Decomposability of polynomial valued 2-forms

Yoshio Agaoka

Department of Mathematics, Faculty of Integrated Arts and Sciences Hiroshima University, Higashi-Hiroshima 739-8521, Japan e-mail address: agaoka@mis.hiroshima-u.ac.jp

Abstract: We give a characterization of decomposable polynomial valued 2-forms in terms of their components. Such 2-forms must satisfy some cubic condition in addition to Plücker’s quadratic relation. Several GL(n, K) × GL(m, K)-invariant varieties naturally appear during this characterization, and we state the mutual relation of these varieties and study their geometric properties in detail.

Key words: polynomial valued 2-form, decomposability, Plücker’s relation, variety

Introduction

Let V be an n-dimensional vector space over the field K of real numbers or complex numbers (n ≥ 2) and V^{∗} be its dual space. As is well known, an element C ∈ ∧^{2}V^{∗} is decomposable i.e., it can be expressed as C = α ∧ β for some α, β ∈ V^{∗} if and only if C satisfies Plücker’s relation

C(v_{1}, v_{2})C(v_{3}, v_{4}) − C(v_{1}, v_{3})C(v_{2}, v_{4}) + C(v_{1}, v_{4})C(v_{2}, v_{3}) = 0

for any v_{i} ∈ V. (For example, see [9], [19]). The main purpose of this paper is to give a similar characterization of decomposable “polynomial valued” 2-forms. This problem is closely related to the existence of local isometric imbeddings of Riemannian manifolds into the Euclidean space with codimension 1 (cf. [2], [7]).

To explain the results, we first fix the notations. We put V = K^{n} (K = R or C) and let A be a polynomial ring over K with m variables x_{1}, ⋯, x_{m}: A = K[x_{1}, ⋯, x_{m}], and A = ∑_{p≥0} A^{p} (A^{0} = K) be the homogeneous decomposition of A. An element α ∈ V^{∗} ⊗ A^{1} may be considered as an A^{1}-valued 1-form on V. Then, for β ∈ V^{∗}, the exterior product α ∧ β ∈ ∧^{2}V^{∗} ⊗ A^{1} is naturally defined as in the scalar valued case, and we say that C ∈ ∧^{2}V^{∗} ⊗ A^{1} is decomposable if it is expressed as α ∧ β for some α ∈ V^{∗} ⊗ A^{1} and β ∈ V^{∗}. In this polynomial valued case, decomposable 2-forms also satisfy Plücker’s relation. But this relation is not sufficient to characterize decomposable 2-forms in contrast to the scalar valued case. In fact the algebraic set of ∧^{2}V^{∗} ⊗ A^{1} defined by only Plücker’s relation is not irreducible and it decomposes into two irreducible components, one of which just coincides with the set of decomposable 2-forms. To obtain a complete characterization of decomposable 2-forms, we must add one cubic condition on C. This additional condition is stated as follows: “For any v_{i} ∈ V, the polynomials C(v_{1}, v_{2}), C(v_{1}, v_{3}), C(v_{2}, v_{3}) are linearly dependent in A^{1}”. We here give one example: Consider the 2-form C = x_{1} ω_{1} ∧ ω_{2} + x_{2} ω_{1} ∧ ω_{3} + x_{3} ω_{2} ∧ ω_{3}, where {ω_{i}} is a basis of V^{∗}. Then, it is easy to see that C satisfies Plücker’s relation, but does not satisfy the above cubic condition, and hence we know that this form C is not decomposable.

The other irreducible component of the algebraic set defined by Plücker’s relation consists of A^{1}-valued 2-forms that can be reduced to some 3-dimensional subspace of V. As in the case of Plücker’s relation, the algebraic set defined by the above cubic condition also decomposes into two irreducible components; one is the variety of decomposable forms, and the other is the variety consisting of 2-forms that take value in two variables x_{1}, x_{2} after some variable transformation.

In order to understand the variety of decomposable 2-forms, it is natural to treat these three varieties simultaneously. All these varieties are characterized by two types of conditions on C, and they are related to each other by possessing one common defining equation for each pair (Theorem 1). In addition, the algebraic set defined by only one type of condition on C splits into two irreducible components (Theorem 2). In considering this mutual relation, another three varieties naturally appear as subsets of the above varieties. In this paper, we characterize these six varieties completely by giving their defining equations, inclusion relations, dimension, and clarify their geometric meaning by introducing a parametrization of each variety (Proposition 3 and Theorem 8).

The space ∧^{2}V^{∗} ⊗ A^{1} may be considered as a sort of 3-tensor space, and the results of this paper possess some resemblance to the case of the 3-tensor space C^{2} ⊗ C^{2} ⊗ C^{2} studied in [3]. It is desirable and also interesting to extend our results to more general 3-tensor spaces such as ∧^{3}V^{∗}, C^{p} ⊗ C^{q} ⊗ C^{r}, etc (cf. [5], [6]).

As stated above, the decomposability of polynomial valued 2-forms C is naturally related to the problem of local isometric imbeddings of Riemannian manifolds through the notion of the partial Gauss equation that was introduced in [2]. By definition, the partial Gauss equation is expressed as

C = α_{1} ∧ β_{1} + ⋯ + α_{r} ∧ β_{r}, (A)

where C ∈ ∧^{2}V^{∗} ⊗ A^{1} is a given 2-form and α_{i} ∈ V^{∗} ⊗ A^{1}, β_{i} ∈ V^{∗}. Roughly speaking, if an n-dimensional Riemannian manifold M^{n} (n = dim V) is locally isometrically imbedded into R^{n+r}, then certain 2-form C constructed from the curvature of M must be expressed in the above form (A). (For the precise statement, see [2].) In particular, the results stated in this paper is related to the case of hypersurfaces of R^{n+1} (the case r = 1), and the conditions on the decomposability of C serve as obstructions to the existence of local isometric imbeddings of M into R^{n+1}. For further applications in geometry, we must obtain a similar characterization of 2-forms C in (A) for larger r.

§ 1. Statement of the main results

In this section, after fixing some notations, we state the main results of this paper. The proof of Theorem 1 and Theorem 2 stated below will be given in the subsequent sections.

Let C be an element of ∧^{2}V^{∗} ⊗ A^{1}. We define two linear maps d_{C} and e_{C} as follows:

d_{C} : V ⟶ V^{∗} ⊗ A^{1}, d_{C}(v) = v⌋C,
e_{C} : ∧^{2}V ⟶ A^{1}, e_{C}(v_{1} ∧ v_{2}) = C(v_{1}, v_{2}),

where v⌋C implies the interior product. In terms of these maps, we define the following five subsets of ∧^{2}V^{∗} ⊗ A^{1}:

Σ_{1} = {C ∈ ∧^{2}V^{∗} ⊗ A^{1} | C = α ∧ β for some α ∈ V^{∗} ⊗ A^{1}, β ∈ V^{∗}},
Σ_{2} = {C ∈ ∧^{2}V^{∗} ⊗ A^{1} | rank d_{C} ≤ 3},
Σ_{3} = {C ∈ ∧^{2}V^{∗} ⊗ A^{1} | rank e_{C} ≤ 2},
Σ_{4} = {C ∈ ∧^{2}V^{∗} ⊗ A^{1} | rank d_{C} ≤ 2},
Σ_{5} = {C ∈ ∧^{2}V^{∗} ⊗ A^{1} | rank e_{C} ≤ 1}.

As we will see later, these five subsets are all irreducible varieties of ∧^{2}V^{∗}⊗A^{1}. We remark that if rank d_{C} ≤ k, then C can be considered as an element of ∧^{2}W^{∗} ⊗ A^{1} where W is a k-dimensional subspace of V. In fact, since dim Ker d_{C} ≥ n − k, there exists a basis {e_{1}, ⋯, e_{n}} of V satisfying e_{k+1}⌋C = ⋯ = e_{n}⌋C = 0. Then, by using the dual basis {ω_{i}}, the 2-form C is expressed as ∑_{i,j=1}^{k} C_{ij} ω_{i} ∧ ω_{j}, where C_{ij} = C(e_{i}, e_{j}). Similarly, it is easy to see that if rank e_{C} ≤ l, the number of variables m can be reduced to l after some variable transformation.

Next, we define several conditions on C ∈ ∧^{2}V^{∗} ⊗ A^{1} in order to describe the defining equations of Σ_{i}. We say that C satisfies condition (C_{P}) if it satisfies classical Plücker’s relation:

C(v_{1}, v_{2})C(v_{3}, v_{4}) − C(v_{1}, v_{3})C(v_{2}, v_{4}) + C(v_{1}, v_{4})C(v_{2}, v_{3}) = 0 ∈ A^{2}

for any vectors v_{i} ∈ V. This condition is equivalent to C ∧ C = 0 ∈ ∧^{4}V^{∗} ⊗ A^{2}. Next, if the polynomials

C(v_{1}, v_{2}), C(v_{1}, v_{3}), C(v_{2}, v_{3})

are linearly dependent in A^{1} for any v_{i}, we say that C satisfies condition (C_{Q}). Using the components of C, this condition is expressed as cubic polynomial relations of C. Finally, for positive integer k, we say that C satisfies condition (C_{k}) if the polynomials

C(v_{1}, v_{2}), C(v_{1}, v_{3}), ⋯, C(v_{1}, v_{k+2})

are linearly dependent in A^{1} for any v_{i}. It is easy to see that this condition is equivalent to

rank (v⌋C) ≤ k for any v ∈ V,

where “rank” means the usual rank of the (m, n)-matrix v⌋C ∈ V^{∗} ⊗ A^{1}. In this paper, we use this condition only in the cases k = 1 and 2. Note that condition (C_{1}) is quadratic and (C_{2}) is cubic, and clearly, condition (C_{1}) implies (C_{2}) and (C_{Q}). By using these four conditions (C_{P}), (C_{Q}), (C_{1}), (C_{2}), we can completely characterize the subset Σ_{i} ⊂ ∧^{2}V^{∗} ⊗ A^{1} in the following way.

Theorem 1. (1) C ∈ Σ_{1} if and only if C satisfies (C_{P}) and (C_{Q}).

(2) C ∈ Σ_{2} if and only if C satisfies (C_{P}) and (C_{2}).

(3) C ∈ Σ_{3} if and only if C satisfies (C_{Q}) and (C_{2}).

(4) C ∈ Σ_{4} if and only if C satisfies (C_{P}) and (C_{1}).

(5) C ∈ Σ_{5} if and only if C satisfies (C_{1}). In addition, each subset Σ_{i} (1 ≤ i ≤ 5) is an irreducible algebraic variety of ∧^{2}V^{∗} ⊗ A^{1}.

In particular, the decomposability of C ∈ ∧^{2}V^{∗} ⊗ A^{1} is completely characterized by two types of conditions (C_{P}) and (C_{Q}). In the case m ≤ 2, we remark that C is decomposable if and only if it satisfies condition (C_{P}) only, because condition (C_{Q}) is automatically satisfied in this case.

By definition, an element C belongs to Σ_{2} if and only if rank d_{C} ≤ 3, and hence, Σ_{2} is defined by quartic polynomials. But, the above theorem asserts that this condition can be reduced to lower degree conditions (C_{P}) and (C_{2}).

By Theorem 1, we have clearly Σ_{1} ∩ Σ_{2} = Σ_{1} ∩ Σ_{3} = Σ_{2} ∩ Σ_{3}, and C belongs to this algebraic set if and only if C satisfies three conditions (C_{P}), (C_{Q}), (C_{2}). In the following, we denote this algebraic set by Σ_{6}.

Next, we characterize the algebraic set of ∧^{2}V^{∗} ⊗ A^{1} defined by one of (C_{P}), (C_{Q}), (C_{2}).

Theorem 2. (1) C satisfies condition (C_{P}) if and only if C ∈ Σ_{1} ∪ Σ_{2}.

(2) C satisfies condition (C_{Q}) if and only if C ∈ Σ_{1} ∪ Σ_{3}.

(3) C satisfies condition (C_{2}) if and only if C ∈ Σ_{2} ∪ Σ_{3}.

By definition, any element C ∈ Σ_{1} can be parametrized by the pair (α, β) ∈ V^{∗} ⊗ A^{1} × V^{∗} as C = α ∧ β. Other varieties Σ_{2} ∼ Σ_{6} also have similar parametrization, by which we can understand their geometric meaning.

Proposition 3. (1) C ∈ Σ_{2} if and only if C = f_{1}β_{1} ∧ β_{2} + f_{2}β_{1} ∧ β_{3} + f_{3}β_{2} ∧ β_{3} for some f_{i} ∈ A^{1} and β_{i} ∈ V^{∗}.

(2) C ∈ Σ_{3} if and only if C = f_{1}Ω_{1} + f_{2}Ω_{2} for some f_{i} ∈ A^{1} and Ω_{i} ∈ ∧^{2}V^{∗}.

(3) C ∈ Σ_{4} if and only if C = fβ_{1} ∧ β_{2} for some f ∈ A^{1} and β_{i} ∈ V^{∗}.

(4) C ∈ Σ_{5} if and only if C = fΩ for some f ∈ A^{1} and Ω ∈ ∧^{2}V^{∗}.

(5) C ∈ Σ_{6} if and only if C = (f_{1}β_{1} + f_{2}β_{2}) ∧ β_{3} for some f_{i} ∈ A^{1} and β_{i} ∈ V^{∗}.

Proof. For the statements (1) and (3), “if” parts are easy to see. The converse parts are already proved after the definition of the varieties Σ_{1} ∼ Σ_{5}, where we show C = ∑_{i,j=1}^{k} C_{ij} ω_{i} ∧ ω_{j} under the condition rank d_{C} ≤ k. The statements (2) and (4) are almost trivial because the condition rank e_{C} ≤ l implies that the number of variables m can be reduced to l, as stated in the same place. For the statement (5), we assume for some time that the definition of Σ_{6} is Σ_{2} ∩ Σ_{3} (since we did not prove Theorem 1 yet). Then, if C ∈ Σ_{6}, C is expressed as f_{1}β_{1} ∧ β_{2} + f_{2}β_{1} ∧ β_{3} + f_{3}β_{2} ∧ β_{3} from the condition C ∈ Σ_{2}. Next, since the number of variables is reducible to two, we may put, by the symmetry, f_{3} = af_{1} +bf_{2} (a, b ∈ K). Then, after substituting this into the above expression, we have C = (β_{1} + bβ_{2} − aβ_{3}) ∧ (f_{1}β_{2} + f_{2}β_{3}). The converse part is trivial from (1) and (2). q.e.d.

This parametrization may be considered as a canonical form of each variety Σ_{i}. This proposition is useful in the proof of Theorems 1 and 2. We summarize the inclusion relations of Σ_{i} in the following figure:

Σ_{4}  Σ_{6}  Σ_{2} Σ_{5}  Σ_{3} Σ_{1} rank d_{C} ≤ 2 (C_{P}), (C_{1})  (C_{P}), (C_{Q}), (C_{2})  rank d_{C} ≤ 3 (C_{P}), (C_{2}) rank e_{C} ≤ 1 (C_{1})  rank e_{C} ≤ 2 (C_{Q}), (C_{2}) decomposable (C_{P}), (C_{Q})

(Note that condition (C_{1}) implies (C_{2}) and (C_{Q}), as stated before.)

Finally we state one remark. The group GL(n, K) × GL(m, K) acts naturally on the space ∧^{2}V^{∗} ⊗ A^{1}, and it is easy to see that the above varieties Σ_{1} ∼ Σ_{6} are invariant under this group action. It is an interesting problem to classify all GL(n, K) × GL(m, K)- invariant subvarieties of ∧^{2}V^{∗} ⊗ A^{1} as in the case of the 3-tensor space C^{2} ⊗ C^{2} ⊗ C^{2} (cf. [3]). Perhaps another new concept is required to solve this problem in addition to (C_{P}), (C_{Q}) and (C_{k}), and to know such fundamental concept is one important step to understand the 3-tensor space ∧^{2}V^{∗} ⊗ A^{1}.

§ 2. Preliminary lemmas

In this section we prepare several lemmas to prove the results in § 1. Each lemma plays a crucial role in the proof of Theorems 1 and 2.

First, we prove the following lemma.

Lemma 4. Assume C ∈ ∧^{2}V^{∗} ⊗ A^{1} satisfies conditions (C_{P}), (C_{Q}), and there exists v ∈ V such that rank (v⌋C) ≥ 2. Then, there exist a basis {e_{1}, ⋯, e_{n}} and a_{2} ∼ a_{n} ∈ K satisfying

C_{ij} = a_{j}C_{1i} − a_{i}C_{1j},

for 1 ≤ i, j ≤ n, where C_{ij} = C(e_{i}, e_{j}) and a_{1} = −1. In addition, such {a_{i}} uniquely exists if we fix a basis {e_{i}}.

Proof. We choose a basis {e_{1}, ⋯, e_{n}} such that e_{1} = v, and let {ω_{1}, ⋯, ω_{n}} be the dual basis of {e_{i}}. Then we have

v⌋C = C_{12} ω_{2} + ⋯ + C_{1n} ω_{n}.

By rearranging the indices if necessary, we may assume that {C_{12}, C_{13}} is linearly independent because rank (v⌋C) ≥ 2. Since {C_{12}, C_{13}, C_{23}} is dependent from condition (C_{Q}), C_{23} is uniquely expressed as C_{23} = a_{3}C_{12} − a_{2}C_{13}. Next, for 4 ≤ i ≤ n, we substitute this equality into Plücker’s relation

C_{12}C_{3i} − C_{13}C_{2i} + C_{1i}C_{23} = 0.

Then we have immediately

C_{12}(C_{3i} + a_{3}C_{1i}) = C_{13}(C_{2i} + a_{2}C_{1i}).

Since {C_{12}, C_{13}} is independent, the above expression is equal to a_{i}C_{12}C_{13} for some a_{i} ∈ K. In particular, we have

C_{2i} = a_{i}C_{12} − a_{2}C_{1i}.

(Note that this equality holds for 1 ≤ i ≤ n.) Uniqueness of a_{4} ∼ a_{n} is clear from this expression. We substitute this equality into

C_{12}C_{ij} − C_{1i}C_{2j} + C_{1j}C_{2i} = 0.

Then we have the desired equality C_{ij} = a_{j}C_{1i} − a_{i}C_{1j} because C_{12} ≠ 0. q.e.d.

Before proving the next lemma, we introduce a notation |f_{1} f_{2} f_{3}| (f_{i} ∈ A^{1}), which we often use in the following arguments. We express f_{i} ∈ A^{1} as ∑_{p=1}^{m} f_{ip}x_{p}, and put

|f_{1} f_{2} f_{3}|_{pqr} = | \begin{matrix}f_{1p} & f_{2p} & f_{3p} \\ f_{1q} & f_{2q} & f_{3q} \\ f_{1r} & f_{2r} & f_{3r} \end{matrix} | ∈ A^{3}.

We define |f_{1} f_{2} f_{3}| by

|f_{1} f_{2} f_{3}| = (|f_{1} f_{2} f_{3}|_{pqr})_{1≤p<q<r≤m} ∈ \overbrace{A^{3} ⊕ ⋯ ⊕ A^{3}}^{\binom{m}{3}},

i.e., |f_{1} f_{2} f_{3}| is the set of \binom{m}{3} polynomials |f_{1} f_{2} f_{3}|_{pqr} (1 ≤ p < q < r ≤ m) arranged in some fixed order. Then, addition and scalar multiplication of |f_{1} f_{2} f_{3}| is naturally defined. For example, we have the equalities

|f_{1} + f_{2} f_{3} f_{4}| = |f_{1} f_{3} f_{4}| + |f_{2} f_{3} f_{4}|,
|af_{1} f_{2} f_{3}| = a|f_{1} f_{2} f_{3}|.

Clearly, |f_{1} f_{2} f_{3}| is skew symmetric with respect to {f_{1}, f_{2}, f_{3}}, and |f_{1} f_{2} f_{3}| = 0 if and only if {f_{1}, f_{2}, f_{3}} is linearly dependent in A^{1}.

Using vectors v_{i} ∈ V, we put C_{ij} = C(v_{i}, v_{j}). Then, in terms of the above notation, condition (C_{Q}) is expressed as

| C_{ij} C_{ik} C_{jk} | = 0.

By replacing the vector v_{k} by v_{k} + v_{l}, we have

| C_{ij} C_{ik} C_{jl} | + | C_{ij} C_{il} C_{jk} | = 0.

In the same way, condition (C_{2}) is expressed in the form

| C_{ij} C_{ik} C_{il} | = 0.

In this equality, we replace v_{i} by v_{i} + v_{j}. Then it follows that

| C_{ij} C_{ik} C_{jl} | + | C_{ij} C_{jk} C_{il} | = 0.

In particular, if C satisfies both conditions (C_{Q}) and (C_{2}), we have from the above two equalities

| C_{ij} C_{ik} C_{jl} | = 0

because | C_{ij} C_{jk} C_{il} | = −| C_{ij} C_{il} C_{jk} |. In addition, by replacing v_{i} by v_{i} + v_{p} in this equality, we have

| C_{ij} C_{pk} C_{jl} | + | C_{pj} C_{ik} C_{jl} | = 0.

Now, using this notation, we prove the following lemma.

Lemma 5. Assume C ∈ ∧^{2}V^{∗} ⊗ A^{1} satisfies condition (C_{P}) or (C_{2}). In addition, there exist v_{1}, v_{2}, v_{3} ∈ V such that {C_{12}, C_{13}, C_{23}} (C_{ij} = C(v_{i}, v_{j})) is linearly independent in A^{1}. Then C is expressed in the form C = f_{1}β_{1} ∧ β_{2} + f_{2}β_{1} ∧ β_{3} + f_{3}β_{2} ∧ β_{3} for some f_{i} ∈ A^{1}, β_{i} ∈ V^{∗}.

Proof. We fix a basis {e_{1}, ⋯, e_{n}} of V such that e_{1} = v_{1}, e_{2} = v_{2}, e_{3} = v_{3}. Since {C_{12}, C_{13}, C_{23}} is linearly independent, we may assume C_{12} = x_{1}, C_{13} = x_{2}, C_{23} = x_{3} after some variable transformation. Now, we divide the proof into two cases.

(i) The case where C satisfies condition (C_{P}). In this case, we have for 4 ≤ i ≤ n,

C_{12}C_{3i} − C_{13}C_{2i} + C_{1i}C_{23} = x_{1}C_{3i} − x_{2}C_{2i} + x_{3}C_{1i} = 0.

From this equality, we have easily

C_{1i} = a_{i}x_{1} − b_{i}x_{2},
C_{2i} = c_{i}x_{1} − b_{i}x_{3},
C_{3i} = c_{i}x_{2} − a_{i}x_{3}

for some a_{i}, b_{i}, c_{i}. Note that by putting a_{2} = 1, b_{3} = c_{1} = −1, a_{1} = a_{3} = b_{1} = b_{2} = c_{2} = c_{3} = 0, the above equalities hold for 1 ≤ i ≤ n. Next, we substitute them into

C_{12}C_{ij} − C_{1i}C_{2j} + C_{1j}C_{2i} = 0.

Then, we have

C_{ij} = (a_{i}c_{j} − a_{j}c_{i})x_{1} + (b_{j}c_{i} − b_{i}c_{j})x_{2} + (a_{j}b_{i} − a_{i}b_{j})x_{3}.

Hence, by putting β_{1} = ∑ a_{i} ω_{i}, β_{2} = ∑ b_{i} ω_{i}, β_{3} = ∑ c_{i} ω_{i} ({ω_{i}} is the dual basis of {e_{i}}), we have

C = x_{1}β_{1} ∧ β_{3} − x_{2}β_{2} ∧ β_{3} − x_{3}β_{1} ∧ β_{2}.

(ii) The case where C satisfies condition (C_{2}). In this case, as prepared above, we have | C_{12} C_{13} C_{1i} | = | C_{21} C_{23} C_{2i} | = | C_{31} C_{32} C_{3i} | = 0, and hence, C_{1i} ∈ ⟨x_{1}, x_{2}⟩, C_{2i} ∈ ⟨x_{1}, x_{3}⟩, C_{3i} ∈ ⟨x_{2}, x_{3}⟩. In addition, from (C_{2}), we have

| C_{12} C_{13} C_{2i} | + | C_{12} C_{23} C_{1i} | = 0,
| C_{23} C_{21} C_{3i} | + | C_{23} C_{31} C_{2i} | = 0,
| C_{31} C_{32} C_{1i} | + | C_{31} C_{12} C_{3i} | = 0.

Using these conditions, we obtain easily

C_{1i} = a_{i}x_{1} − b_{i}x_{2},
C_{2i} = c_{i}x_{1} − b_{i}x_{3},
C_{3i} = c_{i}x_{2} − a_{i}x_{3}.

Next, for 2 ≤ i ≤ n and t ∈ K, we have

(e_{1} + te_{i})⌋C = C_{12} ω_{2} + ⋯ + C_{1n} ω_{n} + t(C_{i1} ω_{1} + ⋯ + C_{in} ω_{n})
= t C_{i1} ω_{1} + (C_{12} + t C_{i2}) ω_{2} + ⋯ + (C_{1n} + t C_{in}) ω_{n}.

Since rank ((e_{1} + te_{i})⌋C) ≤ 2 for any parameter t, we have in particular, dim ⟨C_{12} + t C_{i2}, C_{13} + t C_{i3}, C_{1j} + t C_{ij}⟩ ≤ 2 for 2 ≤ i, j ≤ n. If | t | is sufficiently small, first two elements are linearly independent, and hence

C_{1j} + t C_{ij} ∈ ⟨C_{12} + t C_{i2}, C_{13} + t C_{i3}⟩ ⊂ ⟨x_{1}, x_{2}, x_{3}⟩.

Since C_{1j} ∈ ⟨x_{1}, x_{2}, x_{3}⟩, we have C_{ij} ∈ ⟨x_{1}, x_{2}, x_{3}⟩, and we may put C_{ij} = p_{ij}x_{1} + q_{ij}x_{2} + r_{ij}x_{3}. Next, we take out the coefficient of x_{1}, x_{2}, x_{3} in the above three elements C_{12} +t C_{i2}, C_{13} +t C_{i3}, C_{1j} +t C_{ij}. Then, since these vectors span the space of dimension≤ 2, it follows that

| \begin{matrix}1 − tc_{i} & 0 & tb_{i} \\ 0 & 1 − tc_{i} & ta_{i} \\ a_{j} + tp_{ij} & −b_{j} + tq_{ij} & tr_{ij} \end{matrix} |
= t(tc_{i} − 1){(c_{i}r_{ij} + b_{i}p_{ij} + a_{i}q_{ij})t − r_{ij} + a_{j}b_{i} − a_{i}b_{j}} = 0

for any t. In particular, we have r_{ij} = a_{j}b_{i}−a_{i}b_{j}. Similarly, using the conditions rank((e_{2}+ te_{i})⌋C) ≤ 2, rank ((e_{3} + te_{i})⌋C) ≤ 2, we obtain q_{ij} = b_{j}c_{i} − b_{i}c_{j}, p_{ij} = a_{i}c_{j} − a_{j}c_{i}. Hence, it follows that

C_{ij} = (a_{i}c_{j} − a_{j}c_{i})x_{1} + (b_{j}c_{i} − b_{i}c_{j})x_{2} + (a_{j}b_{i} − a_{i}b_{j})x_{3}.

Then, in the same way as in the case (i), we have the desired result. q.e.d.

We prepare one more lemma for later use.

Lemma 6. Assume that C ∈ ∧^{2}V^{∗} ⊗ A^{1} satisfies condition (C_{Q}) and there exists a vector v ∈ V such that rank (v⌋C) ≥ 3. Then, there exist a basis {e_{1}, ⋯, e_{n}} of V and a_{2} ∼ a_{n} ∈ K satisfying

C_{ij} = a_{j}C_{1i} − a_{i}C_{1j} (a_{1} = −1).

Proof. We fix a basis {e_{1}, ⋯, e_{n}} such that e_{1} = v. Then, in the same way as in the proof of Lemma 4, we may assume that {C_{12}, ⋯, C_{1p}} is linearly independent and C_{1,p+1} ∼ C_{1n} ∈ ⟨C_{12}, ⋯, C_{1p}⟩. (Note that p ≥ 4 because rank(v⌋C) ≥ 3.) From condition (C_{Q}), the set {C_{1i}, C_{1j}, C_{ij}} is linearly dependent for 2 ≤ i ≠ j ≤ p, and hence, we may put

C_{ij} = a_{ij}C_{1i} − a_{ji}C_{1j}

for some a_{ij} ∈ K. (Note that C_{ij} = −C_{ji}.) In addition, we have from condition (C_{Q})

| C_{1i} C_{1j} C_{ik} | + | C_{1i} C_{1k} C_{ij} | = 0

for 2 ≤ i, j, k ≤ p (i, j, k are all distinct). By substituting the above expression into this equality, we have immediately

(a_{ji} − a_{ki})| C_{1i} C_{1j} C_{1k} | = 0.

Since {C_{1i}, C_{1j}, C_{1k}} is linearly independent, we have a_{ji} = a_{ki}. Therefore, we may put a_{ji} = a_{i} for 2 ≤ i ≤ p. Hence, by putting a_{1} = −1, we obtain C_{ij} = a_{j}C_{1i} − a_{i}C_{1j} for 1 ≤ i, j ≤ p.

Next, we express C_{1,p+1} ∼ C_{1n} as

C_{1,p+1} = b_{p+1,2} C_{12} + ⋯ + b_{p+1,p} C_{1p},
· · · · · · · · ·
C_{1n} = b_{n2} C_{12} + ⋯ + b_{np} C_{1p}.

Then, for 2 ≤ i ≠ j ≤ p, p + 1 ≤ λ ≤ n, we have from (C_{Q})

0 = | C_{1i} C_{1j} C_{iλ} | + | C_{1i} C_{1λ} C_{ij} |
= | C_{1i} C_{1j} C_{iλ} | + | C_{1i} C_{1λ} a_{j}C_{1i} − a_{i}C_{1j} |
= | C_{1i} C_{1j} C_{iλ} + a_{i}C_{1λ} |.

In particular, we have C_{iλ} + a_{i}C_{1λ} ∈ ⟨C_{1i}, C_{1j}⟩. Since p ≥ 4, there exists an index k (2 ≤ k ≤ p), different from i, j. Hence, by replacing j by k, we have in the same way, C_{iλ} + a_{i}C_{1λ} ∈ ⟨C_{1i}, C_{1k}⟩, which implies C_{iλ} + a_{i}C_{1λ} ∈ ⟨C_{1i}⟩. Therefore, we may express

C_{iλ} = a_{iλ}C_{1i} − a_{i}C_{1λ}

for 1 ≤ i ≤ p, p + 1 ≤ λ ≤ n. (We may include the case i = 1 because a_{1} = −1.) We will show that the value a_{iλ} does not depend on i. For this purpose, we put v_{1} = e_{1}, v_{2} = e_{i} + e_{j}, v_{3} = e_{k} + te_{λ} (2 ≤ i, j, k ≤ p, i, j, k are all distinct, p + 1 ≤ λ ≤ n and t ∈ K is a parameter). Then, from (C_{Q}), we have

0 = | C(v_{1}, v_{2}) C(v_{1}, v_{3}) C(v_{2}, v_{3}) |
= | C_{1i} + C_{1j} C_{1k} + t C_{1λ} C_{ik} + t C_{iλ} + C_{jk} + t C_{jλ} |
= | C_{1i} + C_{1j} C_{1k} + t C_{1λ} a_{k}C_{1i} − a_{i}C_{1k} + a_{k}C_{1j} − a_{j}C_{1k}
+ t(a_{iλ}C_{1i} − a_{i}C_{1λ} + a_{jλ}C_{1j} − a_{j}C_{1λ}) |
= t | C_{1i} + C_{1j} C_{1k} + t C_{1λ} a_{iλ}C_{1i} + a_{jλ}C_{1j} |
= t | C_{1i} + C_{1j} C_{1k} + t(b_{λ2}C_{12} + ⋯ + b_{λp}C_{1p}) (a_{jλ} − a_{iλ})C_{1j} |.

Then, by taking out the coefficient of C_{1i}, C_{1j}, C_{1k}, we have

t | \begin{matrix}1 & 1 & 0 \\ tb_{λi} & tb_{λj} & 1 + tb_{λk} \\ 0 & a_{jλ} − a_{iλ} & 0 \end{matrix} | = t(a_{iλ} − a_{jλ})(1 + tb_{λk}) = 0

for any t, which implies a_{iλ} = a_{jλ}. In particular, we may put a_{iλ} = a_{λ}, and therefore,

C_{iλ} = a_{λ}C_{1i} − a_{i}C_{1λ}

for 1 ≤ i ≤ p, p + 1 ≤ λ ≤ n.

Finally, we show the equality

C_{λμ} = a_{μ}C_{1λ} − a_{λ}C_{1μ}

for p+1 ≤ λ, μ ≤ n. In the same way as above, we put v_{1} = e_{1}, v_{2} = e_{i} +se_{λ}, v_{3} = e_{j} +te_{μ} (2 ≤ i ≠ j ≤ p, p + 1 ≤ λ ≠ μ ≤ n, and s, t ∈ K are parameters), and apply condition (C_{Q}). Then, we have

0 = | C_{1i} + s C_{1λ} C_{1j} + t C_{1μ} C_{ij} + t C_{iμ} + s C_{λj} + st C_{λμ} |
= | C_{1i} + s C_{1λ} C_{1j} + t C_{1μ} a_{j}C_{1i} − a_{i}C_{1j} + t(a_{μ}C_{1i} − a_{i}C_{1μ})
− s(a_{λ}C_{1j} − a_{j}C_{1λ}) + st C_{λμ} |
= | C_{1i} + s C_{1λ} C_{1j} + t C_{1μ} ta_{μ}C_{1i} − sa_{λ}C_{1j} + st C_{λμ} |
= | C_{1i} + s C_{1λ} C_{1j} + t C_{1μ} st(C_{λμ} − a_{μ}C_{1λ} + a_{λ}C_{1μ}) |
= st | C_{1i} + s C_{1λ} C_{1j} + t C_{1μ} C_{λμ} − a_{μ}C_{1λ} + a_{λ}C_{1μ} |.

Now, assume that st ≠ 0 and | s |, | t | are sufficiently small. Then, since {C_{1i} +s C_{1λ}, C_{1j} + t C_{1μ}} is linearly independent, we have

C_{λμ} − a_{μ}C_{1λ} + a_{λ}C_{1μ} ∈ ⟨C_{1i} + s C_{1λ}, C_{1j} + t C_{1μ}⟩.

In particular, taking the limit s, t → 0, it follows that

C_{λμ} − a_{μ}C_{1λ} + a_{λ}C_{1μ} ∈ ⟨C_{1i}, C_{1j}⟩.

Using an index k (2 ≤ k ≤ p), which is different from i and j, we repeat the same procedure. Then, we have

C_{λμ} − a_{μ}C_{1λ} + a_{λ}C_{1μ} ∈ ⟨C_{1i}, C_{1j}⟩ ∩ ⟨C_{1i}, C_{1k}⟩ ∩ ⟨C_{1j}, C_{1k}⟩ = {0},

which implies C_{λμ} = a_{μ}C_{1λ} − a_{λ}C_{1μ}, and we complete the proof of the lemma. q.e.d.

§ 3. Proof of Theorems

Using the lemmas prepared in § 2, we give a proof of Theorems 1 and 2 in this section. Proof of Theorem 1. (5) If C ∈ Σ_{5}, then C is expressed as fΩ (f ∈ A^{1}, Ω ∈

∧^{2}V^{∗}) by Proposition 3 (4), and hence, condition (C_{1}) clearly holds. Conversely, assume C satisfies condition (C_{1}). Then, for any vector v ∈ V, we have rank (v⌋C) ≤ 1. If C = 0, then the theorem holds trivially, and hence we may assume that there exists v such that rank (v⌋C) = 1. We fix a basis {e_{1}, ⋯, e_{n}} such that e_{1} = v, and by the symmetry, we may put C_{12} = x_{1}, C_{1i} ∈ ⟨x_{1}⟩. From the condition rank (e_{2}⌋C) ≤ 1, we have dim ⟨C_{21}, C_{23}, ⋯, C_{2n}⟩ ≤ 1, in particular, C_{2i} ∈ ⟨x_{1}⟩. Next, for 2 ≤ i ≤ n, we have

(e_{1} + te_{i})⌋C = t C_{i1} ω_{1} + (C_{12} + t C_{i2}) ω_{2} + ⋯ + (C_{1n} + t C_{in}) ω_{n},

as in the proof of Lemma 5 (ii). Since rank((e_{1}+te_{i})⌋C) ≤ 1, we have dim⟨C_{12}+t C_{i2}, C_{1j}+ t C_{ij}⟩ ≤ 1 for 2 ≤ i, j ≤ n. If | t | is sufficiently small, C_{12} + t C_{i2} is not zero, and hence

C_{1j} + t C_{ij} ∈ ⟨C_{12} + t C_{i2}⟩ = ⟨x_{1}⟩.

In particular, we have C_{ij} ∈ ⟨x_{1}⟩ because C_{1j} ∈ ⟨x_{1}⟩. Therefore, the coefficients of C are all contained in the space ⟨x_{1}⟩, and hence, rank e_{C} ≤ 1, i.e., C ∈ Σ_{5}.

(4) If C ∈ Σ_{4}, C is expressed as fβ_{1} ∧ β_{2} by Proposition 3 (3). Then, we have clearly C ∧ C = 0 and C satisfies condition (C_{P}). In addition, from Proposition 3 (4), we have clearly C ∈ Σ_{5}, which implies that C satisfies (C_{1}), just we showed above. Next, assume that C satisfies (C_{P}) and (C_{1}). From condition (C_{1}), we have C ∈ Σ_{5}, and we may express C as fΩ (f ≠ 0 ∈ A^{1}, Ω ∈ ∧^{2}V^{∗}). Then, from condition (C_{P}), we have C ∧ C = f^{2}Ω ∧ Ω = 0, i.e., Ω ∧ Ω = 0, which is equivalent to classical Plücker’s relation. Hence Ω is decomposable, and C is expressed as fβ_{1} ∧ β_{2}. Thus, by Proposition 3 (3), we have C ∈ Σ_{4}.

(1) Assume that C ∈ Σ_{1}. Then C is expressed as α∧β, and hence it satisfies Plücker’s relation C ∧ C = 0. Next, for any vectors v_{i} ∈ V, we put C_{ij} = C(v_{i}, v_{j}), β_{i} = β(v_{i}). Then from the condition β ∧ C = 0, we have β_{1}C_{23} − β_{2}C_{13} + β_{3}C_{12} = 0, which implies that {C_{12}, C_{13}, C_{23}} is linearly dependent in the case (β_{1}, β_{2}, β_{3}) ≠ 0. If β_{1} = β_{2} = β_{3} = 0, we have clearly C_{12} = C_{13} = C_{23} = 0, and we obtain the same conclusion. Now, we prove the converse part. First, assume that there exists v ∈ V such that rank (v⌋C) ≥ 2. Then, by Lemma 4, C_{ij} is expressed as

C_{ij} = a_{j}C_{1i} − a_{i}C_{1j},

for some a_{i} ∈ K. Then, by putting α = ∑ C_{1i} ω_{i} and β = ∑ a_{i}ω_{i}, we have C = α ∧ β, which implies that C is decomposable. Next, assume that rank (v⌋C) ≤ 1 for any v. In this case, the 2-form C satisfies two conditions (C_{P}) and (C_{1}). Hence, by Theorem 1 (4), which we showed above, we have C ∈ Σ_{4}. In particular, from Proposition 3 (3), C is expressed as fβ_{1} ∧ β_{2}, which implies that C is decomposable.

(3) By Proposition 3 (2), “only if” part is clear. We assume that C satisfies conditions (C_{Q}) and (C_{2}). From (C_{2}), we have rank (v⌋C) ≤ 2 for any v ∈ V. If rank (v⌋C) ≤ 1 for any v, then C satisfies condition (C_{1}), and in particular, C ∈ Σ_{5} ⊂ Σ_{3} (cf. Proposition 3 (2), (4)). If there exists v such that rank (v⌋C) = 2, then, as before, we can choose a basis {e_{i}} such that e_{1} = v, {C_{12}, C_{13}} is linearly independent and C_{14} ∼ C_{1n} ∈ ⟨C_{12}, C_{13}⟩. Since two conditions (C_{Q}) and (C_{2}) hold, we have the following two equalities, which we showed in § 2, after the proof of Lemma 4.

| C_{ij} C_{ik} C_{jl} | = 0, (B) | C_{ij} C_{pk} C_{jl} | + | C_{pj} C_{ik} C_{jl} | = 0. (C)

From (B), we have | C_{12} C_{13} C_{2l} | = 0, i.e., C_{2l} ∈ ⟨C_{12}, C_{13}⟩. From (C), we have | C_{21} C_{pk} C_{13} | + | C_{p1} C_{2k} C_{13} | = 0. Since C_{p1}, C_{2k} ∈ ⟨C_{12}, C_{13}⟩, the second term is zero, and hence, we have C_{pk} ∈ ⟨C_{12}, C_{13}⟩, which shows that the 2-form C is ⟨C_{12}, C_{13}⟩- valued. In particular, the number of variables is reducible to two, and hence we have C ∈ Σ_{3}.

(2) If C ∈ Σ_{2}, it is expressed as f_{1}β_{1} ∧ β_{2} + f_{2}β_{1} ∧ β_{3} + f_{3}β_{2} ∧ β_{3} by Proposition 3 (1). And in addition, without loss of generality, we may assume {β_{1}, β_{2}, β_{3}} is linearly independent, by changing f_{i} if necessary. We extend {β_{i}} to a basis of V^{∗} and denote its dual by {e_{i}}. Then, for any vector v = ∑ a_{i}e_{i} ∈ V, we have

v⌋C = a_{1}e_{1}⌋C + a_{2}e_{2}⌋C + a_{3}e_{3}⌋C
= a_{1}(f_{1}β_{2} + f_{2}β_{3}) + a_{2}(−f_{1}β_{1} + f_{3}β_{3}) − a_{3}(f_{2}β_{1} + f_{3}β_{2})
= −(a_{2}f_{1} + a_{3}f_{2})β_{1} + (a_{1}f_{1} − a_{3}f_{3})β_{2} + (a_{1}f_{2} + a_{2}f_{3})β_{3}.

By using the equality

−a_{1}(a_{2}f_{1} + a_{3}f_{2}) + a_{2}(a_{1}f_{1} − a_{3}f_{3}) + a_{3}(a_{1}f_{2} + a_{2}f_{3}) = 0,

we can easily check that rank (v⌋C) ≤ 2, and hence, C satisfies condition (C_{2}). From the above expression of C, Plücker’s relation C ∧ C = 0 is clearly satisfied.

Next, we assume that C satisfies conditions (C_{P}) and (C_{2}). If there exist v_{i} ∈ V such that {C(v_{1}, v_{2}), C(v_{1}, v_{3}), C(v_{2}, v_{3})} is linearly independent, then by Lemma 5 and Proposition 3 (1), we have C ∈ Σ_{2}. If {C(v_{1}, v_{2}), C(v_{1}, v_{3}), C(v_{2}, v_{3})} is dependent for any v_{i}, then C satisfies conditions (C_{P}), (C_{Q}), (C_{2}). Hence, by Theorem 1 (1), (3), it is decomposable and the number of variables can be reducible to two. Using these two facts, it is easy to see that C is in the form (f_{1}β_{1} + f_{2}β_{2}) ∧ β_{3}, and by Proposition 3 (1), we have C ∈ Σ_{2}.

Finally, we show that Σ_{i} is an irreducible variety. By definition and Theorem 1 (1), each Σ_{i} is an algebraic set of ∧^{2}V^{∗} ⊗ A^{1} because it is defined by the vanishing of some polynomials of C. In addition, by Proposition 3, it is just equal to the image of certain polynomial map from some affine space, and hence it is irreducible. q.e.d.

Proof of Theorem 2. For three statements, “if” parts are all clear from Theorem 1. We prove “only if” parts.

(1) Assuming that C satisfies (C_{P}) and C ∉ Σ_{1}, we show C ∈ Σ_{2}. By Theorem 1 (1) and the condition C ∉ Σ_{1}, C does not satisfy condition (C_{Q}), namely, there exist v_{1}, v_{2}, v_{3} such that {C(v_{1}, v_{2}), C(v_{1}, v_{3}), C(v_{2}, v_{3})} is linearly independent. Then, by Lemma 5, C is expressed in the form f_{1}β_{1} ∧ β_{2} + f_{2}β_{1} ∧ β_{3} + f_{3}β_{2} ∧ β_{3}, and hence C ∈ Σ_{2}.

(2) We assume that C satisfies (C_{Q}) and C ∉ Σ_{3}. Then, C does not satisfy (C_{2}), as above. Hence, there exists v such that rank (v⌋C) ≥ 3, and by Lemma 6, we have C_{ij} = a_{j}C_{1i} − a_{i}C_{1j} for some a_{i}. Using this expression, we have immediately C ∈ Σ_{1} as we have done in the proof of Theorem 1 (1).

(3) Assume that the conditions (C_{2}) and C ∉ Σ_{3} hold. Then, since C does not satisfy (C_{Q}), we have C ∈ Σ_{2} by Lemma 5, in the same way as (1). q.e.d.

§ 4. Dimension and the inverse formula

Scalar valued decomposable 2-forms C ∈ ∧^{2}V^{∗} are expressed as β_{1} ∧ β_{2}. But two 1-forms β_{1}, β_{2} ∈ V^{∗} are not uniquely determined from C. In contrast, for A^{1}-valued decomposable 2-forms C = α ∧ β, two 1-forms α ∈ V^{∗} ⊗ A^{1} and β ∈ V^{∗} are essentially uniquely determined if C is sufficiently generic (precisely, if C ∈ Σ_{1}\Σ_{4}). In this section, using this result, we express α and β explicitly in terms of the components of C. In addition, we determine the dimension of each variety Σ_{i} by using the results obtained in previous sections.

Proposition 7. Assume C ∈ Σ_{1}\Σ_{4} and C = α ∧ β = α′ ∧ β′ (α, α′ ∈ V^{∗} ⊗ A^{1}, β, β′ ∈ V^{∗}). Then there exist k (≠ 0) ∈ K and f ∈ A^{1} such that α′ = kα + fβ, β′ = 1/k · β.

Proof. Since C satisfies conditions (C_{P}) and C ∉ Σ_{4}, it does not satisfy (C_{1}), and hence, there exists v ∈ V such that rank (v⌋C) ≥ 2. Then, by Lemma 4, C_{ij} is expressed as

C_{ij} = a_{j}C_{1i} − a_{i}C_{1j},

by using some a_{i} ∈ K, which is uniquely determined. In addition, as stated in the proof of Lemma 4, we may assume that {C_{12}, C_{13}} is linearly independent by changing the indices if necessary. We put α_{i} = α(e_{i}), β_{i} = β(e_{i}), α′_{i} = α′(e_{i}) and β′_{i} = β′(e_{i}). If β_{1} = 0, then we have C_{12} = α_{1}β_{2} and C_{13} = α_{1}β_{3}, which implies that C_{12} and C_{13} are parallel. Hence, we have β_{1} ≠ 0. In the same way, we have β′_{1} ≠ 0. Then, from the condition β ∧ C = β ∧ α ∧ β = 0, we have

β_{1}C_{ij} − β_{i}C_{1j} + β_{j}C_{1i} = 0,

namely,

C_{ij} = \frac{β_{i}}{β_{1}} C_{1j} − \frac{β_{j}}{β_{1}} C_{1i}.

Since the coefficient β_{i}/β_{1} is uniquely determined from Lemma 4, we have β_{i}/β_{1} = β′_{i}/β′_{1}, which implies β′ = β′_{1}/β_{1} · β. Next, from the equality C_{1i} = α_{1}β_{i} − α_{i}β_{1}, we have

α_{i} = \frac{1}{β_{1}} (α_{1}β_{i} − C_{1i}).

Then, in terms of the dual basis {ω_{i}}, we have

α = ∑ α_{i}ω_{i}

= \frac{1}{β_{1}} ∑(α_{1}β_{i} − C_{1i}) ω_{i}

= \frac{1}{β_{1}} (α_{1}β − e_{1}⌋C).

Using this equality, we obtain

α′ = \frac{1}{β′_{1}} (α′_{1}β′ − e_{1}⌋C)

= \frac{α′_{1}}{β′_{1}} · \frac{β′_{1}}{β_{1}} β − \frac{1}{β′_{1}} (α_{1}β − β_{1}α)

= fβ + kα, where f = α′_{1}/β_{1} − α_{1}/β′_{1} ∈ A^{1} and k = β_{1}/β′_{1}. q.e.d.

Remark. As is easy to see, we cannot drop the condition C ∉ Σ_{4} in this proposition. In particular, it is necessary n ≥ 3 and m ≥ 2 to hold the above condition, because rank (v⌋C) ≥ 2 for some v ∈ V. (Note that (v⌋C)(v) = 0.)

Now, we give the explicit inverse formula for generic C. Using a basis {e_{1}, ⋯, e_{n}}, we put C_{ij} = C(e_{i}, e_{j}) = ∑_{p=1}^{m} C_{ijp}x_{p}. Then, since {C_{1i}, C_{1j}, C_{ij}} is dependent, we have

| \begin{matrix}C_{1ip} & C_{1jp} & C_{ijp} \\ C_{1iq} & C_{1jq} & C_{ijq} \\ C_{1ir} & C_{1jr} & C_{ijr} \end{matrix} | = 0,

and this equality implies

| \begin{matrix}C_{1jp} & C_{ijp} \\ C_{1jq} & C_{ijq} \end{matrix} | C_{1i} − | \begin{matrix}C_{1ip} & C_{ijp} \\ C_{1iq} & C_{ijq} \end{matrix} | C_{1j} + | \begin{matrix}C_{1ip} & C_{1jp} \\ C_{1iq} & C_{1jq} \end{matrix} | C_{ij} = 0.

Hence, if | \begin{matrix}C_{1ip} & C_{1jp} \\ C_{1iq} & C_{1jq} \end{matrix} | ≠ 0, we have

C_{ij} = \frac{| \begin{matrix}C_{1ip} & C_{ijp} \\ C_{1iq} & C_{ijq} \end{matrix} | C_{1j} − | \begin{matrix}C_{1jp} & C_{ijp} \\ C_{1jq} & C_{ijq} \end{matrix} | C_{1i}}{| \begin{matrix}C_{1ip} & C_{1jp} \\ C_{1iq} & C_{1jq} \end{matrix} |} .

Then, combining with the expression

C_{ij} = \frac{β_{i}}{β_{1}} C_{1j} − \frac{β_{j}}{β_{1}} C_{1i},

appeared in the proof of Proposition 7, we have

\frac{β_{i}}{β_{1}} = \begin{matrix}| & \begin{matrix}C_{1ip} & C_{ijp} \\ C_{1iq} & C_{ijq} \end{matrix} & | \\ | & \begin{matrix}C_{1ip} & C_{1jp} \\ C_{1iq} & C_{1jq} \end{matrix} & | \end{matrix},

because the coefficient of C_{1j} is uniquely determined from Lemma 4. From this expression, the 1-form β is uniquely determined up to a non-zero constant, and this gives the inverse formula of β. Note that the right hand side of this expression does not depend on the choice of indices j, p, q unless the denominator is zero. The inverse formula for α is already given in the proof of Proposition 7:

α = \frac{1}{β_{1}} (α_{1}β − e_{1}⌋C).

From this expression we know that the 1-form α is essentially equal to e_{1}⌋C up to a non-zero constant. We remark that C ∈ Σ_{1} belongs to Σ_{4} if and only if the determinant

(D)  | \begin{matrix}C_{ijp} & C_{ikp} \\ C_{ijq} & C_{ikq} \end{matrix} |

is zero for all vectors v_{i} and indices p, q, where C_{ij} = C(v_{i}, v_{j}). The denominator of the inverse formula of β is a special case of this determinant (D).

Finally, we determine the dimension of the varieties Σ_{1} ∼ Σ_{6}.

Theorem 8. The dimension of the variety Σ_{i} is given in the following table:

n = 2 m = 1 n ≥ 3 and m ≥ 2
Σ_{1} m 2n − 3  (n − 1)(m + 1)
Σ_{2} m 2n − 3  3(n + m − 3)
Σ_{3} m \binom{n}{2} 2\binom{n}{2} + 2m − 4
Σ_{4} m 2n − 3  2n + m − 4
Σ_{5} m \binom{n}{2} \binom{n}{2} + m − 1
Σ_{6} m 2n − 3  3n + 2m − 7

In the case n = 2, all varieties Σ_{i} are equal to the whole space ∧^{2}V^{∗} ⊗ A^{1} ≃ A^{1}, and in the case m = 1, Σ_{3} = Σ_{5} = ∧^{2}V^{∗} ⊗ A^{1} ≃ ∧^{2}V^{∗} and Σ_{1} = Σ_{2} = Σ_{4} = Σ_{6} coincides with the set of scalar valued decomposable elements of ∧^{2}V^{∗} ⊗ A^{1} ≃ ∧^{2}V^{∗}.

Proof. If n = 2, then any element C ∈ ∧^{2}V^{∗} ⊗ A^{1} can be expressed as fβ_{1} ∧ β_{2}. Hence, by Proposition 3, we have C ∈ Σ_{i} for i = 1 ∼ 6, which implies Σ_{i} = ∧^{2}V^{∗}⊗A^{1}. Next, in the case m = 1, it is easy to see that any element of Σ_{1}, Σ_{2}, Σ_{4}, Σ_{6} (resp. Σ_{3}, Σ_{5}) is expressed in the form x_{1}β_{1}∧β_{2} (resp. x_{1}Ω). (Note that β_{1}∧β_{2}+β_{1}∧β_{3}+β_{2}∧β_{3} = (β_{1}+β_{2})∧(β_{2}+β_{3}), and it is decomposable.) In particular, the variety Σ_{1} = Σ_{2} = Σ_{4} = Σ_{6} coincides with the set of decomposable elements of ∧^{2}V^{∗} and Σ_{3} = Σ_{5} is equal to the whole space. The dimension of Σ_{1} is easily determined by calculating the dimension of the isotropy subgroup of β_{1} ∧ β_{2} (≠ 0) under the action of the general linear group GL(n, K), because GL(n, K) acts transitively on the set Σ_{1}\{0}. We omit the explicit calculations.

Next, we consider the case n ≥ 3 and m ≥ 2. If C = α∧β ∈ Σ_{1}\Σ_{4}, then by Proposition 7, the parametrization of C by α and β has the freedom which is expressed uniquely by the pair (k, f) ∈ K × A^{1}. Hence, we have dim Σ_{1} = dim V^{∗} ⊗ A^{1} + dim V^{∗} − 1 − dim A^{1} = (n − 1)(m + 1).

For the variety Σ_{2}, we first assume m ≥ 3, and C ∈ Σ_{2}\Σ_{6}, i.e., C satisfies (C_{P}), (C_{2}), but not (C_{Q}). Then, using a suitable basis {e_{i}}, the set {C_{12}, C_{13}, C_{23}} is linearly independent, and as stated in the proof of Lemma 5, we have

C_{1i} = a_{i}C_{12} − b_{i}C_{13},
C_{2i} = c_{i}C_{12} − b_{i}C_{23},
C_{3i} = c_{i}C_{13} − a_{i}C_{23}

for 4 ≤ i ≤ n. In addition, other C_{ij} is also expressed in terms of {C_{12}, C_{13}, C_{23}} and {a_{i}, b_{i}, c_{i}}_{4≤i≤n}. Since these parameters are uniquely determined by C, we have dim Σ_{2} = 3m + 3(n − 3) = 3(n + m − 3). If m = 2, any element C ∈ Σ_{2} is contained in Σ_{1} because it satisfies conditions (C_{P}) and (C_{Q}). (Note that condition (C_{Q}) is automatically satisfied in the case m = 2.) Conversely, since any element C ∈ Σ_{1} is expressed as (f_{1}β_{1} + f_{2}β_{2}) ∧ β_{3} in the case m = 2, we have Σ_{1} ⊂ Σ_{2} by Proposition 3 (1). Hence, we have Σ_{1} = Σ_{2}, and in particular, dim Σ_{2} = dim Σ_{1} = 3(n − 1), which is equal to 3(n + m − 3).

For the variety Σ_{3}, we take an element C ∈ Σ_{3}\Σ_{5}. Then, from the condition C ∉ Σ_{5}, we may assume that {C_{12}, C_{13}} is independent, and other C_{ij} is uniquely expressed as a linear combination of {C_{12}, C_{13}} because the number of variables is reducible to two. Hence, we have dim Σ_{3} = 2m + \{\binom{n}{2} − 2\} × 2 = 2\binom{n}{2} + 2m − 4.

Next, any element of Σ_{4} is expressed as fβ_{1}∧β_{2}. As showed above, the dimension of the variety of decomposable elements of ∧^{2}V^{∗} is 2n − 3, and the degree of freedom of f is m. Since the scalar multiplication appears in common, we have dim Σ_{4} = (2n − 3) + m − 1 = 2n + m − 4.

For the variety Σ_{5}, any element of Σ_{5} is expressed in the form fΩ, and by the same reason as above, we have dim Σ_{5} = \binom{n}{2} + m − 1.

Finally, for the variety Σ_{6}, we take an element C ∈ Σ_{6}\Σ_{4}. Then, since it does not satisfy (C_{1}), we can apply Lemma 4. As stated in the proof of this lemma, {C_{12}, C_{13}} is linearly independent, and in addition, we have C_{14} ∼ C_{1n} ∈ ⟨C_{12}, C_{13}⟩ from condition (C_{2}). Hence, we may put C_{1i} = b_{i}C_{12} + c_{i}C_{13} for 4 ≤ i ≤ n. Since other C_{ij} is expressed as

C_{ij} = a_{j}C_{1i} − a_{i}C_{1j}

for some a_{i} (a_{1} = −1), C is parametrized by {C_{12}, C_{13}, a_{2}, ⋯, a_{n}, b_{4}, ⋯, b_{n}, c_{4}, ⋯, c_{n}}. It is easy to check that these parameters are uniquely determined by C, and therefore, we have dim Σ_{6} = 2m + (n − 1) + 2(n − 3) = 3n + 2m − 7. q.e.d.

We remark that the exceptional case n = 2 or m = 1 in this theorem corresponds to the case where the action of the product group GL(n, K) × GL(m, K) on ∧^{2}V^{∗} ⊗ A^{1} reduces to the single group GL(m, K) or GL(n, K), i.e., the case where the 3-tensor space ∧^{2}V^{∗} ⊗ A^{1} is reduced to a 1- or 2-tensor space. And so we must treat separately to determine the dimension of the variety, though two equalities dim Σ_{4} = 2n + m − 4 and dim Σ_{5} = \binom{n}{2} + m − 1 always hold without the assumption n ≥ 3 and m ≥ 2.

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