The transfer ideal under the action of orthogonal group in modular case

Zeng Lingli

doi:10.1515/math-2022-0021

Open Access Published by De Gruyter Open Access May 16, 2022

The transfer ideal under the action of orthogonal group in modular case

Zeng Lingli

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0021

Abstract

In this paper, we study the structures of the invariant subspaces under the action of orthogonal group O 2 ν ( F q , S ) . In particular, we give a detailed description of 2-codimensional invariant subspaces. Moreover, we show that the height of transfer ideal Im ( Tr O 2 ν ( F q , S ) ) is 2 and give a primary decomposition for the radical ideal of this transfer ideal.

Keywords: transfer ideal; transfer variety; invariant subspace; orthogonal group

MSC 2010: 47A15; 20M12

1 Introduction

Let V be a vector space of dimension n over a field F of characteristic p and let F [ V ] be the symmetric algebra of V ∗ (the dual of V ). If { x 1 , … , x n } is a basis for V , then F [ V ] can be identified with the polynomial ring F [ x 1 , … , x n ] . Let G ⊆ G L ( V ) be a finite group. Then the elements of G act on F [ V ] as algebra automorphisms and we form the subring

F [ V ] G ≜ { f ∈ F [ V ] ∣ g f = f , ∀ g ∈ G }

of G -invariant polynomials. The image of transfer map

Tr G : F [ V ] → F [ V ] G ; f ↦ ∑ g ∈ G g ⋅ f

is an ideal of F [ V ] G . We call it the transfer ideal under the action of G and denoted by Im ( Tr G ) . If the order of G is invertible in F , then the transfer map Tr G is a surjection onto F [ V ] G . When the characteristic of F divides the order of G , the transfer ideal is a proper, nonzero ideal in F [ V ] G . The transfer ideal is of considerable interest in modular invariant theory.

In 1999 Shank and Wehlau [1] proved that Im ( Tr G ) is a principal ideal if G is a p -group defined over F p and F [ V ] G is a polynomial ring. They also showed that Im ( Tr G ) are principal for G = SL n ( F q ) and GL n ( F q ) with natural actions. Later, Neusel [2,3] studied the transfer ideal Im ( Tr G ) for permutation group. In addition, she proved that the ideal Im ( Tr G ) is a prime ideal for cyclic p -groups and determined an upper bound of its height. Moreover, Kuhnigt and Smith studied the transfer ideal for the symplectic group Sp 2 ν ( F q ) and showed that the radical ideal of transfer is a principal ideal. These detailed proofs can be found on page 276 of [4].

Along this research route, we focus on the transfer ideal for the orthogonal groups. Let q = p t be a positive odd prime power, F q be the Galois field with q elements. Let S be an n × n nonsingular symmetric matrix over F q . Then the set of all matrices A such that ASA ′ = S forms a group with respect to matrix multiplication, where A ′ denotes the transpose of A . We call it the orthogonal group of degree n with respect to S and denote it by O n ( F q , S ) , i.e.,

O n ( F q , S ) = { A ∈ GL n ( F q ) ∣ ASA ′ = S } .

By [5, Theorem 6.4] we know that the nonsingular symmetric matrix S is one of the following forms:

S 2 ν = I ( ν ) I ( ν ) ; S 2 ν + 1 , 1 = 0 I ( ν ) I ( ν ) 0 1 ; S 2 ν + 1 , z = 0 I ( ν ) I ( ν ) 0 z ; S 2 ν + 2 = 0 I ( ν ) I ( ν ) 0 1 0 0 − z ,

where I ( ν ) is a ν × ν identity matrix and z is a non square element in F q . Then, up to isomorphism, the orthogonal groups are four types. In this paper, we shall focus attention on the orthogonal group O 2 ν ( F q , S ) with respect to the nonsingular symmetric matrix S = S 2 ν . The other cases are similar and are omitted.

The paper is organized as follows. After this introductory section, in Sections 2 and 3 we discuss the structures of invariant subspaces under the action of the orthogonal group O 2 ν ( F q , S ) . In Section 4, we determine the structures of the transfer variety Ω O 2 ν ( F q , S ) and give a primary decomposition for the radical ideal of transfer ideal, and show that the height of this transfer ideal is 2. In addition, we give a detailed example for q = 3 and ν = 2 in Section 5.

2 Types of 2-codimensional invariant subspaces

Let e 1 , e 2 , … , e 2 ν be the standard basis of the vector space V = F q 2 ν . For each v = k 1 e 1 + k 2 e 2 + ⋯ + k 2 ν e 2 ν ∈ V , k i ∈ F q , there is an action of O 2 ν ( F q , S ) on V defined as

V × O 2 ν ( F q , S ) ⟶ V ( ( k 1 , k 2 , … , k 2 ν ) , A ) ⟼ ( k 1 , k 2 , … , k 2 ν ) A .

Then the vector space V together with this action is called the 2 ν -dimensional orthogonal space over F q with respect to S .

Let P be an m -dimensional vector subspace of V . We use the same symbol P to denote the matrix representation of the vector subspace P , i.e., P is an m × 2 ν matrix whose rows form a basis of the vector subspace P . Two n × n matrices A and B are said to be cogredient, if there is a n × n nonsingular matrix Q such that QAQ ′ = B . It is well known that PSP ′ is cogredient to one of the following normal forms [5]:

M ( m , 2 s , s ) = 0 I ( s ) I ( s ) 0 0 ( m − 2 s ) , M ( m , 2 s + 1 , s , 1 ) = 0 I ( s ) I ( s ) 0 1 0 ( m − 2 s − 1 ) , M ( m , 2 s + 1 , s , z ) = 0 I ( s ) I ( s ) 0 z 0 ( m − 2 s − 1 ) , M ( m , 2 s + 2 , s ) = 0 I ( s ) I ( s ) 0 1 − z 0 ( m − 2 s − 2 ) .

We use the symbol M ( m , 2 s + γ , s , Γ ) to represent any one of these four normal forms, where s is its index, γ = 0 , 1 , or 2, and Γ represents the definite part in these normal forms. If PSP ′ is cogredient to M ( m , 2 s + γ , s , Γ ) , then P is called a subspace of type ( m , 2 s + γ , s , Γ ) with respect to S in V . Subspaces of type ( m , 2 s , s ) , ( m , 2 s + 1 , s , 1 ) , ( m , 2 s + 1 , s , z ) , and ( m , 2 s + 2 , s ) are also called subspace of the hyperbolic type, the square type, the nonsquare type, and the elliptic type, respectively.

By the proof of Theorem 6.3 in [5], we have the following lemma

Lemma 2.1

[5] Subspaces of types ( m , 2 s + γ , s , Γ ) exist in the 2 ν -dimensional orthogonal space V = F q 2 ν with respect to the nonsingular symmetric matrix S if and only if

2 s + γ ≤ m ≤ ν + s .

Then, we can determine the types of 2-codimensional subspaces.

Lemma 2.2

There are five types of 2-codimensional subspaces of the 2 ν -dimensional orthogonal space V = F q 2 ν .

Proof

Let m = 2 ν − 2 . By Lemma 2.1, 2 s + γ ≤ m ≤ ν + s , it follows that if r = 0 then s = ν − 2 or ν − 1 ; if r = 1 then s = ν − 2 ; if r = 2 then s = ν − 2 . Hence, we obtain the following five types of 2-codimensional subspaces: ( 2 ν − 2 , 2 ( ν − 2 ) , ν − 2 ) , ( 2 ν − 2 , 2 ( ν − 1 ) , ν − 1 ) , ( 2 ν − 2 , 2 ( ν − 2 ) + 1 , ν − 2 , 1 ) , ( 2 ν − 2 , 2 ( ν − 2 ) + 1 , ν − 2 , z ) , and ( 2 ν − 2 , 2 ( ν − 2 ) + 2 , ν − 2 ) .□

Remark 2.3

For convenience, let type I of 2-codim, type II of 2-codim, type III of 2-codim, type IV of 2-codim, and type V of 2-codim denote the subspaces of type ( 2 ν − 2 , 2 ( ν − 2 ) , ν − 2 ) , ( 2 ν − 2 , 2 ( ν − 1 ) , ν − 1 ) , ( 2 ν − 2 , 2 ( ν − 2 ) + 1 , ν − 2 , 1 ) , ( 2 ν − 2 , 2 ( ν − 2 ) + 1 , ν − 2 , z ) , and ( 2 ν − 2 , 2 ( ν − 2 ) + 2 , ν − 2 ) , respectively.

The following two lemmas celebrated Witt’s transitivity theorem will be often used.

Lemma 2.4

([5], Theorem 6.4) Let P 1 and P 2 be two m -dimensional subspaces of V . Then there is an A ∈ O 2 ν ( F q , S ) such that P 1 = B P 2 A , where B is an m × m nonsingular matrix, if and only if P 1 and P 2 are of the same type with respect to S. In other words, O 2 ν ( F q , S ) acts transitively on each set of subspaces of the same type.

Lemma 2.5

[5, Lemma 6.8] Let P 1 and P 2 be two m × m matrices of rank m. Then there exists an element A ∈ O 2 ν ( F q , S ) such that P 1 = P 2 A if and only if P 1 S P 1 ′ = P 2 S P 2 ′ .

Now, let us study the structures of the type I of 2-codim subspaces.

Definition 2.6

([6], Section 9.2 Definition) An element T ∈ O 2 ν ( F q , S ) is called a 2-transvection if T = I + N , where I is the identity matrix, the rank of N is 2 and NSN ′ = 0 .

Lemma 2.7

([6], Section 9.2 Theorem 1) In the orthogonal group O 2 ν ( F q , S ) , each 2-transvection is similar to

I ( ν ) K I ( ν ) , where K = 0 1 − 1 0 0 ( ν − 2 ) .

Let V A = { v ∈ V ∣ v A = v } where A ∈ GL 2 ν ( F q ) . Then it is easy to check that V A is a subspace of vector space V and V A − 1 B A = V B A for each B ∈ GL 2 ν ( F q ) .

Lemma 2.8

Let T ∈ O 2 ν ( F q , S ) . Then T is a 2-transvection if and only if the invariant subspace V T is a type I of 2-codim subspace.

Proof

Suppose that T is a 2-transvection. Let T 0 = I ( ν ) K I ( ν ) in Lemma 2.7. Then T 0 is also a 2-transvection and A T A − 1 = T 0 for some A ∈ O 2 ν ( F q , S ) . For each v = k 1 e 1 + ⋯ + k 2 ν e 2 ν ∈ V , we have

v T 0 = k 1 e 1 + ⋯ + k ν e ν + ( k ν + 1 − k 2 ) e ν + 1 + ( k ν + 2 + k 1 ) e ν + 2 + k ν + 3 e ν + 3 + ⋯ + k 2 ν e 2 ν .

If v ∈ V T 0 , then v T 0 = v , whence k 1 = k 2 = 0 . Therefore,

V T 0 = { k 3 e 3 + k 4 e 4 + ⋯ + k 2 ν e 2 ν ∣ k i ∈ F q }

and dim V T 0 = 2 ν − 2 . We denote the vector invariant subspace V T 0 as the ( 2 ν − 2 ) × 2 ν matrix

T ˆ 0 = e 3 , e 4 , … , e 2 ν ′ .

By computing, it follows that

T ˆ 0 S T ˆ 0 ′ = 0 2 × ( ν − 2 ) I ( ν − 2 ) 0 ( ν − 2 ) × 2 I ( ν − 2 ) is cogredient to 0 I ( ν − 2 ) I ( ν − 2 ) 0 0 0 0 0 .

Then the type of V T 0 is ( 2 ν − 2 , 2 ( ν − 2 ) , ν − 2 ) . This implies that V T 0 is a type I of 2-codim subspace. Since V T = V A − 1 T 0 A = V T 0 A , the invariant subspace V T has the same type with V T 0 by Lemma 2.4. Consequently, V T is also a type I of 2-codim subspace.

Conversely, suppose that V T is a type I of 2-codim subspace. With the preceding discussion, the invariant subspace V T 0 is a type I of 2-codim subspace and { e 3 , e 4 , … , e 2 ν } ⊆ V T 0 . By Lemma 2.4, there exists an element A ∈ O 2 ν ( F q , S ) such that V T 0 = V T A = V A − 1 T A . Let T 1 = A − 1 T A . Then T 1 ∈ O 2 ν ( F q , S ) and the elements e 3 , e 4 , … , e 2 ν are invariants under the action of T 1 . Therefore, we may assume that

T 1 = a 11 a 12 H 11 a 1 ν + 1 a 1 ν + 2 H 12 a 21 a 22 H 21 a 2 ν + 1 a 2 ν + 2 H 22 0 0 I ( ν − 2 ) 0 0 0 ( ν − 2 ) I ( ν ) ,

where H i j , i , j = 1 , 2 , are 1 × ( ν − 2 ) matrices over F q . Since T 1 ∈ O 2 ν ( F q , S ) , it must satisfy T 1 S T 1 ′ = S , then we get the following equations:

(2.1) a 11 = a 22 = 1 a 1 ν + 2 + a 2 ν + 1 = 0 a 12 = a 21 = a 1 ν + 1 = a 2 ν + 2 = 0 H i j = 0 , i , j = 1 , 2 .

Hence,

T 1 = I ( ν ) a 1 ν + 2 K I ( ν ) ,

where a 1 ν + 2 ∈ F q ∗ .

It is easy to check that ( T 1 − I ) S ( T 1 − I ) ′ = ( 0 ) and rank ( T 1 − I ) = 2 , thus T 1 is a 2-transvection. Consequently, T = A T 1 A − 1 is also a 2-transvection.□

Next, we are going to study the structures of the type II of 2-codim subspaces.

Definition 2.9

([6], Section 9.3 Definition) Let ν ≥ 1 . A subspace P of V is called a hyperbolic place if dim ( P ) = 2 and P has a basis { u , v } such that u S u ′ = v S v ′ = 0 , u S v ′ = 1 . An element R ∈ O 2 ν ( F q , S ) is called hyperbolic motion taking the place P ∗ as axis, if v R = v , ∀ v ∈ P ∗ , and v R ∈ P , ∀ v ∈ P . Furthermore, we call R a hyperbolic rotation if R ∈ O 2 ν + ( F q , S ) .

Lemma 2.10

([6], Section 9.3 Theorem 1) Let ν ≥ 1 . In O 2 ν ( F q , S ) , each hyperbolic motion R is similar to one of the following forms:

R 1 = a I ( ν − 1 ) a − 1 I ( ν − 1 ) or R 2 = 0 a I ( ν − 1 ) 0 ( ν − 1 ) a − 1 0 0 ( ν − 1 ) I ( ν − 1 ) , a ∈ F q ∗ .

If R is a hyperbolic rotation, then R must be similar to R 1 .

Lemma 2.11

If R is a hyperbolic rotation, then ∣ R ∣ ≠ p , i.e., the order of R is not p.

Proof

By Lemma 2.10, ∣ R ∣ = ∣ R 1 ∣ = ∣ a ∣ . Since a ∈ F q ∗ , it implies that ∣ a ∣ ≠ p .□

Lemma 2.12

Let R ∈ O 2 ν ( F q , S ) . Then R is a hyperbolic rotation if and only if the invariant subspace V R is a type II of 2-codim subspace.

Proof

The following proof is similar to Lemma 2.8, thus we just give the main idea. Suppose that R is a hyperbolic rotation. Let R 1 = a I ( ν − 1 ) a − 1 I ( ν − 1 ) , 1 ≠ a ∈ F q ∗ . We have that V R 1 = { k 2 e 2 + ⋯ + k ν e ν + k ν + 2 e ν + 2 + ⋯ + k 2 ν e 2 ν ∣ k i ∈ F q } and dim V R 1 = 2 ν − 2 . Then V R 1 is a type II of 2-codim subspace. By Lemma 2.10, A R A − 1 = R 1 for some A ∈ O 2 ν ( F q , S ) , hence V R = V R 1 A is also a type II of 2-codim subspace by Lemma 2.4.

Conversely, suppose that V R is a type II of 2-codim subspace. Then there exists A ∈ O 2 ν ( F q , S ) such that V R 1 = V R A by Lemma 2.4. Let R 3 = A − 1 R A . We have that V R 1 = V R 3 , R 3 ∈ O 2 ν ( F q , S ) , and the elements e 2 , … , e ν , e ν + 2 , … , e 2 ν are invariant under the action of R 3 . Hence, consider the equations

R 3 S R 3 ′ = S and e i R 3 = e i , i = 2 , … , ν , ν + 2 , … , 2 ν ,

we obtain that

R 3 = a 11 I ( ν − 1 ) a 11 − 1 I ( ν − 1 ) , a 11 ≠ 0 ,

and so R 3 is a hyperbolic rotation. Consequently, R = A R 3 A − 1 is also a hyperbolic rotation.□

Now, we shall consider the cases of the type III and type IV of 2-codim subspaces.

Let

H = H = 1 0 0 − b a 1 b − a b I ( ν − 2 ) 0 ( ν − 2 ) 1 − a 0 1 I ( ν − 2 ) a , b ∈ F q ∗

be a set. The construction of this set is motivated by combining two 2-transvections.

Definition 2.13

An element A ∈ GL 2 ν ( F q ) is called a H 1 -type (resp. H z -type) matrix if there exists a B ∈ GL 2 ν ( F q ) such that B A B − 1 ∈ H and − 2 a b − 1 corresponding with B A B − 1 is a square (resp. nonsquare) element in F q ∗ .

Lemma 2.14

H ⊆ O 2 ν ( F q , S ) .
If H ∈ H , then ∣ H ∣ = p .
H is a H 1 -type (resp. H z -type) matrix if and only if the invariant subspace V H is a type III (resp. type IV) of 2-codim subspace.

Proof

It is easy to check (1) and (2). The proof of (3) is similar to Lemma 2.8, so we just give the main idea. Suppose that H is an H 1 -type (resp. H z -type) matrix. Then we have that V H = { k 1 ( − a b − 1 e 1 + e ν + 1 ) + k 3 e 3 + ⋯ + k ν e ν + k ν + 2 e ν + 2 + ⋯ + k 2 ν e 2 ν ∣ k i ∈ F q } and dim V H = 2 ν − 2 . Thus, the type of the matrix corresponding with V H is I ( ν − 2 ) I ( ν − 2 ) − 2 a b − 1 0 0 0 . Consequently, if − 2 a b − 1 is a square (resp. nonsquare) element in F q ∗ , then V H is a type III (resp. type IV) of 2-codim subspace.

Conversely, we give only the proof for the type III of 2-codim subspace. Suppose that the invariant subspace V A is a type III of 2-codim subspace. Then there exists an element B ∈ O 2 ν ( F q , S ) such that V H = V A B = V B − 1 A B = V C by Lemma 2.4. Let C = B − 1 A B . We have that V C = V H , C ∈ O 2 ν ( F q , S ) , and dim ( V C ) = 2 ν − 2 . Moreover, the elements e 3 , … , e ν , e ν + 2 , … , e 2 ν , and − a b − 1 e 1 + e ν + 1 are invariants under the action of C . Hence, consider the equations

C S C ′ = S , e i C = e i , i = 3 , … , ν , ν + 2 , … , 2 ν , ( − a b − 1 e 1 + e ν + 1 ) C = − a b − 1 e 1 + e ν + 1 ,

we obtain that

C = 1 0 0 − a 2 ν + 1 a 21 1 a 2 ν + 1 − a 21 a 2 ν + 1 I ( ν − 2 ) 0 ( ν − 2 ) 1 − a 21 0 1 I ( ν − 2 ) .

Next, we claim that a 21 ≠ 0 and a 2 ν + 1 ≠ 0 . If a 21 = a 2 ν + 1 = 0 , then C = I ( 2 ν ) , contradicting dim V C = 2 ν − 2 . If a 21 = 0 and a 2 ν + 1 ≠ 0 , then C is a 2-transvection. By Lemma 2.8, V C is a type I of 2-codim subspace, which contradicts that V C = V H is a type III of 2-codim subspace. If a 21 ≠ 0 and a 2 ν + 1 = 0 , then C is also a 2-transvection, a contradiction. Therefore, a 21 ≠ 0 and a 2 ν + 1 ≠ 0 , thus C ∈ H . Since V C = V H is a type III of 2-codim subspace, it follows that − 2 a 21 a 2 ν + 1 − 1 is a square element in F q ∗ . Hence, A = B C B − 1 is a H 1 -type matrix.□

Finally, we shall show the structures of the type V of 2-codim subspaces.

Lemma 2.15

If the order of the matrix A ¯ = A B C D is not p , where A , B , C , D are n × n matrices over F q , then the order of matrix A ¯ ¯ = A ∗ B ∗ 0 I 0 0 C ∗ D ∗ 0 0 0 I is not p either, where I is an m × m identity matrix, and each ∗ is an arbitrary n × m matrix over F q .

Proof

If A ¯ i = A B C D i = A i B i C i D i , then A ¯ ¯ i = A i ∗ B i ∗ 0 I 0 0 C i ∗ D i ∗ 0 0 0 I . Therefore, if A ¯ p ≠ I ( n ) , then we conclude that A ¯ ¯ p ≠ I ( n + m ) .□

Let

Q = 0 0 a 0 0 0 0 b I ( ν − 2 ) 0 ( ν − 2 ) a − 1 0 0 0 0 b − 1 0 0 0 ( ν − 2 ) I ( ν − 2 ) ,

where a , b ∈ F q ∗ , 2 a is a square element, and 2 b = − z . The construction of this element is motivated by combining two hyperbolic motions. It is easily seen that Q ∈ O 2 ν ( F q , S ) , and

V Q = { k 1 ( e 1 + a e ν + 1 ) + k 2 ( e 2 + b e ν + 2 ) + k 3 e 3 + ⋯ + k ν e ν + k ν + 3 e ν + 3 + ⋯ + k 2 ν e 2 ν ∣ k i ∈ F q } .

Then the type of the matrix corresponding with the invariant subspace V Q is I ( ν − 2 ) I ( ν − 2 ) 1 0 0 − z . Hence, V Q is a type V of 2-codim subspace.

Lemma 2.16

If A ∈ O 2 ν ( F q , S ) and the invariant subspace V A is a type V of 2-codim subspace, then ∣ A ∣ ≠ p .

Proof

Since the invariant subspaces V A and V Q are both the type V of 2-codim subspaces, there exists an element B ∈ O 2 ν ( F q , S ) such that V Q = V A B by Lemma 2.4. Let C = B − 1 A B . Then C ∈ O 2 ν ( F q , S ) and the elements e 3 , … , e ν , e ν + 3 , … , e 2 ν , e 1 + a e ν + 1 , and e 2 + b e ν + 2 are all invariants under the action of C . Therefore, we may assume that

C = a 11 a 12 H 11 a 1 ν + 1 a 1 ν + 2 H 12 a 21 a 22 H 21 a 2 ν + 1 a 2 ν + 2 H 22 0 0 I ( ν − 2 ) 0 0 0 ( ν − 2 ) a ν + 1 1 a ν + 1 2 H 31 a ν + 1 ν + 1 a ν + 1 ν + 2 H 32 a ν + 2 1 a ν + 2 2 H 41 a ν + 2 ν + 1 a ν + 2 ν + 2 H 42 0 0 0 ( ν − 2 ) 0 0 I ( ν − 2 ) ,

where H i j , i = 1 , 2 , 3 , 4 , j = 1 , 2 , are 1 × ( ν − 2 ) matrices over F q .

Now we consider the sub-block of matrix C , i.e.,

C ¯ = a 11 a 12 a 1 ν + 1 a 1 ν + 2 a 21 a 22 a 2 ν + 1 a 2 ν + 2 a ν + 1 1 a ν + 1 2 a ν + 1 ν + 1 a ν + 1 ν + 2 a ν + 2 1 a ν + 2 2 a ν + 2 ν + 1 a ν + 2 ν + 2 . .

Since e 1 + a e ν + 1 and e 2 + b e ν + 2 are invariants under the action of C , we have

(2.2) a 11 = 1 − a a ν + 1 1 a 12 = − a a ν + 1 2 a 1 ν + 1 = a − a a ν + 1 ν + 1 a 1 ν + 2 = − a a ν + 1 ν + 2 a 21 = − b a ν + 2 1 a 22 = 1 − b a ν + 2 2 a 2 ν + 1 = − b a ν + 2 ν + 1 a 2 ν + 2 = b − b a ν + 2 ν + 2

Since C ∈ O 2 ν ( F q , S ) , it must satisfy C S C ′ = S , then C ¯ satisfies

C ¯ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 C ¯ ′ = 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 .

And adding (2.2), we can obtain the following equations:

(2.3) a ν + 1 ν + 1 = 1 − a a ν + 1 1

(2.4) a ν + 1 ν + 2 = − b a ν + 1 2

(2.5) a ν + 2 ν + 1 = − a a ν + 2 1

(2.6) a ν + 2 ν + 2 = 1 − b a ν + 2 2

(2.7) a ν + 1 1 ( 1 − a a ν + 1 1 ) = b a ν + 1 2 2

(2.8) a ν + 2 2 ( 1 − b a ν + 2 2 ) = a a ν + 2 1 2

(2.9) a ν + 2 1 ( 2 a a ν + 1 1 − 1 ) + a ν + 1 2 ( 2 b a ν + 2 2 − 1 ) = 0 .

Next, we have to consider the following situations: Whether or not a ν + 1 1 , a ν + 1 2 , a ν + 2 1 , and a ν + 2 2 are 0, respectively. We only give the proof of the most difficult case when a ν + 2 1 ≠ 0 and a ν + 2 2 ≠ 2 − 1 b − 1 .

Since a ν + 2 2 ≠ 2 − 1 b − 1 , 1 − 2 b a ν + 2 2 ≠ 0 . So by (2.9), we obtain

(2.10) 1 − 2 a a ν + 1 1 1 − 2 b a ν + 2 2 = − a ν + 1 2 a ν + 2 1 .

Combining (2.7) with (2.8), we have

(2.11) a ν + 1 1 ( 1 − a a ν + 1 1 ) a ν + 2 2 ( 1 − b a ν + 2 2 ) = b a ν + 1 2 2 a a ν + 2 1 2 .

Combining (2.11) with the square of (2.10), we conclude that

(2.12) a a ν + 1 1 ( 1 − a a ν + 1 1 ) b a ν + 2 2 ( 1 − b a ν + 2 2 ) = 1 = a ν + 1 2 2 a ν + 2 1 2 ,

then a ν + 1 2 = ± a ν + 2 1 .

If a ν + 1 2 = − a ν + 2 1 , then it follows that a ν + 1 1 = a − 1 b a ν + 2 2 according to 2.9. Thus, adding (2.2)–(2.6), we have

C ¯ = 1 − b a ν + 2 2 a a ν + 2 1 a b a ν + 2 2 − a b a ν + 2 1 − b a ν + 2 1 1 − b a ν + 2 2 a b a ν + 2 1 b 2 a ν + 2 2 a − 1 b a ν + 2 2 − a ν + 2 1 1 − b a ν + 2 2 b a ν + 2 1 a ν + 2 1 a ν + 2 2 − a a ν + 2 1 1 − b a ν + 2 2 .

Let

P 1 ≜ 1 0 a 0 0 1 0 b 0 0 a a ν + 2 1 − b a ν + 2 2 0 0 0 1

and

P 2 ≜ P 1 C ¯ P 1 − 1 − I = 0 0 0 0 0 0 0 0 0 − a ν + 2 2 0 2 b a ν + 2 2 a ν + 2 1 a ν + 2 2 − 2 − 4 b a ν + 2 2 .

If we denote E = 0 − a ν + 2 2 a ν + 2 1 a ν + 2 2 , F = 0 2 b a ν + 2 2 − 2 − 4 b a ν + 2 2 , then P 2 i = 0 0 F i − 1 E F i . Since det ( F ) = 4 b a ν + 2 2 , a ν + 2 2 ≠ 0 by (2.8), we have det ( F ) ≠ 0 , thus P 2 p ≠ ( 0 ) . Therefore, ∣ P 1 C ¯ P 1 − 1 ∣ ≠ p , ∣ C ¯ ∣ ≠ p .

If a ν + 1 2 = a ν + 2 1 , then it follows that ∣ C ¯ ∣ = 2 ≠ p according to (2.2)–(2.9).

Using the same method, we finally prove that ∣ C ¯ ∣ ≠ p for all situations. Hence, ∣ C ∣ ≠ p by Lemma 2.15. Since A = B C B − 1 , it follows that ∣ A ∣ ≠ p .□

We have totally described the structures of all types of 2-codimensional invariant subspaces. We summarize the results so far obtained as follows:

Theorem 2.17

Let A ∈ O 2 ν ( F q , S ) , ∣ A ∣ = p , and the codimension of the invariant subspace V A be 2. Then the invariant subspace V A is a type I of 2-codim, type III of 2-codim, or type IV of 2-codim subspace, but neither a type II of 2-codim nor a type V of 2-codim subspace.

Proof

By Lemma 2.2, the 2-codimensional subspaces of orthogonal space V have five types. According to Lemmas 2.11 and 2.12, V A cannot be a type II of 2-codim subspace since ∣ A ∣ = p . Also, V A cannot be a type V of 2-codim subspace by Lemma 2.16. Consequently, V A is a type I of 2-codim, type III of 2-codim, or type IV of 2-codim subspace.□

Remark 2.18

By Lemmas 2.8 and 2.14, the invariant subspaces whose types are type I of 2-codim, type III of 2-codim, and type IV of 2-codim are not empty. And the type of an invariant subspace V A and the type of an element A can be determined by each other.

3 Embedding of 3-codimensional subspaces

In this section, we shall consider the invariant subspaces under the action of the elements of order p in O 2 ν ( F q , S ) , i.e., the set E = { V A ∣ A ∈ O 2 ν ( F q , S ) , ∣ A ∣ = p } . First, if A ∈ O 2 ν ( F q , S ) and the codimension of the invariant subspace V A is 1, then, by Section 9 of Chapter 7 in [6], A is a quasi-symmetry transformation and its order is 2 ≠ p . So we know that the set E cannot contain any 1-codimensional invariant subspace. Furthermore, the invariant subspaces V A whose codimensions are 2 have been studied in Section 2 and every m -codimensional subspace with m ≥ 4 can be embedded into a 3-codimensional subspace. Hence, the remainder work is to study the 3-codimensional subspaces of the orthogonal space V .

Proposition 3.1

Let W be a 3-codimensional subspace of the orthogonal space V . Then W ⊆ U , where U is a type I of 2-codim, type III of 2-codim, or type IV of 2-codim subspace.

Proof

According to Lemma 2.1, there are seven types of the 3-codimensional subspaces in the orthogonal space V . Now we shall choose the most complex type whose corresponding matrix is

M = I ( ν − 3 ) I ( ν − 3 ) 1 0 0 0 − z 0 0 0 0

to prove. Let P be the corresponding matrix of subspace W . Suppose that P S P ′ = M .

If − z is a square (resp. nonsquare) element, then − z is cogredient to 1 (resp. z ). Let W 1 be a type III (resp. type IV) of 2-codim subspace. The corresponding matrix P 1 of subspace W 1 satisfies

P 1 S P 1 ′ = I ( ν − 3 ) I ( ν − 3 ) 0 1 0 0 1 0 0 0 0 0 − z 0 0 0 0 0 .

Let A 1 = I ( ν − 3 ) I ( ν − 3 ) 1 1 2 0 0 0 0 1 0 0 0 0 1 0 1 0 0 . Then A 1 P 1 S P 1 ′ A 1 ′ = I ( ν − 3 ) I ( ν − 3 ) 1 0 0 1 0 − z 0 0 0 0 0 0 1 0 0 0 . Suppose that A 1 P 1 = v 1 ′ ⋮ v 2 ν − 3 ′ v 2 ν − 2 ′ . Let A 1 P 1 ¯ = v 1 ′ ⋮ v 2 ν − 3 ′ be the matrix obtained by deleting the last row vector of A 1 P 1 . Then the corresponding subspace of A 1 P 1 ¯ is embedded in the corresponding subspace of A 1 P 1 , and A 1 P 1 ¯ S A 1 P 1 ¯ ′ = M . By Lemma 2.5, there exists an A ∈ O 2 ν ( F q , S ) such that P = A 1 P 1 ¯ A , thus, the subspace W is embedded in the corresponding subspace of A 1 P 1 . Since A 1 is an invertible matrix, the corresponding subspace of A 1 P 1 is the corresponding subspace of P 1 , i.e., subspace W 1 . Hence, the subspace W is embedded in the type III (resp. type IV) of 2-codim subspace W 1 .

The proofs of the other six types are similar to the above type and are omitted. We summarize all situations that if W be a 3-codimensional subspace, then W is embedded in a type I of 2-codim, type III of 2-codim, or type IV of 2-codim subspace.□

4 Transfer ideal

In this section, we shall determine the structures of the transfer variety and transfer ideal. First, we recall some notations. If J is an ideal of F [ x 1 , … , x n ] , then

V ( J ) = { ( a 1 , … , a n ) ∈ F n ∣ f ( a 1 , … , a n ) = 0 , ∀ f ∈ J }

is called the variety defined by an ideal J . Consider a collection, S , of points of the affine space F n . We define the set I ( S ) of polynomials in F [ x 1 , … , x n ] by

I ( S ) = { f ∈ F [ x 1 , … , x n ] ∣ f ( a 1 , … , a n ) = 0 , ∀ ( a 1 , … , a n ) ∈ S } .

It is easy to verify that the set I ( S ) is an ideal in F [ x 1 , … , x n ] and V ( I ( S ) ) = S .

Lemma 4.1

([7, Hilbert Nullstellensatz]) Let F be an algebraically closed field. If J is an ideal of F [ x 1 , … , x n ] , then I ( V ( J ) ) = J .

Let ρ : G ↪ G L ( n , F ) be a faithful representation of a finite group over the field F . The transfer variety, denoted by Ω G , is defined by ([4, Section 6.4])

Ω G = { v ∈ V ∣ Tr G ( f ) ( v ) = 0 , ∀ f ∈ Tot ( F [ V ] ) } .

Since Im ( Tr G ) is an ideal of F [ V ] G , F [ V ] G ⊆ F [ V ] is a ring extension, we have

Ω G = { v ∈ V ∣ f ( v ) = 0 , ∀ f ∈ ( Im ( Tr G ) ) e } = V ( ( Im ( Tr G ) ) e ) ,

where ( Im ( Tr G ) ) e denotes the extension ideal of Im ( Tr G ) in F [ V ] .

Lemma 4.2

([Corollary 2.6] and [4, Corollary 6.4.6]) Let ρ : G ↪ G L ( n , F ) be a representation of a finite group over the field F of characteristic p . Then

Ω G = ⋃ g ∈ G , ∣ g ∣ = p V g ,

i.e., transfer variety is the union of the fixed-point sets of the elements in G of order p .

By Theorem 6.4.7 in [4] and its proof, we have

Lemma 4.3

[4] Let ρ : G ↪ G L ( n , F ) be a representation of a finite group over the field F of characteristic p . Then

Im ( Tr G ) = ( Im ( Tr G ) ) e ∩ F [ V ] G ,

and ht ( Im ( Tr G ) ) = ht ( Im ( Tr G ) ) = n − max { dim F ( V g ) ∣ g ∈ G and ∣ g ∣ = p } < n .

Now we can obtain the main results for the transfer variety.

Theorem 4.4

The transfer variety Ω O 2 ν ( F q , S ) of the orthogonal group O 2 ν ( F q , S ) is

Ω O 2 ν ( F q , S ) = ⋃ U 1 is type I of 2-codim U 1 ∪ ⋃ U 2 is type III of 2-codim U 2 ∪ ⋃ U 3 is type IV of 2-codim U 3 ,

i.e., Ω O 2 ν ( F q , S ) is the union of all type I of 2-codim, type III of 2-codim, and type IV of 2-codim subspaces (the same notations in Remark 2.3).

Proof

Let U be the right side of the above equality. For each U 1 , let T ∈ O 2 ν ( F q , S ) be a 2-transvection. By Lemmas 2.8 and 2.4, there exists an element A 1 ∈ O 2 ν ( F q , S ) such that U 1 = V T A 1 = V A 1 − 1 T A 1 and ∣ A 1 − 1 T A 1 ∣ = ∣ T ∣ = p . For each U 2 (resp. U 3 ), let H ¯ 1 (resp. H ¯ z ) be a H 1 -type (resp. H z -type) matrix. By Lemmas 2.14 and 2.4, there exists an element A 2 (resp. A 3 ) ∈ O 2 ν ( F q , S ) such that U 2 = V H ¯ 1 A 2 = V A 2 − 1 H ¯ 1 A 2 and ∣ A 2 − 1 H ¯ 1 A 2 ∣ = ∣ H ¯ 1 ∣ = p (resp. U 3 = V H ¯ z A 3 = V A 3 − 1 H ¯ z A 3 and ∣ A 3 − 1 H ¯ z A 3 ∣ = ∣ H ¯ z ∣ = p ). So we have U ⊆ ⋃ A ∈ O 2 ν ( F q , S ) , ∣ A ∣ = p V A . On the other hand, according to Theorem 2.17 and Section 3, it follows that ⋃ A ∈ O 2 ν ( F q , S ) , ∣ A ∣ = p V A ⊆ U . Hence, U = ⋃ A ∈ O 2 ν ( F q , S ) , ∣ A ∣ = p V A . Then Ω O 2 ν ( F q , S ) = U by Lemma 4.2.□

Remark 4.5

By the discussions in Section 3, we deduce that the whole space V is contained in transfer variety. Then Ω O 2 ν ( F q , S ) = V over F q . Let F q ¯ be the algebraic closure of F q and Ω ¯ O 2 ν ( F q , S ) be the transfer variety over F q ¯ . By the similar argument, we have that

Ω ¯ O 2 ν ( F q , S ) = ⋃ U 1 is type I of 2-codim U 1 ⊗ F q F q ¯ ∪ ⋃ U 2 is type III of 2-codim U 2 ⊗ F q F q ¯ ∪ ⋃ U 3 is type IV of 2-codim U 3 ⊗ F q F q ¯ ≠ V ⊗ F q F q ¯ .

Now, we shall determine the structures of radical ideal of transfer. Let T = I ( ν ) K I ( ν ) , where K = 0 1 − 1 0 0 ( ν − 2 ) . By the proof of Lemma 2.8, T is a 2-transvection. Moreover, the invariant subspace V T is a type I of 2-codim subspace and

V T = { k 3 e 3 + ⋯ + k 2 ν e 2 ν ∣ k i ∈ F q } .

Thus, the ideal

I ( V T ) = ⟨ x 1 , x 2 ⟩ F q [ V ] ,

where ⟨ x 1 , x 2 ⟩ F q [ V ] is the ideal generated by x 1 and