →
∧2 V , from which we define
ϑ : Hom
(∧2 V, V
)
→ Hom
(∧2 F, V/F
)
by ϑ = πW ◦ ι∗∧2 F
, i.e.,
ϑ(A) = A|∧2 F mod F. (2.10)
From [11, Section 3.1, Proposition 4] and Remark 2.5, we deduce that for a C-class ODE of
order ≥ 4, the A2-component A2 of its harmonic curvature κH satisfies ϑ(A2) ∈ im(δ), which
we refer to as the Doubrov–Medvedev condition. Correspondingly, for n ≥ 3 we formulate the
algebraic condition
ϑ(A) ∈ im(δ), ∀A ∈ A2, (2.11)
which we refer to as the DM condition. (This condition is not present for 3rd order ODE.)
3 Lowest weight vectors for irreducible C-class modules
The g0-module structure for irreducible C-class modules U ⊂ E was stated in Table 2. While this
abstract structural information proved useful in our previous study of symmetry gaps [16], more
precise information is needed in our current study. Namely, viewing U as harmonic 2-cochains
via the G0-equivariant identification (2.5), we may ask for concrete realizations of lowest weight
vectors ΦU ∈ U (from which a full basis of U may be obtained by applying raising operators).
These realizations are not found in the existing literature, and our main goal in this section is
to provide them. This information will provide the starting point in subsequent sections for our
classification of submaximally symmetric structures.
Given the notation introduced in Section 2.2, and letting Ei,a denote the dual basis elements
to Ei,a, we have:
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 11
Theorem 3.1. Fix n,m ≥ 2 and an irreducible C-class module U ⊂ E, viewed as a G0-submodule
of ker□ ⊂ C2(g−, g) via (2.5). Then the unique lowest weight vector ΦU ∈ U, up to a scaling,
is given in Table 4.
n U Lowest weight vectorΦU ∈ U
2 B4
E2,1 ∧ E1,1 ⊗ X− 1
2E
2,1 ∧ E0,1 ⊗ H− 1
2E
1,1 ∧ E0,1 ⊗ Y
+
∑m
a=1
(
E2,1 ∧ E0,a − E1,1 ∧ E1,a + E0,1 ∧ E2,a
)
⊗ ea
1
≥ 3 Atr
2
α
∑n
i=0
[
Φ2,i +
(
n
2 − i
)
Φ1,i − 1
2 i(n+ 1− i)Φ0,i
]
+β
∑n
i=0
[
(n+ 1− i)
(
Φi,0 − Φ0,i
)
+Φi,1 − Φ1,i
]
,
where Φi,j :=
∑m
a=1E
i,1 ∧ Ej,a ⊗ Ei+j−1,a
and α = −6(n−1)(m+1)
mn(n+1)+6 β
≥ 2 Atf
2
∑n
j=0
[
(n+ 1− j)Φ0,j +Φ1,j
]
,
where Φi,j := Ei,1 ∧ Ej,1 ⊗ Ei+j−1,m
Table 4. Classification of lowest weight vectors ΦU for irreducible C-class modules U ⊂ E.
Let us give a brief summary of the computations to follow. For ΦU ∈ U lying in the ap-
propriate module given in (2.7), we use the bi-grade and sl(W )-lowest weight data for U from
Table 2 to first write a general form for ΦU. (The reader should recall the bi-grades given in
Section 2.2, e.g., Ei,a has bi-grade (−i,−1), and so Ei,a ∈ g−i−1 ⊂ g−i−1.) We then further
constrain this form by imposing additional linear conditions coming from harmonicity, strong
regularity, and the DM condition (2.11). (For example, since ΦU ∈
∧2 V ∗ ⊗ V in the Atr
2 , Atf
2
cases, then ∂ΦU = 0 if and only if X · ΦU = 0 by Lemma 2.2. Imposing X-annihilation will be
a detailed calculation involving the relations X · Ei,a = Ei+1,a and X · Ei,a = −Ei−1,a.) This
calculation will be involved, but we remark that in fact not all such conditions will need to be
explicitly imposed:
Remark 3.2. If ΦU can be constrained to a 1-dimensional subspace by imposing some of the
conditions above, then ΦU necessarily satisfies all the remaining linear conditions (harmonicity,
strong regularity, and (2.11)). This follows from existence of the module U ⊂ E for ODE systems,
which was established in [11, 22].
Let us now carry out the indicated computations and establish Theorem 3.1 above.
3.1 B4 case
Since U := B4 ⊂
∧2 V ∗ ⊗ q has bi-grade (2, 2), then ΦU must be a linear combination of
E2,a ∧ E1,b ⊗ X, E1,a ∧ E0,b ⊗ Y, E2,a ∧ E0,b ⊗ H, E1,a ∧ E1,b ⊗ H,
E2,a ∧ E0,b ⊗ ec
d, E1,a ∧ E1,b ⊗ ec
d, 1 ≤ a, b, c, d ≤ m.
Since U has sl(W )-lowest weight λ = −2ϵ1 (Table 2), then ΦU lies in the subspace spanned by
E2,1 ∧ E1,1 ⊗ X, E2,1 ∧ E0,1 ⊗ H, E1,1 ∧ E0,1 ⊗ Y,
E2,1 ∧ E0,1 ⊗ e1
1, E2,1 ∧ E0,1 ⊗ ea
a, E2,1 ∧ E0,a ⊗ ea
1,
E2,a ∧ E0,1 ⊗ ea
1, E1,a ∧ E1,1 ⊗ ea
1. (3.1)
For ann(ΦU) from Table 3, requiring ann(ΦU) · ΦU = 0 further constrains ΦU to lie in span of
E2,1 ∧ E1,1 ⊗ X, E2,1 ∧ E0,1 ⊗ H, E1,1 ∧ E0,1 ⊗ Y,
12 J.A. Kessy and D. The
m∑
a=1
E2,1 ∧ E0,1 ⊗ ea
a,
m∑
a=1
E2,1 ∧ E0,a ⊗ ea
1,
m∑
a=1
E2,a ∧ E0,1 ⊗ ea
1,
m∑
a=1
E1,a ∧ E1,1 ⊗ ea
1. (3.2)
Let us briefly explain this. From Table 3, slm−1 embeds into ann(ΦU) via A 7→ diag(0, A), which
acts trivially on the first 4 elements of (3.1). The remaining tensors in (3.1) lie in a direct
sum of 4 slm−1-reps equivalent to the sum of 4 copies of glm−1. (Namely, consider the span of
E2,1 ∧ E0,1 ⊗ ea
b, E2,1 ∧ E0,b ⊗ ea
1, etc.) Since glm−1
∼= R ⊕ slm−1, then the aforementioned
subspace contains a 4-dimensional subspace annihilated by slm−1. This is clearly spanned by
the last 4 elements of (3.2) except taking the sum over 2 ≤ a ≤ m. Finally, forcing annihilation
with respect to ef
1 for f ≥ 2 yields (3.2).
Let ΦU be a general linear combination of all elements of (3.2), with µi denoting the coefficient
of the i-th term, i.e., ΦU = µ1E
2,1∧E1,1⊗X+µ2E
2,1∧E0,1⊗H+ · · ·+µ7
∑m
a=1E
1,a∧E1,1⊗ea1.
We conclude our computation by imposing ∂-closedness for ΦU using Lemma 2.2:
� X-annihilation: This yields µ2 = µ3 = −µ1
2 , µ4 = 0 and µ7 = µ5 = −µ6.
� ∂V -closedness: 0 = ∂V ΦU(E1,2, E2,1, E1,1) = (µ5 − µ1)E2,2, and hence µ1 = µ5.
This uniquely pins down ΦU (as stated in Table 4), up to a nonzero scaling. From Remark 3.2,
we in particular have that ΦU is normal and strongly regular. (The condition (2.11) does not
apply for 3rd order ODE systems.)
3.2 Atr
2 case
This case proceeds similarly, but is more involved than the B4 case. In particular, more condi-
tions are required to pin down the lowest weight vector (up to scale).
Let n ≥ 3. Since U := Atr
2 ⊂
∧2 V ∗⊗V has bi-grade (1, 1) and sl(W )-lowest weight λ = −ϵ1,
then ΦU must be a linear combination of
Ei,1 ∧ Ej,a ⊗ Ei+j−1,a, 0 ≤ i, j ≤ n, 1 ≤ i+ j ≤ n+ 1, 1 ≤ a ≤ m.
Moreover, ann(ΦU) from Table 3 annihilates ΦU, so ΦU is in fact constrained to be a linear
combination of
Φi,j :=
m∑
a=1
Ei,1 ∧ Ej,a ⊗ Ei+j−1,a. (3.3)
Recalling our convention in Section 2.2 that Ek = 0 for k < 0 or k > n, we have:
Proposition 3.3. Fix n ≥ 3 and m ≥ 2. Let U = Atr
2 and define ΦU =
∑n
i,j=0 ci,jΦ
i,j for Φi,j
as in (3.3), where we may assume that c0,0 = 0 = ci,j for i + j > n + 1. Since ΦU is ∂-closed
and satisfies the strong regularity and DM conditions, then we have
ci+1,j + ci,j+1 = ci,j ; (XA): annihilation by X;
ci,j = 0, for min(i, j) ≥ 3; (SR): strong regularity;
cn−1,2 = 0, for n ≥ 4. (DM): DM conditions beyond (SR).
(3.4)
Proof. By Lemma 2.2, ∂-closedness of ΦU is equivalent to its X-annihilation, so using X ·Ei,a =
Ei+1,a and X·Ei,a = −Ei−1,a, Leibniz rule, and re-indexing the summation, we straightforwardly
obtain
0 = X · ΦU =
n∑
i=0
n∑
j=0
(ci,j − ci+1,j − ci,j+1)Φ
i,j .
This proves the first relations.
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 13
Next, recall that Ek,a ∈ g−k−1. Strong regularity (2.3) of Φi,j forces that we have Ei+j−1,a ∈
gmin(−i−1,−j−1)−1, i.e.,
− i− j ≥ min(−i− 1,−j − 1)− 1 ≥ −max(i, j)− 2 ⇐⇒ i+ j ≤ max(i, j) + 2,
or equivalently min(i, j) ≤ 2. All other terms are not present in the summation.
Finally, for the last relations we force (2.11) for A = ΦU, i.e., ϑ(ΦU) ∈ im(δ). Recall the
maps δ and ϑ given in (2.9) and (2.10), and F = span{E0, . . . , En−1} ⊗W ⊂ V . Modulo F ,
� ϑ(ΦU) =
n∑
i,j=0
ci,jϑ(Φ
i,j) ≡
n∑
i,j=0
ci,jΦ
i,j |∧2 F ≡
n−1∑
i=2
ci,n+1−iΦ
i,n+1−i
(SR)
≡ c2,n−1Φ
2,n−1 + cn−1,2Φ
n−1,2.
� δ(Ei,a ⊗ X) ≡
m∑
b=1
Ei,a ∧ En−1,b ⊗ En,b, i.e., bi-grade (i− 1, 1) tensors for 0 ≤ i ≤ n− 1.
Since ϑ(ΦU) only consists of bi-grade (1, 1) tensors, it suffices to examine the (1, 1) subspace
of im(δ). From above, this always contains Φ2,n−1 (modulo F ), but does not contain Φn−1,2
when n ≥ 4. Hence, beyond (SR), DM condition implies ϑ(ΦU) ∈ im(δ), which forces cn−1,2 = 0
for n ≥ 4. ■
We now solve (3.4):
Proposition 3.4. Fix n ≥ 3. Then (ci,j)0≤i,j≤n from Proposition 3.3 is of the following form:
ci,0 =
(n− i+ 1)β, 3 ≤ i ≤ n;
α+ (n− 1)β, i = 2;
nα
2 + (n− 1)β, i = 1;
c0,i =
{
−c1,0, i = 1;
(i− n− 1)(β + iα
2 ), 2 ≤ i ≤ n;
c1,i =
(n
2
− i
)
α+
(
δi
1 − 1
)
β, 1 ≤ i ≤ n; ci,1 = β + δi
2α, 2 ≤ i ≤ n;
c2,i = (1− δi
n)α, 2 ≤ i ≤ n; ci,2 = 0, 3 ≤ i ≤ n,
(3.5)
where α := c2,n−1 and β := cn,1, and all other coefficients are trivial.
Proof. Since (DM) is only present for n ≥ 4, we split our proof into two cases:
� n = 3: The system (3.4) becomes
c1,0 + c0,1 = 0, c1,1 + c0,2 = c0,1, c2,0 + c1,1 = c1,0,
c1,2 + c0,3 = c0,2, c2,1 + c1,2 = c1,1, c3,0 + c2,1 = c2,0,
c1,3 = c0,3, c2,2 + c1,3 = c1,2, c3,1 + c2,2 = c2,1, c3,1 = c3,0.
Solving this in terms of α = c2,2 and β = c3,1 gives (3.5).
� n ≥ 4:
Step 1: Start with the assumed conditions c0,0 = 0 = ci,j for i + j > n + 1, the (SR)
relations, as well as the (DM) relation cn−1,2 = 0. Using (XA), determine the entries above
cn−1,2 = 0 and left of c2,n−1 =: α (until the (2, 2)-position), as shown below:
0 ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ · · · ∗ ∗ α 0
∗ ∗ ∗ 0 0 · · · 0 0 0 0
.
.
.
.
.
.
.
.
. 0 0 · · · 0 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
..
.
.
.
.
.
.
.
.
.
.
∗ ∗ ∗ 0 0
.
.
. 0 0 0 0
∗ ∗ ∗ 0 0 · · · 0 0 0 0
∗ ∗ 0 0 0 · · · 0 0 0 0
∗ ∗ 0 0 0 · · · 0 0 0 0
;
0 ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ ∗
∗ ∗ α α α · · · α α α 0
∗ ∗ 0 0 0 · · · 0 0 0 0
.
.
.
.
.
. 0 0 0 · · · 0 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
∗ ∗ 0 0 0 · · · 0 0 0 0
∗ ∗ 0 0 0 · · · 0 0 0 0
∗ ∗ 0 0 0 · · · 0 0 0 0
∗ ∗ 0 0 0 · · · 0 0 0 0
.
14 J.A. Kessy and D. The
Step 2: Using (XA), we have β := cn,1 = cn,0. Use (XA) to determine the entries
above cn,1 and left of γ := c1,n (until the (1, 1)-position), as shown below:
0 ∗ ∗ ∗ ∗ · · · ∗ ∗ ∗ ∗
∗ (n − 1)α + β + γ (n − 2)α + γ (n − 3)α + γ (n − 4)α + γ · · · 3α + γ 2α + γ α + γ γ
∗ α + β α α α · · · α α α 0
∗ β 0 0 0 · · · 0 0 0 0
.
.
. β 0 0 0 · · · 0 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
∗ β 0 0 0 · · · 0 0 0 0
∗ β 0 0 0 · · · 0 0 0 0
∗ β 0 0 0 · · · 0 0 0 0
β β 0 0 0 · · · 0 0 0 0
.
More precisely, we have
c1,i = (n− i)α+ δ1i β + γ, 1 ≤ i ≤ n. (3.6)
Step 3: Using (XA), determine all entries above cn,0 = β. This yields
ci,0 =
(n− i+ 1)β, 3 ≤ i ≤ n;
α+ (n− 1)β, i = 2;
n(α+ β) + γ, i = 1.
(3.7)
Step 4: Impose c0,1
(XA)
= −c1,0 = −n(α + β) − γ. For 2 ≤ i ≤ n, we have the telescoping
sum
c0,i − c0,1 =
i∑
k=2
(c0,k − c0,k−1)
(XA)
= −
i∑
k=2
c1,k−1
(3.6)
= −
i∑
k=2
[(n− k + 1)α+ δ1k−1β + γ],
c0,i = c0,1 − β − (i− 1)γ − α[(n− 1) + · · ·+ (n− i+ 1)]
= −(n+ 1)β − iγ − α
[(
n+ 1
2
)
−
(
n− i+ 1
2
)]
. (3.8)
Step 5: Impose c0,n
(XA)
= c1,n = γ. Solving this yields γ = −β − nα
2 . Substituting this into
(3.6), (3.7) and (3.8) then gives the stated result. ■
We conclude our computation by imposing the coclosedness condition, i.e., ∂∗ΦU = 0.
Proposition 3.5. Let n ≥ 3 and m ≥ 2. Take ΦU from Proposition 3.3 with coefficients (3.5).
Then
α =
−6(n− 1)(m+ 1)
mn(n+ 1) + 6
β. (3.9)
Proof. From Lemma B.2, we have
0 =
n−1∑
k=0
(n− k)(k + 1)
n(n− 1)
(ck,2 −mc2,k) +
n∑
k=0
2k − n
n
(mc1,k − ck,1)
+
n∑
k=1
(mc0,k − ck,0). (3.10)
We now substitute (3.5) into (3.10) and simplify. The computations are straightforward but
tedious, and for the respective summations above, this leads to
0 = − (n+ 2)Ω
6n(n− 1)
− (n+ 2)Ω
6n
− (n+ 2)Ω
12
= −(n+ 2)(n+ 1)Ω
12(n− 1)
,
where Ω := (mn(n+1)+6)α+6(n− 1)(m+1)β. This implies Ω = 0, and hence the result. ■
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 15
Combining (3.9), (3.5), and Proposition 3.3 uniquely determines ΦU (given in Table 4 after
some simplification/reorganization), up to a nonzero scaling. Note that (3.10) (derived in Ap-
pendix B) was only a small part of the coclosedness condition, but using Remark 3.2, we deduce
that indeed ∂∗ΦU = 0.
3.3 Atf
2 case
The trace-free case proceeds analogously to the trace case. Since U := Atf
2 ⊂
∧2 V ∗ ⊗ V has
bi-grade (1, 1) and sl(W )-lowest weight λ = ϵm − 2ϵ1, then ΦU is a linear combination of
Φi,j := Ei,1 ∧ Ej,1 ⊗ Ei+j−1,m, 0 ≤ i, j ≤ n, 1 ≤ i+ j ≤ n+ 1. (3.11)
Note that Φi,j = −Φj,i is annihilated by all elements of ann(ΦU) given in Table 3.
Proposition 3.6. Fix n,m ≥ 2. Let U = Atf
2 and define ΦU =
∑n
i,j=0 ci,jΦ
i,j for Φi,j as
in (3.11), where we may assume that ci,j = −cj,i, and ci,j = 0 for i + j > n + 1. Since ΦU is
∂-closed and satisfies the strong regularity and DM conditions, then we have
ci+1,j + ci,j+1 = ci,j ; (XA): annihilation by X;
ci,j = 0, for min(i, j) ≥ 3; (SR): strong regularity;
cn−1,2 = c2,n−1 = 0, for n ≥ 3. (DM ′): DM conditions beyond (SR).
(3.12)
Proof. The proof is very similar to Proposition 3.3, as we now explain. Recall that X · Ei,a =
Ei+1,a and X · Ei,a = −Ei−1,a, i.e., the X-action on these basis elements is independent of the
second index. Consequently, comparing (3.11) and (3.3), it is immediate that X · ΦU = 0 yields
the same conditions (XA). Strong regularity similarly does not involve the second index, and so
we obtain the same conditions (SR).
Finally, let us focus on (2.11). As in the proof of Proposition 3.3,
� ϑ(ΦU) ≡
∑n−1
i=2 ci,n+1−iΦ
i,n+1−i mod F , which consists of bi-grade (1, 1) tensors;
� the bi-grade (1, 1) tensors in im(δ) are spanned by
∑m
b=1E
2,a ∧ En−1,b ⊗ En,b mod F .
Since m ≥ 2, then ϑ(ΦU) ∈ im(δ) forces ϑ(ΦU) ≡ 0. This is automatic for n = 2, while for n ≥ 3,
we have ci,n+1−i = 0 for 2 ≤ i ≤ n− 1. Beyond (SR), we have merely c2,n−1 = cn−1,2 = 0. ■
Proposition 3.7. Fix n ≥ 2. Then (ci,j)0≤i,j≤n from Proposition 3.6 is of the following form:
ci,0 = −c0,i =
{
(n− i+ 1)β, 2 ≤ i ≤ n
(n− 1)β, i = 1
and ci,1 = −c1,i = β, 2 ≤ i ≤ n (3.13)
and all other coefficients are trivial.
Proof. We split our proof into two cases:
� n = 2: the system (3.12) reduces to
c1,0 + c0,1 = 0, c2,0 = c1,0, c0,2 = c0,1, c1,2 = c0,2, c2,1 = c2,0,
and solving the system in terms of c2,1 proves the claim.
� n ≥ 3: The conditions on ci,j in Proposition 3.6 can be viewed as (3.4) with additional-
ly ci,j = −cj,i (and c2,n−1 = 0 when n = 3). Consequently, the solution to (3.12) can be
obtained from the solution (3.5) to (3.4) by merely imposing α := c2,n−1 = 0. ■
Combining (3.13) and Proposition 3.6 uniquely determines ΦU (given in Table 4), up to
a nonzero scaling. As before, using Remark 3.2, we deduce that ∂∗ΦU = 0. This completes our
proof of Theorem 3.1.
16 J.A. Kessy and D. The
4 Homogeneous structures and Cartan-theoretic descriptions
Our method for proving Theorems 1.1 and 1.2 will rely on the fact that Cartan geometries
(see Section 2.3) associated to submaximally symmetric vector ODEs are locally homogeneous
(see Lemma 4.2). In this section, we summarize all relevant symmetry-based facts about such
geometries and their corresponding algebraic models of ODE type. We will use G, P , G0 and g
from Section 2.2, and the filtration and grading on g defined there.
4.1 Symmetry gaps for ODE
An infinitesimal symmetry of a given Cartan geometry (G → E , ω) of type (G,P ) is a P -
invariant vector field ξ ∈ X(G)P on G that preserves ω under Lie differentiation, i.e., Lξω = 0.
The collection of all such vector fields forms a Lie algebra, which we denote by
inf(G, ω) :=
{
ξ ∈ X(G)P : Lξω = 0
}
⊂ X(G).
The submaximal symmetry dimension is
S := max
{
dim inf(G, ω) : (G → E , ω) strongly regular, normal of type (G,P )
associated to a vector ODE E (1.1), with κH ̸≡ 0
}
.
Recall that E decomposes into G0-irreducible submodules U ⊂ E. Analogous to S above, we
define:
SU := max
{
dim inf(G, ω) : (G → E , ω) strongly regular, normal of type (G,P )
associated to a vector ODE E (1.1), with 0 ̸≡ im(κH) ⊂ U
}
.
To define suitable algebraic upper bounds, we will need the following notion from [18]:
Definition 4.1. Given a subspace a0 ⊂ g0, the graded subalgebra a = pr(g−, a0) := a− ⊕ a0 ⊕
a1 ⊂ g, where a− := g− = g−n−1 ⊕ · · · ⊕ g−1 and a1 := {x ∈ g1 : [x, g−1] ⊂ a0}, is called the
Tanaka prolongation algebra. For ϕ in some g0-module, we define aϕ := pr(g−, ann(ϕ)), where
ann(ϕ) ⊂ g0 is the annihilator of ϕ.
Now, we define
U := max
{
dim aϕ : 0 ̸= ϕ ∈ E
}
and UU := max
{
dim aϕ : 0 ̸= ϕ ∈ U
}
. (4.1)
By [16, Theorem 2.11], we conclude that
S ≤ U < dim g and SU ≤ UU for all G0- irreducible modules U ⊂ E. (4.2)
Note that U = maxU⊂E UU. In fact, by [16, Theorem 1.2], in all of the vector cases we have
equality:
S = U and SU = UU.
Examples of some vector ODEs realizing these can be found in [16, Tables 8 and 10].
4.2 Local homogeneity and algebraic models of ODE type
Recall that a Cartan geometry (G → E , ω) of type (G,P ) is said to be locally homogeneous if there
exists a (left) action by a local Lie group F on G by principal bundle morphisms preserving ω
that projects onto a transitive action down on E . We then have [16, Lemma A.1]:
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 17
Lemma 4.2. Fix a G0-irrep U ⊂ E. Then any regular, normal Cartan geometry (G → E , ω)
of type (G,P ) with 0 ̸≡ im(κH) ⊂ U and dim(inf(G, ω)) = UU is locally homogeneous about any
point u ∈ G with κH(u) ̸= 0.
By [16, Section A.1], such a homogeneous Cartan geometry can be encoded Cartan-theoreti-
cally by:
Definition 4.3. An algebraic model (f; g, p) of ODE type is a Lie algebra (f, [·, ·]f) such that:
(i) f ⊂ g is a filtered subspace whose associated graded s := gr(f) ⊂ g has s− = g−;
(ii) f0 inserts trivially into κ(x, y) := [x, y]− [x, y]f, i.e., κ(z, ·) = 0 for all z ∈ f0;
(iii) κ is normal and strongly regular : ∂∗κ = 0 and κ(gi, gj) ⊂ gi+j+1 ∩ gmin(i,j)−1, ∀i, j.
Let N denote the set of all algebraic models (f; g, p) of ODE type for fixed (G,P ). Then N
admits a P -action and is partially ordered:
(1) P -action: for p ∈ P and f ∈ N , we have p · f := Adp(f). We will regard all algebraic models
(f; g, p) of ODE type in the same P -orbit to be equivalent.
(2) Partial order relation ≤: for f, f̃ ∈ N regard f ≤ f̃ if there exists a map f ↪→ f̃ of Lie
algebras. We will focus on maximal elements in (N ,≤).
Combining (4.2), Lemma 4.2 and Definition 4.3, we obtain the following key existence result:
Theorem 4.4. Fix an irreducible G0-module U in the effective part E for vector ODEs (1.1)
of order ≥ 3. Then there exists an algebraic model (f; g, p) of ODE type with 0 ̸≡ im(κH) ⊂ U
and dim f = UU = SU.
Remark 4.5. Conversely, by [18, Lemma 4.1.4], for a given algebraic model (f; g, p) of ODE
type, there exists a locally homogeneous strongly regular, normal Cartan geometry (G → E , ω)
of type (G,P ) with inf(G, ω) containing a subalgebra isomorphic to f.
We caution that such a geometry may not arise from an ODE (1.1). (For instance, the
Doubrov–Medvedev condition must additionally hold.) Consequently, our strategy involves:
(i) classifying (up to the P -action) the corresponding algebraic models (f; g, p) of ODE type,
and then
(ii) providing vector ODEs of C-class realizing these algebraic models.
A filtered linear space f ⊂ g can be described as the graph of some linear map on s
into g as follows. Let s⊥ ⊂ g be a graded subalgebra such that g = s ⊕ s⊥. Then f :=⊕
i span {x+D(x) : x ∈ si}, for some unique linear (deformation) map D : s → s⊥ such that
D(x) ∈ s⊥ ∩ gi+1 for x ∈ si.
We will use the following results from [16, Section A.1] in carrying out the classifications.
Lemma 4.6. Let T ∈ f0 and suppose that the complementary graded subspaces s, s⊥ ⊂ g are
adT -invariant, then the map D : s → s⊥ is adT -invariant, i.e., T ·D = 0 ⇐⇒ adT ◦D = D◦adT .
Recall from Section 2.3 that κH := κ mod im ∂∗, where ∂∗ is the codifferential. We then
have:
Proposition 4.7. Let (f; g, p) be an algebraic model of ODE type. Then
(i) (f, [·, ·]f) is a filtered Lie algebra.
(ii) f0 · κ = 0, i.e., [z, κ(x, y)]f = κ([z, x]f, y) + κ(x, [z, y]f), ∀x, y ∈ f, ∀z ∈ f0.
(iii) s ⊂ aκH .
Following [28, Section 2.2], we shall refer to f as a (constrained) filtered sub-deformation of s.
18 J.A. Kessy and D. The
4.3 Characterizing maximality of the Tanaka prolongation
Fix a G0-irrep U ⊂ E, and recall UU defined in (4.1). For vector ODEs (1.1), UU were computed
in [16, Section 3.4] using the fact that UU = dim aΦU , where ΦU ∈ U is an extremal (lowest
or highest) weight vector. For the purpose of our goal in Section 5, we next prove that UU is
achieved precisely in this way:
Lemma 4.8. Let U ⊂ E be a G0-irrep and ΦU ∈ U be a lowest weight vector. Then, UU =
dim aΦU. Moreover, if 0 ̸= ϕ ∈ U, then dim aϕ = UU iff [ϕ] is contained in the G0-orbit
of [ΦU] ∈ P(U).
Proof. The proof used in [18, Proposition 3.1.1] can be applied for our purposes here. (We
note that the initial hypothesis of G complex semisimple Lie group and P ≤ G a parabolic
subgroup is not necessary. We use our G0 here for the G0 appearing there.) Over C, the same
proof yields the result. Over R, the essential fact used in the proof is that the split-real Lie
group SLmR ⊂ G0 acts with a unique closed orbit O (of minimal dimension) in P(U), where U
is an SLmR irrep. (See [30, Corollary 1].) For U ⊂ E in Table 2, the explicit orbits are
U sl(W )-module structure O ⊂ P(U)
B4 S2W ∗ {[
η2
]
: [η] ∈ P(W ∗)
}
Atr
2 W ∗ P(W ∗)
Atf
2
(
S2W ∗ ⊗W
)
0
{[
η2 ⊗ w
]
: [η] ∈ P(W ∗), [w] ∈ P(W ), η(w) = 0
}
This finishes the proof. ■
4.4 Prolongation-rigidity
In terms of the Tanaka prolongation algebra aϕ (see Definition 4.1), we define:
Definition 4.9. A G0-module U ⊂ E is said to be prolongation-rigid (PR) if aϕ1 = 0 for all
non-zero ϕ ∈ U.
Let U ⊂ E be an irreducible C-class module (see Section 2.5). To study prolongation-rigidity,
it suffices by Lemma 4.8 to consider the lowest weight vector ϕ = ΦU ∈ U. By [16, Lemma 3.3],
we have aΦU
1 = RY if and only if U has bi-grade that is a multiple of (n, 2). From Table 2, the
bi-grade of U is a multiple of (1, 1), so U is not PR if and only if n = 2. A summary is given in
Table 5, with aΦU in each case, and ann(ΦU) stated in Table 3.
n U U PR? aΦU
2 B4 × g− ⊕ ann(ΦU)⊕ RY
≥ 3 Atr
2 ✓ g− ⊕ ann(ΦU)
2 Atf
2 × g− ⊕ ann(ΦU)⊕ RY
≥ 3 Atf
2 ✓ g− ⊕ ann(ΦU)
Table 5. Prolongation-rigidity for irreducible C-class modules U ⊂ E.
5 Embeddings of filtered sub-deformations
By Section 4.2 above, all submaximally symmetric vector ODEs (1.1) can be encoded using
algebraic models of ODE type. Consequently, proving our main results (see Theorems 1.1
and 1.2) boils down to classifying these corresponding algebraic models (see Theorem 4.4).
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 19
More precisely, in view of Lemma 4.8, for each irreducible C-class module U ⊂ EC ⊊ E (see
Definition 2.7), our goal is to classify (up to the P -action) all algebraic models (f; g, p) of ODE
type with κH = ΦU ∈ U, where ΦU is the lowest weight vector from Table 4, and dim f = SU.
In this section, we classify all possible (filtered) linear embeddings f ⊂ g for such (f; g, p).
The possibilities for curvature κ of (f; g, p) are then classified in Section 6. Recall the canonical
filtration and the grading structure on g from Section 2.2. Having computed graded subalgebras
aΦU ⊂ g in Table 5, we next classify, up to the P -action, possible filtered linear subspaces f ⊂ g
for algebraic models (f; g, p) satisfying gr(f) = aΦU :
Proposition 5.1. Fix an irreducible C-class module U = B4, Atf
2 or Atr
2 in EC ⊊ E, viewed
as a G0-submodule of ker□ ⊂ C2(g−, g) via (2.5), and consider an algebraic model (f; g, p) of
ODE type with κH = ΦU ∈ U, for ΦU from Table 4, and dim f = SU. Then using the P -action,
f 7→ Adp f, we may normalize to
(a) f = aΦU when U = B4 or Atf
2 ;
(b) f = span{En,a, . . . , E2,a, E1,1 + (n − 2)ζZ1, E1,b, E0,1 + ζY, E0,b, X : ζ ∈ R, 1 ≤ a ≤ m,
2 ≤ b ≤ m} ⊕ ann(ΦAtr
2
) when U = Atr
2 .
Proof. Since U has a bi-grade that is a multiple of (1, 1) (see Table 2), then T := Z1 − Z2 ∈
ann(ΦU) = aΦU
0 . We note that aΦU ⊂ g is a graded subalgebra, and denote by T̂ the element
in f0 with the leading part T , i.e., gr0
(
T̂
)
= T . Since g1 = RY and gi = 0 for all i ≥ 2 (see
Section 2.2), then necessarily T̂ = T + sY ∈ f0. We claim that without loss of generality, i.e.,
using the P -action (for P defined in Section 2.2), we may assume that T ∈ f0. This is immediate
when U is not PR, since Y ∈ f0 and therefore T = T̂ − sY is a linear combination of T̂ and Y.
Otherwise, when U is PR
(
Y ̸∈ f0
)
, using the P+-action and [Y, T ] = −Y, we have
Adexp(tY)
(
T̂
)
= exp(adtY)
(
T̂
)
= T̂ +
[
tY, T̂
]
+
1
2!
[
tY,
[
tY, T̂
]]
+ · · · = Z1 − Z2 + (s− t)Y,
then choosing t = s normalizes the right-hand side to T . So, relabeling the left hand side by T̂ ,
gives T = T̂ ∈ f0.
By Definition 4.3 (ii), we have κ(T, z) = 0, i.e., [T, z]f = [T, z], ∀z ∈ f. Then, by exploiting the
semi-simplicity of adT , we next determine the remaining basis elements x̂ ∈ fi with the leading
parts x ∈ aΦU
i , i.e., gri(x̂) = x.
We first consider x̂ ∈ f0. We claim that without loss of generality, as it was for T̂ ∈ f0 above,
we may assume that x ∈ f0 for all x ∈ aΦU
0 . We let x̂ = x + cxY. Then, for U that is not
PR the claim holds, since Y ∈ f0, and so x = x̂ − cxY ∈ f0. In order to give the argument
for the case when U is PR, we recall that [T,Y] = Y and [T, x] = 0 for all x ∈ aΦU
0 . So,
[T, x̂]f = [T, x̂] = cxY ∈ f0. Now, since (f, [·, ·]f) is a Lie algebra and Y ̸∈ f0, then the closure
condition [T, x̂] ∈ f0 implies that cx = 0. So, x = x̂ ∈ f0.
Next, we similarly consider x̂ ∈ fi for i < 0. Recall that by Definition 4.1 for these cases we
have aΦU
i = gi, for gi as was defined in Section 2.2. In view of Lemma 4.6, we fix adT -invariant
subspaces s⊥ ⊂ g in Table 6 such that g = s ⊕ s⊥, where s := aΦU , and define the deformation
map D : s → s⊥ (see Section 4.2). Let Ei,a and X∗ denote the dual basis elements to Ei,a and X,
respectively, and recall bi-grades for the basis elements from Figure 1. Since, for 0 ≤ i ≤ n,
1 ≤ a, b, c ≤ m, the eigenvalues of adT on
Ei,a ⊗ Z1, Ei,a ⊗ ec
b, Ei,a ⊗ Y, X∗ ⊗ Z1, X∗ ⊗ ec
b, X∗ ⊗ Y (5.1)
are i− 1, i− 1, i, 1, 1, 2 respectively, then we have zero eigenvalues only when i = 0 or 1. Then
T ·D = 0 (see Lemma 4.6) implies X = X̂ ∈ f and Ei,a = Êi,a ∈ f for all i except possibly when
i = 0 or 1.
20 J.A. Kessy and D. The
n Irreducible C-class module U Generators for s⊥ ⊂ g Ranges
2 B4 Z1, e1
b 2 ≤ b ≤ m
≥ 3 Atr
2 Z1, e1
b, Y 2 ≤ b ≤ m
2 Atf
2 Z1, e1
b, ed
m 2 ≤ b ≤ m, 2 ≤ d ≤ m− 1
≥ 3 Atf
2 Z1, e1
b, ed
m, Y 2 ≤ b ≤ m, 2 ≤ d ≤ m− 1
Table 6. adT -invariant subspace s⊥ ⊂ g complementary to s = aΦU .
Now, consider the above exceptional cases. Based on the eigenvalues for adT given in (5.1),
we must have
Ê0,a = E0,a + λaY.
Since U is a C-class module, then κ(X, ·) = 0 (see Definition 2.4), which implies [X, ·]f = [X, ·].
Recall that (f, [·, ·]f) is a Lie algebra and [X, E0,a] = E1,a. Then for U that is
(a) not PR
(
U = B4 or Atf
2 when n = 2
)
: we have that E0,a = Ê0,a − λaY ∈ f, since Y ∈ f
and Ê0,a ∈ f. Since X ∈ f, then [X, E0,a]f = [X, E0,a] = E1,a ∈ f. Hence, E1,a = Ê1,a and so
f = aΦU .
(b) PR
(
U = Atf
2 or Atr
2 when n ≥ 3
)
: Recall from our discussion above that for any x̂ ∈ f0
with the leading part x ∈ aΦU
0 , we may assume without loss of generality that x ∈ f0.
Hence, Table 3 yields
q := e1
1 − e2
2 + Z2 ∈ f0 for Atr
2 ,
p := e1
1 − e2
2 + (2 + δm−1
1)Z2 ∈ f0 for Atf
2 .
Recall that by Definition 4.3 (ii) we have κ
(
f0, ·
)
= 0, which implies [z, ·]f = [z, ·] for all
z ∈ f0. Now, since both p and q commute with Y, [Z2, Ei,a] = −Ei,a and
[
e1
1−e22, Ei,a
]
=(
δa
1 − δa
2
)
Ei,a (see Section 2.2), then for
(i) U = Atf
2 : we have
[
p, Ê0,a
]
f
=
[
p, Ê0,a
]
=
(
δa
1− δa
2− δm−1
1− 2
)
E0,a. So, the closure
condition
[
p, Ê0,a
]
∈ f forces λa = 0, i.e., E0,a = Ê0,a ∈ f. Then [X, E0,a] = E1,a ∈ f,
which implies E1,a = Ê1,a ∈ f. Hence, f = a
ΦAtf2 . This completes the proof for (a).
(ii) U = Atr
2 : we have
[
q, Ê0,a
]
f
=
[
q, Ê0,a
]
=
(
δ1a − δ2a − 1
)
E0,a. So,
[
q, Ê0,a
]
∈ f implies
that λa = 0 except for a = 1, so for these cases we have E0,a = Ê0,a ∈ f. Consequently,
[X, E0,a] = E1,a ∈ f implies that E1,a = Ê1,a ∈ f except for a = 1.
Finally, we consider the case when a = 1. Based on the eigenvalues for adT given
in (5.1) and setting λ1 = ζ, we necessarily have
Ê0,1 = E0,1 + ζY and Ê1,1 = E1,1 + βZ1 +
m∑
b=2
αbe1
b.
Since from (2.2) we have H = nZ2 − 2Z1 and T = Z1 − Z2, then we have[
X, Ê0,1
]
f
= [X, E0,1 + ζY] = E1,1 + ζH = E1,1 + ζ(nZ2 − 2Z1)
= E1,1 + ζ(n− 2)Z1 − ζnT.
So, the closure condition [X, Ê0,1]f ∈ f holds only if Ê1,1 = E1,1 + ζ(n − 2)Z1 − ζnT .
This implies β = (n − 2)ζ and αb = 0 for all b, which proves (b) and concludes our
proof. ■
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 21
We have the following result for the curvature κ of such an algebraic model. This result is
essential for our study of curvatures in Section 6.
Corollary 5.2. Fix an irreducible C-class module U ⊂ EC ⊊ E, viewed as a G0-submodule
of ker□ ⊂ C2(g−, g) via (2.5), and consider an algebraic model (f; g, p) of ODE type with
κH = ΦU ∈ U, for ΦU from Table 4, and dim f = SU, normalized according to Proposition 5.1.
Then X · κ = 0.
Proof. Since U is a C-class module, then, κ ∈
∧2 V ∗⊗g (see Remark 2.5) for V from Section 2.2.
So, for X ∈ f we have [X, z]f = [X, z] for all z ∈ f. Then, as a consequence of the Jacobi identity
we get the claim as follows:
(X · κ)(x, y) = [X, κ(x, y)]− κ([X, x], y)− κ(x, [X, y])
= [X, [x, y]]− [X, [x, y]f]︸ ︷︷ ︸
[X,[x,y]f]f
+[[X, x]︸ ︷︷ ︸
[X,x]f
, y]f − [[X, x], y] + [x, [X, y]︸ ︷︷ ︸
[X,y]f
]f − [x, [X, y]] = 0. ■
6 Classification of submaximally symmetric vector ODEs
of C-class
In this section, we classify (up to the P -action) all algebraic models of ODE type for submaxi-
mally symmetric vector ODEs (1.1) of C-class (see the introduction to Section 5) and establish
Theorems 1.1 and 1.2.
6.1 Algebraic curvature constraints
For the algebraic models (f; g, p) whose possible filtered linear subspaces f ⊂ g have been classified
in Proposition 5.1, we classify their possible curvatures κ below.
Proposition 6.1. Fix an irreducible C-class module U = B4, Atf
2 or Atr
2 in EC ⊊ E, viewed
as a G0-submodule of ker□ ⊂ C2(g−, g) via (2.5), and consider an algebraic model (f; g, p) of
ODE type with κH = ΦU ∈ U, for ΦU from Table 4, and dim f = SU, normalized according to
Proposition 5.1. Then κ is
(a) U = B4 : κ = ±ΦU (over C, we can take κ = ΦU);
(b) U = Atf
2 : κ = ΦU;
(c) U = Atr
2 : κ = ΦU + κ4, where
κ4 = µ1E
3,1 ∧ E0,1 ⊗ X+ µ2E
2,1 ∧ E1,1 ⊗ X
− µ1 + µ2
2
(
E2,1 ∧ E0,1 ⊗ H+ E1,1 ∧ E0,1 ⊗ Y
)
+ µ3
m∑
a=1
(
E2,1 ∧ E0,a − E2,a ∧ E0,1 + E1,a ∧ E1,1
)
⊗ ea
1 (6.1)
for some µ1, µ2, µ3 ∈ R.
Proof. The majority of the proof will consist of evaluating the annihilation conditions f0 ·κ = 0.
Recall from Table 2 that U has bi-grade either (1, 1) or (2, 2), so Z1−Z2 ∈ ann(ΦU), which is
contained in f0 by Proposition 5.1. Hence, (Z1 − Z2) · κ = 0 implies that κ is the sum of terms
with bi-grades that are multiples of (1, 1). But κ is regular, lies in
∧2(g/p)∗ ⊗ g, and Z2 acts on
the latter with eigenvalues 0, 1 or 2. Thus, the terms in κ can only have bi-grades (1, 1) or (2, 2).
By Theorem A.3, κH can be identified with the lowest Z-degree component of κ. Moreover, κH is
22 J.A. Kessy and D. The
a nonzero multiple of ΦU. Using the G0-action by exp(Zt), where Z ∈ z(g0) is the grading ele-
ment (see Section 2.2), this multiple can be re-scaled to ±1. For U = Atf
2 or Atr
2 , we can further
normalize this multiple to +1. (Use the diagonal elements in g = diag(a1, . . . , am) ∈ GLm ⊂ G0,
i.e., g · Φi,j = 1
a1
Φi,j for Φi,j from (3.3), while g · Φi,j = am
(a1)2
Φi,j for Φi,j from (3.11).) Summa-
rizing, we have
κ =
{
±ΦU, when U = B4,
ΦU + κ4, when U = Atf
2 or Atr
2 ,
where κ4 is the bi-grade (2, 2) component of κ. The B4 case is complete, and we turn to the
remaining cases.
Since U is a C-class module, then by Remark 2.5 we have κ ∈
∧2 V ∗ ⊗ g in the notation
of Section 2.2. Recall g = q ⋉ V , and q and V have Z2-degrees 0 and −1 respectively (see
Figure 1). In particular, ΦU ∈
∧2 V ∗ ⊗ V and κ4 ∈
∧2 V ∗ ⊗ q. More precisely, since κ4 has
bi-grade (2, 2), then in terms of the dual basis elements Ei,a to Ei,a, having bi-grades (i, 1) and
(−i,−1) respectively, κ4 must lie in the subspace K4 ⊂
∧2 V ∗ ⊗ q spanned by
E1,a ∧ E0,b ⊗ Y, E3,a ∧ E0,b ⊗ X, E2,a ∧ E1,b ⊗ X,
E2,a ∧ E0,b ⊗ H, E1,a ∧ E1,b ⊗ H, E2,a ∧ E0,b ⊗ ec
d, E1,a ∧ E1,b ⊗ ec
d, (6.2)
where 1 ≤ a, b, c, d ≤ m. We will further constrain κ4 as follows. Using Proposition 5.1, we
have ann(ΦU) ⊂ f0. Such elements annihilate both ΦU and κ, and so
z · κ4 = 0, ∀z ∈ ann(ΦU).
Let us use these to find more explicit conditions on κ4.
(1) U = Atf
2 : from Table 3, we have p := e1
1− e22+
(
2+ δm−1
1
)
Z2 ∈ ann(ΦU). From 0 = p ·κ4
and Z2 · κ4 = 2κ4, we have that κ4 has eigenvalue λ = −2
(
2 + δm−1
1
)
for e1
1 − e2
2. We
conclude that κ4 = 0 (hence κ = ΦU) from the following considerations:
(i) m = 2: We have λ = −6. Noting that e1
1 − e2
2 commutes with {X,H,Y}, and(
e1
1 − e2
2
)
· Ei,a =
(
δa
2 − δa
1
)
Ei,a,(
e1
1 − e2
2
)
· eab = δa
1e1
b − δ1
bea
1 − δa
2e2
b + δ2
bea
2.
From (6.2), we conclude that the eigenvalues of e1
1− e22 in K4 lie between −4 and 4.
Since −6 is not an eigenvalue, then κ4 = 0.
(ii) m ≥ 3: We have λ = −4. Proceeding as in (a), we observe that K4 has −4-
eigenspace for e1
1− e22 spanned by E2,1 ∧E0,1⊗ e21. But from Table 3, we also have
em−1
m−1 − em
m + Z2 ∈ ann(ΦU), which must similarly annihilate κ and κ4. But its
eigenvalue on E2,1 ∧ E0,1 ⊗ e2
1 is δm−1
2 + 2, which is nonzero, so κ4 = 0 follows.
(2) U = Atr
2 : from Table 3, we have qd := ed
d − ed+1
d+1 + δ1
dZ2 ∈ ann(ΦU), so 0 = qd · κ4
for 1 ≤ d ≤ m − 1. Letting h ⊂ slm denote the standard Cartan subalgebra consisting of
diagonal trace-free matrices, and ϵa ∈ h∗ the standard weights for h, we observe
(i) 0 = qd · κ4 for 1 ≤ d ≤ m− 1 is equivalent to κ4 having weight −2ϵ1,
(ii) the first five elements of (6.2) have weight −ϵa − ϵb,
(iii) the last two elements of (6.2) have weight −ϵa − ϵb + ϵc − ϵd.
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 23
Matching these weights with −2ϵ1, we deduce that κ4 lies in the span of the following:
E3,1 ∧ E0,1 ⊗ X, E2,1 ∧ E1,1 ⊗ X, E2,1 ∧ E0,1 ⊗ H,
E1,1 ∧ E0,1 ⊗ Y, E2,1 ∧ E0,1 ⊗ e1
1, E2,1 ∧ E0,1 ⊗ ea
a,
E2,1 ∧ E0,a ⊗ ea
1, E2,a ∧ E0,1 ⊗ ea
1, E1,a ∧ E1,1 ⊗ ea
1,
where 2 ≤ a ≤ m. Similarly as in Section 3.1, we conclude that imposing annihilation by
all of ann(ΦAtr
2
) ⊂ f0 forces κ4 to lie in the subspace spanned by
E3,1 ∧ E0,1 ⊗ X, E2,1 ∧ E1,1 ⊗ X, E2,1 ∧ E0,1 ⊗ H, E1,1 ∧ E0,1 ⊗ Y,
m∑
a=1
E2,1 ∧ E0,1 ⊗ ea
a,
m∑
a=1
E2,1 ∧ E0,a ⊗ ea
1,
m∑
a=1
E2,a ∧ E0,1 ⊗ ea
1,
m∑
a=1
E1,a ∧ E1,1 ⊗ ea
1. (6.3)
Finally, we complete the proof by imposing X ·κ = 0 (see Corollary 5.2). Since X ·ΦAtr
2
= 0
(see Lemma 2.2), then X · κ = 0 implies that X · κ4 = 0. Now let κ4 be a general linear
combination of all elements of (6.3), i.e., κ4 = ν1E
3,1∧E0,1⊗X+ ν2E
2,1∧E1,1⊗X+ · · ·+
ν8
∑m
a=1E
1,a ∧ E1,1 ⊗ ea
1, and impose 0 = X · κ4 using the actions given in Section 2.2.
Namely, X · Y = H, X · H = −2X, and X · eab = 0. Also, X · Ei,a = Ei+1,a, and so
X · Ei,a = −Ei−1,a. We find that 0 = X · κ4 is equivalent to
ν3 = ν4 = −ν1 + ν2
2
, ν5 = 0, ν6 = ν8 = −ν7.
Setting (ν1, ν2, ν8) = (µ1, µ2, µ3) then yields the result. ■
Corollary 6.2. All parameters involved in an algebraic model (f; g, p) of ODE type from Propo-
sition 6.1 for U = Atr
2 are uniquely determined.
Proof. Recall from Table 2 that U = Atr
2 arises for n ≥ 3. By Propositions 5.1 (b) and 6.1 (c),
any algebraic model (f; g, p) of ODE type with κH = ΦU ∈ U, for ΦU from Table 4, and
dim f = SU has
f = span
{
En,a, . . . , E2,a, Ê1,1, E1,b, Ê0,1, E0,b,X : 1 ≤ a ≤ m, 2 ≤ b ≤ m
}
⊕ ann(ΦU), (6.4)
where ann(ΦU) was given in Table 3, and
Ê1,1 := E1,1 + (n− 2)ζZ1 ∈ f, Ê0,1 := E0,1 + ζY ∈ f
for some ζ ∈ R. Curvature is κ = ΦAtr
2
+ κ4, for κ4 given in (6.1), and [·, ·]f = [·, ·]− κ(·, ·).
Let us now impose the Jacobi identity. We define
Jacf(x, y, z) := [x, [y, z]f]f − [[x, y]f, z]f − [y, [x, z]f]f, ∀x, y, z ∈ f.
We calculate[
Ê1,1, [E0,2, E3,1]f
]
f
= −(n− 2)2
(
2ζ +
3(2m+ 3)− n(4m+ 3)
mn(n+ 1) + 6
)
E2,2,[
E0,2,
[
Ê1,1, E3,1
]
f
]
f
= −(n− 2)2
(
3ζ +
3(3m+ 5)− n(5m+ 3)
mn(n+ 1) + 6
)
E2,2,[[
Ê1,1, E0,2
]
f
, E3,1
]
f
= −(n− 1)(n− 2)2(mn− 3)
mn(n+ 1) + 6
E2,2,
24 J.A. Kessy and D. The
so that
Jacf
(
Ê1,1, E0,2, E3,1
)
= 0 implies ζ =
(
2n− n2 − 3
)
m+ 3n− 9
mn(n+ 1) + 6
. (6.5)
Continuing in a similar manner, we find that
Jacf
(
Ê0,1, E1,2, E3,1
)
= 0 implies µ1 =
6(n− 1)(n− 2)(m+ 1)
mn(n+ 1) + 6
, (6.6)
Jacf
(
Ê1,1, E2,2, E2,1
)
= 0 implies µ2 = −
6(n− 1)(m+ 1)
(
m
(
n3 + n2 − 6n+ 6
)
+ 6
)
(mn(n+ 1) + 6)2
.
Using ζ and µ1 above, we then have
Jacf(Ê0,1, E2,2, E3,1) = 0 implies µ3 = 1− n. (6.7)
As claimed, the parameters ζ, µ1, µ2, µ3 are uniquely determined functions of (n,m).
We remark that the remaining Jacobi identities for f are necessarily satisfied because the
existence of a submaximally symmetric ODE model in the Atr
2 -branch (see Table 1) guarantees
the existence of a corresponding algebraic model of ODE type. (Necessarily, this is equivalent
to the one found above.) ■
6.2 Conclusion
Let us now complete the proofs for Theorems 1.1 and 1.2. Fix an irreducible C-class module
U = B4,Atr
2 , or Atf
2 in the effective part E, and recall the respective lowest weight vectors ΦU ∈ U
from Table 4. By Propositions 5.1 and 6.1, the classification of algebraic models (f; g, p) of ODE
type with 0 ̸≡ im(κH) ⊂ U and dim f = SU is given in Table 7. This completes step (i) of the
classification strategy given in Remark 4.5.
n
Irreducible C-class
moduleU ⊂ E
f κ
2 B4 aΦU
{
ΦU, over C
±ΦU, over R
≥ 3 Atr
2
f in (6.4) with
ζ in (6.5)
ΦU + κ4, with β = 1, κ4 in (6.1),
and µ1, µ2, µ3 in (6.6) and (6.7).
≥ 2 Atf
2 aΦU ΦU
Table 7. Classification of algebraic models of ODE type with 0 ̸≡ im(κH) ⊂ U and dim f = SU.
We now turn to step (ii) of Remark 4.5 and discuss how the ODE model classification in
Table 1 is deduced from the abstract classification in Table 7. Using fundamental invariants
described in Section 2.4, we confirm that these ODE lie in the claimed branches. In [16, Table 10],
the point symmetries were given for all of these models with the exception of the second B4
model. (See below for this case.) We confirm submaximal symmetry dimensions and deduce
the associated algebraic models. (The latter is immediate by uniqueness in the Atr
2 , Atf
2 cases,
as well as the B4 case over C.)
To complete the proof of Theorem 1.2, we establish point-inequivalence over R of the following
submaximally symmetric B4 models:
ua3 =
3u12u
a
2
2u11
or ua3 =
3u11u
1
2u
a
2
1 + (u11)
2
for 1 ≤ a ≤ m. (6.8)
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 25
The point symmetry algebra S of the former is given in [16, Table 10]. On J0(R,Rm), the
distributions ker
(
du1
)
and ker(dt) are each S-invariant, so these determine S-invariant foliations
by level sets of u1 and t, respectively. Total differentiation of the former implies that the level
set
{
u11 = 0
}
⊂ J1(R,Rm) is S-invariant. Hence, the prolonged action of S on J1(R,Rm) is not
locally transitive.
In contrast, we now establish that the latter ODE in (6.8) has symmetry algebra that acts
locally transitively on J1(R,Rm). Its point symmetries are (for 1 ≤ a ≤ m and 2 ≤ b ≤ m)
∂t, ∂ua , t∂ub , ua∂ub ,
(
t2 +
(
u1
)2)
∂ub , u1∂t − t∂u1 ,
t∂t + u1∂u1 + 2
m∑
b=2
ub∂ub ,
(
t2 −
(
u1
)2)
∂t + 2t
m∑
a=1
ua∂ua ,
tu1∂t +
1
2
((
u1
)2 − t2
)
∂u1 + u1
m∑
b=2
ub∂ub .
In particular over R, transitivity immediately follows from prolonging some of them to J1(R,Rm):
∂t, ∂ua , t∂ub + ∂ub
1
, u1∂t − t∂u1 −
(
1 +
(
u11
)2)
∂u1
1
− u11
m∑
b=2
ub1∂ub
1
.
Thus, the symmetry algebras of (6.8) are point-inequivalent, and hence the ODEs are point-
inequivalent.
An alternate method is to establish that the symmetry algebras are abstractly non-isomorphic.
Indeed, for m ≥ 2, their semisimple parts are respectively sl2 × sl2 × slm−1 and so1,3 × slm−1.
(The m = 1 case was remarked in [27, p. 18].) However, this requires more details, while our
argument given above is more direct. Moreover:
Remark 6.3. Considering invariant foliations also gives the added bonus of suggesting a complex
point-equivalence between the two ODE systems in (6.8). If we regard the latter ODE in (6.8)
over C, then on J0(C,Cm), we find two invariant foliations by levels sets of u1 + it and u1 − it
respectively. The invariant foliations discussed above in the first case now suggest considering
the following (complex) point transformation(
t̃, ũ1, ũ2, . . . , ũm
)
=
(
u1 + it, u1 − it, u2, . . . , um
)
.
We can straightforwardly verify that its prolongation pulls back the former ODE in (6.8) (written
in tilded variables) to the latter ODE in (6.8).
This completes the proof of Theorem 1.2. Following the remarks preceding Theorem 1.1,
we have also proven the remaining Theorem 1.1(b) since for vector ODEs (1.1) of C-class of
order n+ 1 ≥ 3, we have S = M− 2 = SB4 = SAtf
2
only when (n,m) = (2, 2). This completes
our proofs for Theorems 1.1 and 1.2.
A Harmonic curvature as the lowest degree component
of curvature
Fix G and P as in Section 2.2 and recall from Section 2.3 some basic notions of Cartan geome-
tries (G → E , ω) of type (G,P ) associated to ODEs (1.1). We formulate Theorem A.3 below
stating that the harmonic curvature κH can be identified with the lowest degree component (with
respect to the grading element) of the curvature κ. (We note that this is used in the proof of
Proposition 6.1, which is essential in proving Theorems 1.1 and 1.2.)
26 J.A. Kessy and D. The
Definition A.1. Let (G → E , ω) be a Cartan geometry of type (G,P ), let ρ : G → GL(V ) be
a G-representation, and ρ ◦ ι : P → GL(V ) its restriction, where ι : P ↪→ G is the canonical
inclusion. A tractor bundle is an associated vector bundle G ×P V with respect to the P -
representation ρ ◦ ι. Given the adjoint representation ρ = Ad: G → GL(g), the tractor bundle
AE := G ×P g is called the adjoint tractor bundle (see [4, Section 1.5.7] for further details).
Using the Cartan connection ω, the tangent bundle TE can be identified with the bundle
G ×P (g/p). Then, the P -invariant quotient map from g onto g/p gives rise to the natural
projection Π: AE → TE . Using this identification, we can regard the curvature as κ ∈ Ω2(E ,AE),
i.e., AE-valued 2-form on E [4, Proposition 1.5.7].
Definition A.2. Given a Cartan geometry (G → E , ω) of type (G,P ) with curvature κ ∈
Ω2(E ,AE). Then
(a) ω is called regular if κ ∈
(
Ω2(E ,AE)
)1
, i.e., κ(T iE , T jE) ⊂ Ai+j+1E , ∀i, j < 0.
(b) ω is called normal if ∂∗κ = 0.
(c) If ω is both regular and normal, then the harmonic curvature is κH := κ mod im(∂∗),
which is a section of G ×P
ker ∂∗
im ∂∗ .
Then, we have the following result.
Theorem A.3. Fix G and P as in Section 2.2. Let (G → E , ω) be a regular, normal Cartan
geometry of type (G,P ) whose curvature κ ∈
(
Ω2(E ,AE)
)ℓ
for some ℓ ≥ 1, i.e., κ
(
T iE , T jE
)
⊂
Ai+j+ℓE for all i, j < 0. Then the induced section grℓ(κ) ∈ grℓ
(
Ω2(E ,AE)
)
coincides with the
degree ℓ component of the harmonic curvature κH . Consequently, κH ≡ 0 implies κ ≡ 0.
Proof. The statement was proved in [4, Theorem 3.1.12] for parabolic geometries. The same
proof works for our non-parabolic Cartan geometries associated to vector ODEs (1.1) of or-
der ≥ 3. ■
B A necessary condition for coclosedness of ΦAtr
2
From Section 3.2, our strategy for computing a sl(W )-lowest weight vector ΦAtr
2
∈ Atr
2 involves
imposing coclosedness, i.e., ∂∗ΦAtr
2
= 0, where ∂∗ was defined in Section 2.3. By adjointness of ∂
and ∂∗ with respect to the inner product ⟨·, ·⟩ on cochains induced from Definition 2.1, we have
∂∗ΦAtr
2
= 0 ⇐⇒
〈
ΦAtr
2
, ∂ψ
〉
= 0, ∀ψ ∈ g∗− ⊗ g. (B.1)
In order to pin down ΦAtr
2
in Proposition 3.5, only a small part of the conditions in (B.1) will be
in fact required. In this section, we identify a key condition (see Lemma B.2) that is essential
to the proof of Proposition 3.5.
Recalling ann(ΦAtr
2
) given in Table 3, let us restrict attention to ψ lying in the subspace
below.
Lemma B.1. Suppose that ψ ∈ g∗−⊗ g has bi-grade (1, 1), with X ·ψ = 0 and ann(ΦAtr
2
) ·ψ = 0.
Then ψ is a multiple of
Ψ := −2E2,1 ⊗ X+ E1,1 ⊗ H+ E0,1 ⊗ Y. (B.2)
Proof. Any ψ ∈ g∗− ⊗ g with bi-grade (1, 1) lies in the span of
E2,a ⊗ X, E1,a ⊗ H, E0,a ⊗ Y, E1,a ⊗ eb
c, 1 ≤ a, b, c ≤ m.
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 27
Since X ·Ei,a = Ei+1,a, then X ·Ei,a = −Ei−1,a. Imposing X ·ψ = 0 forces ψ to lie in the span of
−2E2,a ⊗ X+ E1,a ⊗ H+ E0,a ⊗ Y, 1 ≤ a ≤ m. (B.3)
Let us now impose ann(ΦAtr
2
) · ψ = 0. Recall from Table 3 that qd = ed
d − ed+1
d+1 + δ1
dZ2 ∈
ann(ΦAtr
2
) for 1 ≤ d ≤ m− 1. Let h ⊂ slm denote the standard Cartan subalgebra consisting of
diagonal trace-free matrices, and ϵa ∈ h∗ the standard weights for h. Since Z2 · ψ = ψ, then
qd · ψ = 0 for 1 ≤ d ≤ m− 1 ⇐⇒ ψ has weight− ϵ1.
Since each element of (B.3) has weight −ϵa, then being of weight −ϵ1 implies that ψ is a multiple
of (B.2). We note that (B.2) is annihilated by all off-diagonal elements ef
d ∈ ann
(
ΦAtr
2
)
, since
we have f ≥ 2 and ef
d commutes with {X,H,Y}. This completes the proof. ■
In terms of Φi,j =
∑m
a=1E
i,1 ∧ Ej,a ⊗ Ei+j−1,a defined in (3.3), and using ∂ (2.4), we get
∂Ψ = −2
n−1∑
k=0
Φ2,k +
n∑
k=0
(2k − n)Φ1,k +
n∑
k=1
k(n+ 1− k)Φ0,k. (B.4)
We then have the following necessary condition, which will be used in the proof of Proposition 3.5.
Lemma B.2. Take ΦAtr
2
defined in Proposition 3.3, i.e., ΦAtr
2
=
∑n
i,j=0 ci,jΦ
i,j with ci,j satisfy-
ing (3.4). Then
0 =
n−1∑
k=0
(n− k)(k + 1)
n(n− 1)
(ck,2 −mc2,k) +
n∑
k=0
2k − n
n
(mc1,k − ck,1) +
n∑
k=1
(mc0,k − ck,0).
Proof. We evaluate (B.1) for ψ = Ψ given in (B.2). In preparation for this, note that from
Definition 2.1, we have ⟨Ek,a, Ek,a⟩ = k!
(n−k)! and ⟨Ek,a, Ek,a⟩ = (n−k)!
k! , and so
∣∣∣∣E2,1 ∧ Ek,a ⊗ Ek+1,a
∣∣∣∣2 = ∣∣∣∣E2,1
∣∣∣∣2∣∣∣∣Ek,a
∣∣∣∣2∣∣∣∣Ek+1,a
∣∣∣∣2 = (n− k)(k + 1)(n− 2)!
2
.
Hence, by bilinearity of ⟨·, ·⟩ and orthogonality of the basis elements for g (see Definition 2.1),
we have
〈
Φ2,k,ΦAtr
2
〉
=
n∑
i,j=0
m∑
a,b=1
ci,j
〈
E2,1 ∧ Ek,b ⊗ Ek+1,b, E
i,1 ∧ Ej,a ⊗ Ei+j−1,a
〉
=
n∑
i,j=0
m∑
a=1
(
δi
2δj
k − δi
kδj
2δa
1
)
ci,j
∣∣∣∣E2,1 ∧ Ek,a ⊗ Ek+1,a
∣∣∣∣2
=
m∑
a=1
(
c2,k − ck,2δa
1
)(n− k)(k + 1)(n− 2)!
2
= (mc2,k − ck,2)
(n− k)(k + 1)(n− 2)!
2
.
Similarly, we have〈
Φ1,k,ΦAtr
2
〉
= (n− 1)!(mc1,k − ck,1),
〈
Φ0,k,ΦAtr
2
〉
=
n!
k(n+ 1− k)
(mc0,k − ck,0).
We use these relations and (B.4) to evaluate 0 = ⟨∂Ψ,ΦAtr
2
⟩ and obtain the claimed result. ■
28 J.A. Kessy and D. The
Acknowledgements
The authors acknowledge the use of the DifferentialGeometry package in Maple. We also ac-
knowledge helpful conversations with Boris Kruglikov, Andreu Llabres, and Eivind Schneider.
The research leading to these results has received funding from the Norwegian Financial Mech-
anism 2014–2021 (project registration number 2019/34/H/ST1/00636), the Tromsø Research
Foundation (project “Pure Mathematics in Norway”), and the UiT Aurora project MASCOT,
and this article/publication is based upon work from COST Action CaLISTA CA21109 sup-
ported by COST (European Cooperation in Science and Technology), https://www.cost.eu.
References
[1] Bryant R.L., Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geo-
metry and Lie Theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., Vol. 53, American Mathematical
Society, Providence, 1991, 33–88.
[2] Čap A., On canonical Cartan connections associated to filtered G-structures, arXiv:1707.05627.
[3] Čap A., Doubrov B., The D., On C-class equations, Comm. Anal. Geom., to appear, arXiv:1709.01130.
[4] Čap A., Slovák J., Parabolic geometries I. Background and general theory, Math. Surveys Monogr., Vol. 154,
American Mathematical Society, Providence, 2009.
[5] Cartan É., Les espaces généralisés et l’intégration de certaines classes d’équations différentielles, C. R. Hebd.
Séances Acad. Sci. 206 (1938), 1689–1693.
[6] Casey S., Dunajski M., Tod P., Twistor geometry of a pair of second order ODEs, Comm. Math. Phys. 321
(2013), 681–701, arXiv:1203.4158.
[7] Doubrov B., Three-dimensional homogeneous spaces with non-solvable transformation groups,
arXiv:1704.04393.
[8] Doubrov B., Contact trivialization of ordinary differential equations, in Differential Geometry and Its Ap-
plications (Opava, 2001), Math. Publ., Vol. 3, Silesian University Opava, Opava, 2001, 73–84.
[9] Doubrov B., Generalized Wilczynski invariants for non-linear ordinary differential equations, in Symmetries
and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer,
New York, 2008, 25–40, arXiv:math.DG/0702251.
[10] Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J.
Math. 3 (1999), 39–71.
[11] Doubrov B., Medvedev A., Fundamental invariants of systems of ODEs of higher order, Differential Geom.
Appl. 35 (2014), suppl., 291–313, arXiv:1312.0574.
[12] Dunajski M., Tod P., Paraconformal geometry of nth-order ODEs, and exotic holonomy in dimension four,
J. Geom. Phys. 56 (2006), 1790–1809, arXiv:math.DG/0502524.
[13] Godlinski M., Nurowski P., Geometry of third-order ODEs, arXiv:0902.4129.
[14] Godlinski M., Nurowski P., GL(2,R) geometry of ODE’s, J. Geom. Phys. 60 (2010), 991–1027,
arXiv:0710.0297.
[15] Grossman D.A., Torsion-free path geometries and integrable second order ODE systems, Selecta Math. (N.S.)
6 (2000), 399–442.
[16] Kessy J.A., The D., Symmetry gaps for higher order ordinary differential equations, J. Math. Anal. Appl.
516 (2022), 126475, 23 pages, arXiv:2110.03954.
[17] Kostant B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. 74 (1961),
329–387.
[18] Kruglikov B., The D., The gap phenomenon in parabolic geometries, J. Reine Angew. Math. 723 (2017),
153–215, arXiv:1303.1307.
[19] Kryński W., Paraconformal structures, ordinary differential equations and totally geodesic manifolds,
J. Geom. Phys. 103 (2016), 1–19, arXiv:1310.6855.
[20] Lie S., Classification und integration von gewöhnlichen differentialgleichungen zwischen x, y, die eine Gruppe
von Transformationen gestatten, Math. Ann. 32 (1888), 213–281.
https://www.cost.eu
https://doi.org/10.1090/pspum/053/1141197
https://doi.org/10.1090/pspum/053/1141197
https://arxiv.org/abs/1707.05627
https://arxiv.org/abs/1709.01130
https://doi.org/10.1090/surv/154
https://doi.org/10.1007/s00220-013-1729-7
https://arxiv.org/abs/1203.4158
https://arxiv.org/abs/1704.04393
https://doi.org/10.1007/978-0-387-73831-4_2
https://arxiv.org/abs/math.DG/0702251
https://doi.org/10.1016/j.difgeo.2014.07.001
https://doi.org/10.1016/j.difgeo.2014.07.001
https://arxiv.org/abs/1312.0574
https://doi.org/10.1016/j.geomphys.2005.10.007
https://arxiv.org/abs/math.DG/0502524
https://arxiv.org/abs/0902.4129
https://doi.org/10.1016/j.geomphys.2010.03.003
https://arxiv.org/abs/0710.0297
https://doi.org/10.1007/PL00001394
https://doi.org/10.1016/j.jmaa.2022.126475
https://arxiv.org/abs/2110.03954
https://doi.org/10.2307/1970237
https://doi.org/10.1515/crelle-2014-0072
https://arxiv.org/abs/1303.1307
https://doi.org/10.1016/j.geomphys.2016.01.003
https://arxiv.org/abs/1310.6855
https://doi.org/10.1007/BF01444068
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations 29
[21] Lie S., Vorlesungen über continuirliche Gruppen mit geometrischen und anderen Anwendungen, Teubner,
Leipzig, 1893.
[22] Medvedev A., Geometry of third order ODE systems, Arch. Math. (Brno) 46 (2010), 351–361.
[23] Medvedev A., Third order ODEs systems and its characteristic connections, SIGMA 7 (2011), 076, 15 pages,
arXiv:1104.0965.
[24] Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
[25] Schneider E., Projectable Lie algebras of vector fields in 3D, J. Geom. Phys. 132 (2018), 222–229,
arXiv:1803.08878.
[26] Se-ashi Y., On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J.
17 (1988), 151–195.
[27] Shah S.W., Mahomed F.M., Azad H., Symmetry classification of scalar nth order ordinary differential
equations, arXiv:2208.10395.
[28] The D., On uniqueness of submaximally symmetric parabolic geometries, arXiv:2107.10500.
[29] The D., A Cartan-theoretic classification of multiply-transitive (2, 3, 5)-distributions, arXiv:2205.03387.
[30] Winther H., Minimal projective orbits of semi-simple Lie groups, arXiv:2302.12138.
https://doi.org/10.3842/SIGMA.2011.076
https://arxiv.org/abs/1104.0965
https://doi.org/10.1017/CBO9780511609565
https://doi.org/10.1016/j.geomphys.2018.05.025
https://arxiv.org/abs/1803.08878
https://doi.org/10.14492/hokmj/1381517803
https://arxiv.org/abs/2208.10395
https://arxiv.org/abs/2107.10500
https://arxiv.org/abs/2205.03387
https://arxiv.org/abs/2302.12138
1 Introduction
2 Cartan geometries and vector ODEs of C-class
2.1 ODE geometry and symmetry
2.2 Structure underlying the trivial ODE
2.3 Cartan geometries associated to ODE
2.4 Vector ODEs of C-class
2.5 C-class modules
2.6 The Doubrov–Medvedev condition
3 Lowest weight vectors for irreducible C-class modules
3.1 B_4 case
3.2 A_2^{tr} case
3.3 A_2^{tf} case
4 Homogeneous structures and Cartan-theoretic descriptions
4.1 Symmetry gaps for ODE
4.2 Local homogeneity and algebraic models of ODE type
4.3 Characterizing maximality of the Tanaka prolongation
4.4 Prolongation-rigidity
5 Embeddings of filtered sub-deformations
6 Classification of submaximally symmetric vector ODEs of C-class
6.1 Algebraic curvature constraints
6.2 Conclusion
A Harmonic curvature as the lowest degree component of curvature
B A necessary condition for coclosedness of Phi_{A_2^{tr}}
References
|
| id | nasplib_isofts_kiev_ua-123456789-211971 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T12:31:13Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kessy, Johnson Allen The, Dennis 2026-01-20T16:11:53Z 2023 On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class. Johnson Allen Kessy and Dennis The. SIGMA 19 (2023), 058, 29 pages 1815-0659 2020 Mathematics Subject Classification: 35B06; 53A55; 17B66; 57M60 arXiv:2301.09364 https://nasplib.isofts.kiev.ua/handle/123456789/211971 https://doi.org/10.3842/SIGMA.2023.058 The fundamental invariants for vector ODEs of order ≥ 3 considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant 𝒰, we give a local (point) classification for all submaximally symmetric ODEs of C-class with 𝒰 ≢ 0 and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain. The authors acknowledge the use of the DifferentialGeometry package in Maple. We also acknowledge helpful conversations with Boris Kruglikov, Andreu Llabres, and Eivind Schneider. The research leading to these results has received funding from the Norwegian Financial Mechanism 2014–2021 (project registration number 2019/34/H/ST1/00636), the Tromsø Research Foundation (project “Pure Mathematics in Norway”), and the UiT Aurora project MASCOT, and this article/publication is based upon work from COST Action CaLISTA CA21109 supported by COST (European Cooperation in Science and Technology), https://www.cost.eu. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class Article published earlier |
| spellingShingle | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class Kessy, Johnson Allen The, Dennis |
| title | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class |
| title_full | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class |
| title_fullStr | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class |
| title_full_unstemmed | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class |
| title_short | On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class |
| title_sort | on uniqueness of submaximally symmetric vector ordinary differential equations of c-class |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211971 |
| work_keys_str_mv | AT kessyjohnsonallen onuniquenessofsubmaximallysymmetricvectorordinarydifferentialequationsofcclass AT thedennis onuniquenessofsubmaximallysymmetricvectorordinarydifferentialequationsofcclass |