2, 3, n-independency of tangential weights of G/K
岡山理科大学理学部応用数学科 黒木 慎太郎
Shintarô Kuroki
Dep. of Applied Math. Faculty of Science, Okayama University of Science

1. Introduction
This article is the research announcement of the progress work [KS]. In this article, we
introduce the main result of [KS].
1.1. k-independence. Let S := {α1 , . . . , αm } ⊂ Zn be a set of vectors. The set S is said tobe k-independent for some 2 ≤ k ≤ n if every k vectors αi1 , . . . , αik ∈ S are linearly independent
but there exists k + 1 vectors αj1 , . . . , αjk+1 ∈ S which are linearly dependent. In this case, wekalso say that the set of vectors S has the property Um, i.e., the uniform matroid with rank k andwith m elements, see [O06].
Example 1.1. Let S := {e1 , e2 , −e1 − e2 , e1 + e2 + e3 } ⊂ R3 be the set of 4 vectors in the
3-dimensional real vector space R3 , where ei , i = 1, 2, 3, is the standard basis of R3 . Then,
one can easily check that S has the property U42 , i.e., S is 2-independent. Note that the subset{e1 , e2 , e1 + e2 + e3 } is linearly independent but {e1 , e2 , −e1 − e2 } is linearly dependent.
1.2. Problem and motivation. Let M be a 2m-dimensional, smooth T -manifold with isolated ﬁxed points M T , where dim T = n. Assume that the T -action is almost eﬀective, i.e., thekernel of the T -action is ﬁnite. Under this assumption, the diﬀerentiable slice theorem tells usthat the inequality n ≤ m holds.
If we choose an invariant complex structure on Tp M ≃ R2m for a ﬁxed point p ∈ M T , thereis the following irreducible decomposition(1.1)

Tp M ≃mM

Vαp,i ,

i=1

where Vαp,i is the complex one-dimensional representation of T with the weight vector (called atangential weight) αp,i ∈ t∗Z ≃ Zn for i = 1, . . . , m. Here, the symbol t∗Z represents the characterlattice of the dual of the Lie algebra t∗ of T .
Set Sp := {αp,i | i = 1, . . . , m} ⊂ Zn . Assume that Sp is k(p)-inependent. By deﬁnition,
2 ≤ k(p) ≤ m for all p ∈ M T ; therefore, this is equivalent to that we assume M is a GKM manifold
in [GKM98, GZ01]. Notice that if k(p) = m for all p ∈ M T , M is called a torus manifold, deﬁned
in [HM03]. If a GKM manifold satisﬁes H odd (M ) = 0, we call it an equivariantly formal GKM
manifold.
Mikiya Masuda has asked the following question in the private communication:
Problem 1.2. Assume that M 2m is an equivariantly formal GKM manifold with 4-independent
T -action. Then, does the T -action on M extend to the almost eﬀective T m -action? Equivalently,
is M a torus manifold?
nThis article answers this problem for the homogeneous GKM manifolds. Roughly, the main
theorem states that k(p) can be taken only 2, 3 or n. We will prepare to describe the main theorem
from the next section.
12. Remarks from GKM theory
We ﬁrst introduce the following proposition. This answers Toshio Sumi’s question during theauthor’s talk in RIMS conference 2024.
Proposition 2.1 (Proposition 4.3 [Ku19]). Let M 2m be a GKM manifold with T n -action.
Assume that M 2m has the extended T l -action, i.e., the restricted T n ⊂ T l action is the original
torus action. Then, the T l -action on M 2m also satisﬁes the condition of a GKM manifold.
So, the GKM condition is preserved after extending the torus action.
We can construct a GKM manifold M such that there are ﬁxed points p, q ∈ M T with
k(p)
k(q)
diﬀerent properties Um and Um .
Example 2.2. Let CP 1 be the complex projective space with the natural T 1 -action. We
assume that T 1 ⊂ T 3 is the ﬁrst coordinate. Deﬁne the following symbols:• pi : T 3 → S 1 for i = 0, 1, 2, 3 is the projection onto the ith coordinate, where p0 is thetrivial homomorphism;• ϵi = CP 1 × C for i = 0, 1, 2, 3 is the T 3 -equivariant trivial line bundle over CP 1 with the
T 3 -action on the ﬁber C by pi : T 3 → S 1 ;• γ is the tautological line bundle over CP 1 .
Let M 8 := P((γ ⊗ ϵ2 ⊗ ϵ3 ) ⊕ ϵ2 ⊕ ϵ3 ⊕ ϵ0 ) be the projectivization of the rank 4 equivariant complex
vector bundle over CP 1 , i.e., M 8 is equivariantly diﬀeomorphic to a CP 3 -bundle over CP 1 with
T 3 -action (see e.g. [KS]). Then, by computing the tangential weights around 8 ﬁxed points, one
can check that M 8 is a GKM manifold, i.e., for every p, the tangential weights have the propertyk(p)U4with k(p) ≥ 2. Moreover, the tangential weights on the ﬁxed points ([1 : 0], [0 : 0 : 0 : 1])and ([0 : 1], [0 : 0 : 0 : 1]) have the property U42 and the property U43 respectively.
In Proposition 3.1, we prove that the number k(p) does not depend on the ﬁxed points for thecase of homogeneous GKM manifolds; therefore, we may write it by k(G/K).
For the general GKM manifold which is deﬁned by the 1-skeleton has the structure of a graph,
there are several ways to deﬁne the axial function on edges. Here, we state the position of thisarticle about the axial functions in the following remark.
Remark 2.3. Let M be a 2m-dimensional GKM manifold with the n-dimensional torus T action. If there is a T -invariant almost complex structure J on M (e.g. M = G/Z, where G is acompact, connected Lie group and Z is the centralizer of a maximal torus of G, see [KKLS20]),then the identiﬁcation (1.1) can be induced from J, see [GZ01, GHZ06]. For the case when m =n, called a torus manifold (e.g. M = S 2n ≃ SO(2n + 1)/SO(2n) with the standard T n -action for
n ≥ 2), the identiﬁcation (1.1) is determined by choosing the omni-orientation of the characteristic
submanifolds in the torus manifold, see [MMP07, Ku16]. For the other cases (e.g. M = HP n ≃Sp(n+1)/Sp(n)×Sp(1) for n ≥ 2, see [GL]), there is no canonical way to determine the sign of therepresentations in (1.1). To avoid the sign ambiguities of the representations, we may choose some
identiﬁcation (1.1) for each ﬁxed point. These deﬁne the label on edges of GKM graphs without
sign ambiguities (called an axial function). In the purpose of works in some literature, the axial
functions are deﬁned up to signs (e.g. this is called a pre-axial in [GHZ06]). In particular, thisdeﬁnition is enough to deﬁne the graph equivariant cohomology of a GKM graph. In this article,
we deﬁne the axial function without sign ambiguities by choosing some identiﬁcation (1.1) for eachﬁxed point.
3. The maximal torus action on the homogeneous space G/K
Let G be a compact, connected, semi-simple Lie group, T be its maximal torus, and K
be a closed, connected subgroup of G such that T ⊂ K ⊂ G, i.e., a maximal rank subgroupof G. In this setting, we have that the fundamental group π1 (G) is ﬁnite. It is known thatthe homogeneous space G/K is a 2m-dimensional, simply connected and equivariantly formal
manifold, i.e., H odd (G/K) = 0, see e.g. [MT79, B13]. Due to [GHZ06], G/K satisﬁes the GKM
2condition. Below we recall the description of the tangential weights of the T -action on G/K byfollowing [GHZ06, Theorem 2.4], also see [MT79, B13].
We ﬁrst recall some basic facts from the theory of Lie groups. We denote the Lie algebras
of T ⊂ K ⊂ G by t ⊂ k ⊂ g, respectively. Let exp : g → G be the exponential map, andΛG := exp−1 (e) ⊂ t be the integer lattice, where e ∈ T ⊂ G is the identity element. Then, we
have an identiﬁcation T ≃ t/ΛG and Hom(T, S 1 ) ≃ Λ∗G := Hom(ΛG , Z) ≃ Zn . The Z-moduleΛ∗G ⊂ t∗ is called a weight (or character) lattice of G, where t∗ = Hom(t, R) is the dual of t.
Note that if G is simply connected, then Λ∗G is spanned by the fundamental weight (see [MT79,
Chapter 5 Theorem 6.36]). If G is not simply connected, then by taking the universal covering
e → G and the dual dp∗ of its diﬀerential dp on the identities of maximal tori, we may regardp:G∗ΛG ⊂ Λ∗Ge for two weight lattices (see [MT79, Chapter 5 Theorem 4.9] or [B13, Section 19]). Let∆G ⊂ Λ∗G ⊂ t∗ be the root systems of G with respect to the maximal torus T . Because the linear
relations among the root systems do not change by taking the ﬁnite covering of G, it is enough toconsider the case when G is simply connected for the purpose of this article, i.e., computing thek(p)-independence for the tangent wights on every p ∈ (G/K)T .
Assume that G is simply connected, i.e., π1 (G) = 0. The set of the ﬁxed points (G/K)T canbe identiﬁed with the ﬁnite set of the quotient WG /WK of the Weyl groups for the maximal torus
T , where WG := NG (T )/T and WK := NK (T )/T . Let p0 = eK ∈ (G/K)T ≃ WG /WK be thecoset of the identity e ∈ G. In other words, WG acts on (G/K)T transitively, where the isotropy
subgroup of p0 is WK . Namely, we may write(G/K)T = {wp0 | w ∈ WG }.
Then, one has the irreducible decomposition
Tp0 G/K = g/k =M

V[β] ,[β]of the tangential T -representation at p0 , where [β] runs over ∆G,K /{±1} := (∆G \ ∆K )/{±1}. If
one can choose a section sp0 : ∆G,K /{±1} → ∆G,K of the natural projection ∆G,K → ∆G,K /{±1}such that WK acts on Im(sp0 ) ⊔ −Im(sp0 ), called a WK -equivariant section, then [β] may be
regarded as the weight vector sp0 ([β]) in ∆G,K ⊂ Λ∗G ⊂ t∗ . This is equivalent to choosing the
T -invariant complex structure on Tp0 G/K. Then, the set of the tangential weights on p0 may be
regarded as ∆p0 := sp0 (∆G,K /{±1}) ⊂ Λ∗G ≃ Zn . Let p = wp0 with [w] ∈ WG /WK . Then, thetangential representation around wp0 isM
Twp0 G/K =V[wβ] .
[β]∈∆G,K /{±1}Similarly, we may deﬁne the set of tangential weights around the ﬁxed point wp0 by ∆wp0 :=Im(swp0 ) for some section swp0 : ∆G,K /{±1} → ∆G,K .
Note that ∆p0 and ∆wp0 for every w ∈ WG /WK are isomorphic to ∆G,K /{±1}. This implies
the following proposition:
Proposition 3.1. Let G be a compact, connected semi-simple Lie group, K be its maximal
rank subgroup and T ⊂ K ⊂ G be a maximal torus. Then, there exists an integer k(G/K) suchthat 2 ≤ k(G/K) ≤ dim T and the tangential weights of Tp (G/K) are k(G/K)-independent forevery p ∈ (G/K)T ≃ WG /WK .
+This proposition shows that if we compute the k(G/K)-independence on ∆+G \∆K ≃ ∆G,K /{±1}++for positive roots ∆K ⊂ ∆G , then we have the k(G/K)-independence of all tangent spaces on theﬁxed points of G/K. (Cf. Example 2.2) In this paper, we determine the k-independence for all
maximal rank homogeneous spaces G/K.

3.1. Reducing to the simple cases and main theorem. If we assume G is simply connected, then the standard T -action on G/K is equivariantly diﬀeomorphic to the following productof the maximal rank homogeneous spaces (see e.g. [Ku10, Section 2.2]):(3.1)

G/K ≃ G1 /K1 × · · · × Gr /Kr ,3

where Gi is a compact, simply connected simple Lie group, Ki be its maximal rank subgroupand there is an isomorphism T ≃ T1 × · · · × Tr such that Ti ⊂ Ki ⊂ Gi is a maximal torus for
i = 1, . . . , r. Moreover, we can easily show the following proposition:
Proposition 3.2. The following equality holds for the splitting (3.1):
k(G/K) = min{k(Gi /Ki ) | 1 ≤ i ≤ r}.
This proposition says that to compute the independency of G/K, it is enough to compute theindependency of the simple factors Gi /Ki . The classiﬁcation of each pair of (Gi .Ki ) is known by
the work of Borel-deSiebenthal [BS49]. The following list shows their classiﬁcation of the simple
Lie group G and its maximal, maximal rank subgroup K, where the locally isomorphic means the
Lie algebras are isomorphic.
Type An+1 : If G is of type An+1 for n ≥ 0, then K is locally isomorphic to one of thefollowings:
Case 1: An × T 1 ;Case 2: Ai−1 × An+1−i × T 1 (1 < i < n + 1).
Type Dn : If G is of type Dn for n ≥ 4, then K is locally isomorphic to one of the followings,
where D1 = T 1 , D2 = A1 × A1 , D3 = A3 :Case 1: An−1 × T 1 .
Case 2: Dn−1 × T 1 ;Case 3: Di × Dn−i (1 < i < n − 1);
Type Bn : If G is of type Bn for n ≥ 2, then K is locally isomorphic to one of the followings,
where B1 = A1 :Case 1: Bn−1 × T 1 ;Case 2: Di × Bn−i (1 < i < n);Case 3: Dn .
Type Cn : If G is of type Cn for n ≥ 3, then K is locally isomorphic to one of the followings,
where C1 = A1 , C2 = B2 :Case 1: Ci × Cn−i (1 ≤ i < n);Case 2: An−1 × T 1 .
Type E6 : If G is of type E6 , then K is locally isomorphic to one of the followings:
Case 1: D5 × T 1 ;Case 2: A1 × A5 ;Case 3: A2 × A2 × A2 .
Type E7 : If G is of type E7 , then K is locally isomorphic to one of the followings:
Case 1: D6 × A1 ;Case 2: A2 × A5 ;Case 3: A7 ;Case 3: E6 × T 1 .
Type E8 : If G is of type E8 , then K is locally isomorphic to one of the followings:
Case 1: D8 ;Case 2: A4 × A4 ;Case 3: A8 ;Case 4: E6 × A2 ;Case 5: E7 × A1 .
Type F4 : If G is of type F4 , then K is locally isomorphic to one of the followings:
Case 1: C3 × A1 ;Case 2: A2 × A2 ;Case 3: B4 .
Type G2 : If G is of type G2 , then K is locally isomorphic to one of the followings:
Case 1: A2 ;Case 2: A1 × A1 .
The following lemma is also useful to reduce the cases.
4Lemma 3.3. Let H ⊂ K ⊂ G be a sequence of compact connected maximal rank Lie groups.
If k(G/K) = 2, then k(G/H) = 2.
Proof. There is the ﬁbration K/H → G/H → G/K. Let p0 = eH. Then, there is thefollowing decomposition:
Tp0 G/H ≃ g/h ≃ g/k ⊕ k/h.
Because k(G/K) = 2, the subspace g/k is 2-independent. Since the lowest independency is 2, wehave the statement of the lemma.
□
Now we may state the main theorem:
Theorem 3.4. Assume that G is simply connected, simple Lie group and K be its maximal
rank subgroup such that rank(G) = rank(K) = dim T = n. Then, k(G/K) = 2, 3, n and thefollowing holds:• k(G/K) = n if and only if (G, K) is one of the following types:
Type An : (An , An−1 × T 1 );Type Bn : (Bn , Dn ).
• k(G/K) = 3(̸= n) if and only if (G, K) is one of the following types:
Type An : (An , Ai−1 × An−i × T 1 );Type Cn : (Cn , Ci−1 × Cn−i+1 );Type Dn : (Dn , Dn−1 × T 1 ), (Dn , Di−1 × Dn−i+1 );Type F4 : (F4 , B4 );Type E6 : (E6 , D5 × T 1 ), (E6 , A5 × A1 ), (E6 , A4 × T 1 × A1 );Type E7 : (E7 , A7 ), (E7 , D6 × A1 ), (E7 , E6 × T 1 ), (E7 , D5 × T 2 ), (E7 , A5 × A1 × T 1 ),(E7 , A4 × A1 × T 2 );Type E8 : (E8 , D8 ).
• k(G/K) = 2, otherwise.
As a corollary of this theorem, we have the following characterization of the homogeneous
space by using the independence of tangential weights:
Corollary 3.5. Assume that G is simple and of classical types, i.e., type An , Bn , Cn and
Dn . Then, the following two facts hold:• if the tangential representations on eH ∈ G/H is more than 4-independent, then G/H ≃CP n or S 2n ;f
• if the tangential representations on eH ∈ G/H is 3-independent, then G/H ≃ Gr(i, n; C), Gr(i−1, n; R), Gr(i − 1, n; H), Q2n .
In [Ku10], we classify all homogeneous torus manifolds. Together with this, Theorem 3.4
gives the partial answer to Problem 1.2 for the case when a GKM manifold is a homogeneous
space. In [KS], we will prove the main theorem by using the signed graphs.
Remark 3.6. In [So23], Solomadin constructs the GKM graph which is 4-independent but
there is no extension. This may be regarded as the combinatorial counter-example for Problem 1.2. However, there is no GKM manifold whose GKM graph is the example in [So23]. Hence,
Problem 1.2 is still open.
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number 21K03262 and the ResearchInstitute for Mathematical Sciences, an International Joint Usage/Research Center located in
Kyoto University. The author would like to thank Mikiya Masuda, Grigory Solomadin, Toshio
Sumi for their valuable comments.
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Department of Applied Mathematics Faculty of Science, Okayama University of Science, 1-1 Ridaicho Kita-ku Okayama-shi Okayama 700-0005, Okayama, Japan
Email address: kuroki@ous.ac.jp6