On invariant fields of vectors and covectors
aa r X i v : . [ m a t h . A C ] S e p ON INVARIANT FIELDS OF VECTORS AND COVECTORS
YIN CHEN AND DAVID L. WEHLAUA
BSTRACT . Let F q be the finite field of order q . Let G be one of the three groups GL ( n , F q ) ,SL ( n , F q ) or U ( n , F q ) and let W be the standard n -dimensional representation of G . For non-negative integers m and d we let mW ⊕ dW ∗ denote the representation of G given by the directsum of m vectors and d covectors. We exhibit a minimal set of homogenous invariant polynomials { ℓ , ℓ , . . . , ℓ ( m + d ) n } ⊆ F q [ mW ⊕ dW ∗ ] G such that F q ( mW ⊕ dW ∗ ) G = F q ( ℓ , ℓ , . . . , ℓ ( m + d ) n ) for allcases except when md = G = GL ( n , F q ) or SL ( n , F q ) .
1. I
NTRODUCTION
Let F q be the finite field with q elements and GL ( n , F q ) be the general linear group of degree n >
1. Suppose W is the standard representation of GL ( n , F q ) and W ∗ is the dual space of W .We denote by SL ( n , F q ) the special linear group, and by U ( n , F q ) the unipotent group of uppertriangular matrices with 1’s on the diagonal. Write mW ⊕ dW ∗ to denote the direct sum of m copies of W and d copies of W ∗ for m , d ∈ N and let G be one of the groups GL ( n , F q ) , SL ( n , F q ) or U ( n , F q ) .We are concerned with the invariant rings F [ mW ⊕ dW ∗ ] G = { f ∈ F [ mW ⊕ dW ∗ ] | σ ( f ) = f , ∀ σ ∈ G } and the invariant fields F ( mW ⊕ dW ∗ ) G = { f ∈ F ( mW ⊕ dW ∗ ) | σ ( f ) = f , ∀ σ ∈ G } .In particular, the purpose of this paper is to study the following question: Does there exist ℓ , ℓ , . . . , ℓ ( m + d ) n ∈ F q [ mW ⊕ dW ∗ ] G such that F q ( mW ⊕ dW ∗ ) G = F q ( ℓ , ℓ , . . . , ℓ ( m + d ) n ) ?( ∗ )Here we will show that the answer to Question ( ∗ ) is positive for F q [ mW ⊕ dW ∗ ] G and G ∈{ GL ( n , F q ) , SL ( n , F q ) , U ( n , F q ) } in all cases except possibly for the cases where G = GL ( n , F q ) or SL ( n , F q ) and either m or d is 0.An affirmative answer to Question ( ∗ ) for F q ( W ) GL ( n , F q ) and F q ( W ) SL ( n , F q ) dates back to Dick-son [7] who proved that both rings of invariants F q [ W ] GL ( n , F q ) and F q [ W ] SL ( n , F q ) are polynomialalgebras. In 1975, Mui [12] proved that the invariant ring F q [ W ] U ( n , F q ) is also a polynomialalgebra, thus showing that Question ( ∗ ) has a positive answer for F q ( W ) U ( n , F q ) . Date : September 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Invariant fields; vector invariants; rationality.
These results are especially significant in light of the so called "No-name Lemma" which wenow state, see for example Jensen-Ledet-Yui [8, Section 1.1, page 22].L
EMMA
Let G be a finite group acting faithfully on a finite dimensionalvector space V over a field F , and let U be a faithful F ( G ) -submodule of V . Then the extensionof invariant fields F ( V ) G / F ( U ) G is purely transcendental. Thus the results of Dickson and Mui imply that each of the fields F q ( mW ⊕ dW ∗ ) G is purelytranscendental over the base field F q for each of the groups G = GL ( n , F q ) , SL ( n , F q ) or U ( n , F q ) .It is well-known that for any finite p -group P and any modular representation V , the invariantfield is always purely transcendental, see Miyata [11, Theorem 1] or Kang [9, Theorem]. In 2007,Campbell-Chuai [2, Theorem 2.4] gave an inductive method to find a polynomial generating setfor the invariant field of P . In particular this showed that F q ( mW ⊕ dW ∗ ) U ( n , F q ) can be generatedby ( m + d ) n homogeneous invariant polynomials. However, Campbell-Chuai [2] did not giveany explicit polynomial generating sets for F q ( mW ⊕ dW ∗ ) U ( n , F q ) and for all m , d , n .For a special case where n = q = p is prime, Richman [13, Corollary, page 38] exhib-ited a polynomial generating set for F p ( mW ) U ( , F p ) , which is used to study the invariant ring F p [ mW ] U ( , F p ) , a classical object of study in modular invariant theory.The structure of F p [ mW ] U ( , F p ) is more complicated than that of F p ( mW ) U ( , F p ) , see for ex-ample, Campbell-Hughes [3], Campbell-Shank-Wehlau [4], Wehlau [14], and Campbell-Wehlau[5].In 2011, Bonnafé-Kemper [1] initiated a study of the modular invariant ring of a vector anda covector, showing that the invariant ring F q [ W ⊕ W ∗ ] U ( n , F q ) is a complete intersection algebraas well as raising a conjecture on generating sets for of F q [ W ⊕ W ∗ ] GL ( n , F q ) . Recently, Chen-Wehlau [6] confirmed this conjecture of Bonnafé-Kemper’s. As the first step of the proof of thisconjecture, we proved that F q ( W ⊕ W ∗ ) GL ( n , F q ) can be generated by 2 n homogeneous invariantpolynomials from F q [ W ⊕ W ∗ ] GL ( n , F q ) , answering the Question ( ∗ ) affirmatively in this specialcase.The purpose of this paper is to generalize the above results to the invariant fields of any numberof vectors and covectors. Our main result is the following Theorem 1.2 whose proof will beseparated into three parts below: Theorems 3.3, 3.4, and 3.5.T HEOREM . For all m , d ∈ N + and G ∈ { GL ( n , F q ) , SL ( n , F q ) , U ( n , F q ) } with standard rep-resentation W , there exist ( m + d ) n invariant polynomials ℓ , ℓ , . . . , ℓ ( m + d ) n ∈ F q [ mW ⊕ dW ∗ ] G such that F q ( mW ⊕ dW ∗ ) G = F q ( ℓ , ℓ , . . . , ℓ ( m + d ) n ) . N INVARIANT FIELDS OF VECTORS AND COVECTORS 3
Furthermore, we will explicitly exhibit such invariant polynomials ℓ , ℓ , . . . , ℓ ( m + d ) n generat-ing the field of invariants in each case.R EMARK ∗ ) for F q ( mW ) U ( n , F q ) has an affirmative answer. 2. S PECIAL E XAMPLE : m = d = m = d =
1. In the next section wewill give proofs for the general cases using the result for this special case.2.1.
Generators of the invariant fields F q ( W ⊕ W ∗ ) GL ( n , F q ) and F q ( W ⊕ W ∗ ) SL ( n , F q ) . We choose { x , x , . . . , x n } as a basis of W and { y , y , . . . , y n } as the dual basis in W ∗ . Then F q [ W ⊕ W ∗ ] = F q [ x , x , . . . , x n , y , y , . . . , y n ] is a polynomial algebra of Krull dimension 2 n , which can be endowed with an involution whichis an algebra endomorphism ∗ : F q [ W ⊕ W ∗ ] −→ F q [ W ⊕ W ∗ ] , f f ∗ determined by x y n , x y n − , . . . , x n − y , x n y . There are also two F q -algebra homo-morphisms: F : F q [ W ⊕ W ∗ ] −→ F q [ W ⊕ W ∗ ] by x i x qi , y i y i F ∗ : F q [ W ⊕ W ∗ ] −→ F q [ W ⊕ W ∗ ] by x i x i , y i y qi . We have a natural invariant u in F q [ W ⊕ W ∗ ] GL ( n , F q ) corresponding to the natural pairing be-tween W and W ∗ : u : = x y + x y + · · · + x n y n . For i ∈ N + , we define u i : = F i ( u ) = x q i y + x q i y + · · · + x q i n y n u − i : = ( F ∗ ) i ( u ) = x y q i + x y q i + · · · + x n y q i n which are also GL ( n , F q ) -invariants, since F and F ∗ commute with the action of GL ( n , F q ) . Weobserve that u ∗− i = u i for all i ∈ N .Suppose F q [ W ] GL ( n , F q ) = F q [ c , c , . . . , c n − ] and F q [ W ] SL ( n , F q ) = F q [ d n , c , . . . , c n − ] denote theDickson invariants. Note that c = d q − n . We write F q [ W ] U ( n , F q ) = F q [ f , f , . . . , f n ] for the Muiinvariants. See Bonnafé-Kemper [1] or Chen-Wehlau [6] for details. YIN CHEN AND DAVID L. WEHLAU P ROPOSITION F q ( W ⊕ W ∗ ) GL ( n , F q ) = F q ( c , u − n , u − n , . . . , u , u , . . . , u n − )= F q ( c ∗ , u − n , u − n , . . . , u , u , . . . , u n − )= F q ( c , c , c , . . . , c n − , u , u , . . . , u n − ) . We can use the above result to compute the corresponding field for the action SL ( n , F q ) on W ⊕ W ∗ .P ROPOSITION F q ( W ⊕ W ∗ ) SL ( n , F q ) = F q ( d n , u − n , u − n , . . . , u , u , . . . , u n − )= F q ( d ∗ n , u − n , u − n , . . . , u , u , . . . , u n − )= F q ( d n , c , . . . , c n − , u , u , . . . , u n − ) . Proof.
Put K : = F q ( c , u − n , u − n , . . . , u , u , . . . , u n − ) = F q ( W ⊕ W ∗ ) GL ( n , F q ) and L : = K ( d n ) .Since d n is SL ( n , F q ) -invariant we have K ( d n ) ⊆ M : = F q ( W ⊕ W ∗ ) SL ( n , F q ) . Thus K = F q ( W ⊕ W ∗ ) GL ( n , F q ) ⊆ L = K ( d n ) ⊆ M . Since c = d q − n we have L = K ( d n ) = F q ( d n , u − n , u − n , . . . , u , u , . . . , u n − ) . Furthermore, itis clear that the minimal polynomial of d n over F q [ W ⊕ W ∗ ] GL ( n , F q ) is X q − − c . Thus [ L : K ] = q − = [ GL ( n , F q ) : SL ( n , F q )] = [ M : K ] . Therefore L = M proving the first equality.The other two equalities follow similarly. (cid:3) R EMARK d n d ∗ n ∈ F q [ u − n , u − n , . . . , u , u , . . . , u n − ] .R EMARK c by d n and c ∗ by d ∗ n throughout.2.2. F q ( W ⊕ W ∗ ) U ( n , F q ) . The rest of this section is devoted to finding a minimal generating setfor F q ( W ⊕ W ∗ ) U ( n , F q ) .P ROPOSITION
For n = , F q ( W ⊕ W ∗ ) U ( n , F q ) = F q ( W ⊕ W ∗ ) . (2) For n = , F q ( W ⊕ W ∗ ) U ( n , F q ) = F q ( f , f ∗ , f ∗ , u ) = F q ( f , f ∗ , f , u ) . N INVARIANT FIELDS OF VECTORS AND COVECTORS 5 (3)
For any n > , the invariant field is given by F q ( W ⊕ W ∗ ) U ( n , F q ) = F q ( f , f , f ∗ , f ∗ , . . . , f ∗ n − , u , u , u − , . . . , u − n )= F q ( f ∗ , f ∗ , f , f , . . . , f n − , u − , u , u , . . . , u n − ) . Proof. (1) Since U ( , F q ) = { } is the trivial group, F q ( W ⊕ W ∗ ) U ( , F q ) = F q ( x , y ) U ( , F q ) = F q ( x , y ) = F q ( f , f ∗ ) .(2) By Bonnafé-Kemper [1, Equation (2.11), page 106], the invariant ring F q [ W ⊕ W ∗ ] U ( , F q ) = F q [ u , f , f , f ∗ , f ∗ ] , is a hypersurface ring with the only relation u q − ( f f ∗ ) q − u − f q f ∗ − f ∗ q f = . Thus F q ( W ⊕ W ∗ ) U ( , F q ) = F q ( f , f ∗ , f ∗ , u ) = F q ( f , f ∗ , f , u ) .(3) First of all, we rewrite the relations ( R + ), ( R ), ( R − ), ( R ), ( R − ), ( R ), . . . , ( R − n − ), ( R n − )and ( R − n ), in Bonnafé-Kemper [1, page 105] as follows: f q · f ∗ n = u q − n + α − n · u q − n + · · · + α · u q + α · u , ( R + ) where all α k ∈ F q [ f ∗ , . . . , f ∗ n − ] are non-zero . f · f ∗ n − = u − n + α − n · u − n + · · · + α · u ( R ) + u q − n + β − n · u q − n + · · · + β · u q + β · u , where all α k , β k ∈ F q [ f , f ∗ , . . . , f ∗ n − ] are non-zero . f · f ∗ qn − = u − n + α − n · u − n + · · · + α − · u − ( R − ) + u q − n + β − n · u q − n + · · · + β · u q + u q − n + γ − n · u q − n + · · · + γ · u q + γ · u q , where all α k , β k , γ k ∈ F q [ f , f , f ∗ , . . . , f ∗ n − ] are non-zero . f · f ∗ n − = u − n + α − n · u − n + · · · + α · u ( R ) + u q − n + β − n · u q − n + · · · + β · u q + β · u + u q − n + γ − n · u q − n + · · · + γ · u q + γ · u q + γ · u , where all α k , β k , γ k ∈ F q [ f , f , f ∗ , . . . , f ∗ n − ] are non-zero .... ... ...... ... ... f ∗ q · f n = u qn − + α n − · u qn − + · · · + α · u q + α − · u − , ( R − n ) where all α k ∈ F q [ f , . . . , f n − ] are non-zero . YIN CHEN AND DAVID L. WEHLAU
Define L : = F q ( f , f , f ∗ , f ∗ , . . . , f ∗ n − , u , u , u − , . . . , u − n ) . Clearly, L ⊆ F q ( W ⊕ W ∗ ) U ( n , F q ) and L is generated by 2 n polynomials. The transcendence degree of F q ( W ⊕ W ∗ ) U ( n , F q ) is equalto 2 n . Thus it suffices to show that F q ( W ⊕ W ∗ ) U ( n , F q ) ⊆ L . By Bonnafé-Kemper [1, Proposition1.1]) we have F q ( W ⊕ W ∗ ) U ( n , F q ) = F q ( f , . . . , f n , f ∗ , . . . , f ∗ n , u ) . Hence, it suffices to show(2.1) f ∗ n − , f ∗ n , f , . . . , f n ∈ L which we will show using the above relations. Indeed, it follows from the relation ( R ) that(2.2) f ∗ n − ∈ L . Thus ( R + ) implies that(2.3) f ∗ n ∈ L . The relation ( R − ) yields(2.4) f ∈ L which combining with ( R ) gives(2.5) u ∈ L . The fact that u ∈ L together with ( R − ) shows that f ∈ L , and f ∈ L , together with ( R ) , impliesthat u ∈ L . Proceeding in this way and in this order, we deduce f k ∈ L from ( R − k ) and then using ( R k ) we conclude(2.6) u k − ∈ L which in turn serves to show that(2.7) f k + ∈ L . Continuing we finally have f , . . . , f n , u , . . . , u n − ∈ L . This finishes the proof of the first equationof Proposition 2.5 (3).By Bonnafé-Kemper [1, Example 2.1], U ( n , F q ) is a ∗ -stable subgroup of GL ( n , F q ) . We applythe involution ∗ to F q ( W ⊕ W ∗ ) U ( n , F q ) to demonstrate the second equation. (cid:3)
3. G
ENERAL C ASES
Basic constructions.
Let m , d > W (cid:27) W (cid:27) · · · (cid:27) W m (cid:27) W and W ∗ (cid:27) W ∗ (cid:27) · · · (cid:27) W ∗ d (cid:27) W ∗ are n -dimensional vector spaces over F q . N INVARIANT FIELDS OF VECTORS AND COVECTORS 7
For any 1 j m and any 1 k d , we choose a basis X j : = { x j , x j , . . . , x jn } for W j andchoose a basis Y k : = { y k , y k , . . . , y kn } for W ∗ k such that any X j and Y k are dual bases to oneanother. Then we have F q [ mW ⊕ dW ∗ ] = F q [ W ⊕ · · · ⊕ W m ⊕ W ∗ ⊕ · · · ⊕ W ∗ d ] (cid:27) F q [ x j , x j , . . . , x jn , y k , y k , . . . , y kn : 1 j m , k d ] which is a polynomial algebra of Krull dimension ( m + d ) n .For each pair ( j , k ) with 1 ≤ j ≤ m and 1 ≤ k ≤ d there is an invariant P ni = x ji y ki correspond-ing to u ∈ F q [ W ⊕ W ∗ ] GL ( n , F q ) . Similarly for each ( j , k ) there are invariants corresponding to u i for all i . However, in the proof for the general cases we will work exclusively with thoseinvariants associated to a pair ( j , ) or ( , k ) . We introduce notation for the invariants associatedto these pairs as follows.For 1 j m and i ∈ N , we define u ji : = x q i j · y + x q i j · y + · · · + x q i jn · y n . (3.1) u j , − i : = x j · y q i + x j · y q i + · · · + x jn · y q i n . (3.2)For 1 k d and i ∈ N , we define v ki : = y q i k · x + y q i k · x + · · · + y q i kn · x n . (3.3) v k , − i : = y k · x q i + y k · x q i + · · · + y kn · x q i n . (3.4)Note that u = v .For any 1 j m and any 1 k d , Dickson’s Theorem [7] yields(3.5) F q [ W j ] GL ( n , F q ) = F q [ x j , x j , . . . , x jn ] GL ( n , F q ) = F q [ c j , c j , . . . , c j , n − ] . The polynomial ring F q [ W j ⊕ W ∗ k ] can be endowed with an involutive algebra endomorphism ∗ jk : F q [ W j ⊕ W ∗ k ] −→ F q [ W j ⊕ W ∗ k ] , f f ∗ determined by x j y kn , x j y k , n − , . . . , x j , n − y k , x jn y k . As in the previous section, wehave F q [ W ∗ k ] GL ( n , F q ) = F q [ y k , y k , . . . , y kn ] GL ( n , F q ) = F q [ c ∗ k , c ∗ k , . . . , c ∗ k , n − ] where c ∗ ki = ∗ jk ( c ji ) for 0 i n −
1. Moreover, Mui’s Theorem [12] yields(3.6) F q [ W j ] U ( n , F q ) = F q [ x j , x j , . . . , x jn ] U ( n , F q ) = F q [ f j , f j , . . . , f jn ] where(3.7) f ji = Y v ∈ W ∗ j , i − ( x ji + v ) YIN CHEN AND DAVID L. WEHLAU and W ∗ j , i − denotes the subspace of W ∗ j with the basis { x j , x j , . . . , x j , i − } . Similarly, we have F q [ W ∗ k ] U ( n , F q ) = F q [ y k , y k , . . . , y kn ] U ( n , F q ) = F q [ f ∗ k , f ∗ k , . . . , f ∗ kn ] such that f ∗ ki = ∗ jk ( f ji ) for 1 i n .3.2. An application of Galois theory.
The following result directly generalizes Bonnafé-Kemper[1, Proposition 1.1].P
ROPOSITION
Let G GL ( n , F q ) be any subgroup. Write F q [ W j ] G = F q [ g j , . . . , g js ] and F q [ W ∗ k ] G = F q [ h k , . . . , h kt ] for j m and k d. Then F q ( mW ⊕ dW ∗ ) G is generated by { g j , . . . , g js , u j : 1 j m } ∪ { h k , . . . , h kt : 1 k d } ∪ { v k : 2 k d } .Proof. The group G : = G × G × · · · × G | {z } m + d acts on mW ⊕ dW ∗ in the obvious way. Furthermore F q [ mW ⊕ dW ∗ ] G = F q [ g j , . . . , g js , h k , . . . , h kt : 1 j m , k d ] . For σ ∈ G we write diag ( σ ) to denote the element diag ( σ ) : = ( σ , σ , . . . , σ ) ∈ G ⊂ G .Let L be the subfield of F q ( mW ⊕ dW ∗ ) generated by { g j , . . . , g js , u j : 1 j m } ∪ { h k , . . . , h kt : 1 k d } ∪ { v k : 2 k d } over F q .Let L denote the field L : = F q ( mW ⊕ dW ∗ ) G = F q ( g j , . . . , g js , h k , . . . , h kt : 1 j m , k d ) . By Artin’s theorem [10, page 264, Theorem 1.8], F q ( mW ⊕ dW ∗ ) is Galois over L with group G . Thus F q ( mW ⊕ dW ∗ ) is also Galois over L with the Galois group G L , say. Now we have thefollowing situation: L ✤✤✤ (cid:31) (cid:127) / / L ✤✤✤ (cid:31) (cid:127) / / F q ( mW ⊕ dW ∗ ) G ✤✤✤ (cid:31) (cid:127) / / F q ( mW ⊕ dW ∗ ) ✤✤✤ G G L ? _ o o G ? _ o o ? _ o o By Galois theory, to show F q ( mW ⊕ dW ∗ ) G = L , it suffices to show that G L = G . By definition G ⊆ G L .Conversely, let α = ( σ , · · · , σ m , τ , · · · , τ d ) be an arbitrary element of G L ⊆ G . Fix j with1 ≤ j ≤ m . Since u j ∈ L and diag ( σ − j ) ∈ G ⊂ G L and α ∈ G L , both α and diag ( σ − j ) fix u j .Thus we have u j = α · ( diag ( σ − j ) · u j ) = ( α diag ( σ − j )) · u j
0N INVARIANT FIELDS OF VECTORS AND COVECTORS 9 = ( σ σ − j , . . . , id , . . . , σ m σ − j , τ σ − j , . . . , τ d σ − j ) · u j . Since { x j , x j , . . . , x jn } is an algebraically independent set, this forces ( τ σ − j ) · y i = y i for all i = , , . . . , n . Therefore τ = σ j and this holds for all j = , , . . . , m .Similarly since α and diag ( τ − k ) both fix v k we have σ = τ k for all k = , , . . . , d . Therefore, α = diag ( σ ) ∈ G as required. (cid:3) Proposition 3.1 immediately yields some large generating sets for F q ( mW ⊕ dW ∗ ) G for G = GL ( n , F q ) , SL ( n , F q ) or U ( n , F q ) :C OROLLARY
The invariant field F q ( mW ⊕ dW ∗ ) GL ( n , F q ) is generated by { c j , c j , . . . , c j , n − , u j : 1 j m } ∪ { c ∗ k , c ∗ k , . . . , c ∗ k , n − : 1 k d } ∪ { v k : 2 k d } . (2) The invariant field F q ( mW ⊕ dW ∗ ) SL ( n , F q ) is generated by { d jn , c j , . . . , c j , n − , u j : 1 j m } ∪ { d ∗ kn , c ∗ k , . . . , c ∗ k , n − : 1 k d } ∪ { v k : 2 k d } where d jn and d ∗ kn are defined by c j = d q − jn and c ∗ k = ( d ∗ kn ) q − respectively. (3) The invariant field F q ( mW ⊕ dW ∗ ) U ( n , F q ) is generated by { f j , . . . , f jn , u j : 1 j m } ∪ { f ∗ k , . . . , f ∗ kn : 1 k d } ∪ { v k : 2 k d } . Generators for F q ( mW ⊕ dW ∗ ) GL ( n , F q ) and F q ( mW ⊕ dW ∗ ) SL ( n , F q ) . We define A : = { c , u i : 1 − n i n − } B : = { u ji : 2 j m , − n i } C : = { v ki : 2 k d , i n − } . Note that | A | + | B | + | C | = n + ( m − ) n + ( d − ) n = ( m + d ) n . The following theorem is thefirst main result.T
HEOREM . The invariant field F q ( mW ⊕ dW ∗ ) GL ( n , F q ) is generated by A ∪ B ∪ C.Proof.
Let L : = F q ( A ∪ B ∪ C ) denote the subfield generated by A ∪ B ∪ C over F q . Since A ∪ B ∪ C ⊂ F q [ mW ⊕ dW ∗ ] GL ( n , F q ) , it suffices to prove that that F q ( mW ⊕ dW ∗ ) GL ( n , F q ) ⊆ L . ByCorollary 3.2 (1), it is sufficient to show the following three statements: c , . . . , c , n − , c ∗ , c ∗ , . . . , c ∗ , n − ∈ L († ) c j , c j , . . . , c j , n − ∈ L for 2 j m († ) c ∗ k , c ∗ k , . . . , c ∗ k , n − ∈ L for 2 k d . († ) Firstly, consider the invariant field F q ( W ⊕ W ∗ ) GL ( n , F q ) = F q ( x , . . . , x n , y , . . . , y n ) GL ( n , F q ) .By Proposition 2.1, F q ( W ⊕ W ∗ ) GL ( n , F q ) = F q ( c , u , − n , . . . , u , − , u , , u , . . . , u , n − ) ⊂ L . Thus, c , . . . , c , n − , c ∗ , c ∗ , . . . , c ∗ , n − ∈ L . This proves († ).Secondly, for 2 j m , we consider F q ( W j ⊕ W ∗ ) GL ( n , F q ) = F q ( x j , . . . , x jn , y , . . . , y n ) GL ( n , F q ) .It follows from Proposition 2.1 that c j , c j , . . . , c j , n − ∈ F q ( c j , u j , − n , . . . , u j , − , u j , u j , . . . , u j , n − ) ⊂ L ( c j , u j , . . . , u j , n − ) . Thus to show († ) it suffices to show that(3.8) c j , u j , . . . , u j , n − ∈ L for 2 ≤ j ≤ m . Recall the relations ( T ∗ ), ( T ∗ ), and ( T ∗ n − ) in the invariant ring F q [ W j ⊕ W ∗ ] GL ( n , F q ) (see Chen-Wehlau [6]): c ∗ u j − c ∗ u qj + c ∗ u qj , − + · · · + ( − ) n c ∗ , n − u qj , − n + ( − ) n u qj , − n = T ∗ ) c ∗ u j − c ∗ u qj + c ∗ u q j + · · · + ( − ) n c ∗ , n − u q j , − n + ( − ) n u q j , − n = T ∗ ) ... ... ... ... ... c ∗ u j , n − − c ∗ u qj , n − + c ∗ u q j , n − − · · · + ( − ) n − c ∗ , n − u q n − j + ( − ) n u q n − j , − = . ( T ∗ n − )Since c ∗ , c ∗ , . . . , c ∗ , n − ∈ L by († ), the relation ( T ∗ ) implies that u j ∈ L which, together with the relation ( T ∗ ), implies that u j ∈ L . Proceeding in this way and using the relations ( T ∗ ), ( T ∗ ), and ( T ∗ n − ) (in this order), we eventuallyobtain(3.9) u j , u j , . . . , u j , n − ∈ L . To show c j ∈ L , we recall that c j = d q − jn and c ∗ = ( d ∗ n ) q − . Furthermore d jn · d ∗ n = det x j x j · · · x jn x qj x qj · · · x qjn ... ... ... ... x q n − j x q n − j · · · x q n − jn · det y y q · · · y q n − y y q · · · y q n − ... ... ... ... y n y q n · · · y q n − n N INVARIANT FIELDS OF VECTORS AND COVECTORS 11 = det u j u j , − u j , − · · · u j , − n u j u qj u qj , − · · · u qj , − n u j u qj . . . . . . ...... . . . . . . . . . u q n − j , − u j , n − u qj , n − · · · u q n − j u q n − j ∈ L , (3.10)see Chen-Wehlau [6]. Hence c j = ( c ∗ ) − · ( c j · c ∗ ) = ( c ∗ ) − · ( d jn · d ∗ n ) q − ∈ L . This com-pletes the proof of († ).Finally, we show († ). For any 2 k d , we consider F q ( W ⊕ W ∗ k ) GL ( n , F q ) = F q ( x , . . . , x n , y k , . . . , y kn ) GL ( n , F q ) . Therefore c ∗ k , c ∗ k , . . . , c ∗ k , n − ∈ F q ( c , v k , − n , . . . , v k , − , v k , v k , . . . , v k , n − ) ⊂ L ( v k , − n , . . . , v k , − ) . Thus it suffices to show that(3.11) v k , − , v k , − , . . . , v k , − n ∈ L . In the invariant ring F q [ W ⊕ W ∗ k ] GL ( n , F q ) , we have the following relations ( T ), ( T ), and ( T n − )described in [6]: c v k , − − c v qk + c v qk + · · · + ( − ) n c , n − v qk , n − + ( − ) n v qk , n − = T ) c v k , − − c v qk , − + c v q k + · · · + ( − ) n c , n − v q k , n − + ( − ) n v q k , n − = T ) · · · · · · · · · · · · · · · · · · · · · · · · · · · c v k , − n − c v qk , − n + c v q k , − n − · · · + ( − ) n − c , n − v q n − k + ( − ) n v q n − k = . ( T n − )As before, these relations ( T ), ( T ), and ( T n − ) (in this order), together with († ), imply that v k , − , v k , − , . . . , v k , − n ∈ L .This completes the proof. (cid:3) A similar argument showsT
HEOREM . The invariant field F q ( mW ⊕ dW ∗ ) SL ( n , F q ) is generated by { d n , u i : 1 − n i n − } ∪ { u ji : 2 j m , − n i } ∪ { v ki : 2 k d , i n − } . Generators for the invariant field F q ( mW ⊕ dW ∗ ) U ( n , F q ) . We define D : = { f , f , f ∗ s , u i : 1 s n − , − n i } E : = { f j , u ji : 2 j m , − n i } F : = { f ∗ k , v ki : 2 k d , i n − } . T HEOREM . (1) For n = , F q ( mW ⊕ dW ∗ ) U ( n , F q ) = F q ( mW ⊕ dW ∗ ) . (2) For n = , F q ( mW ⊕ dW ∗ ) U ( n , F q ) is generated by { f , f ∗ , f ∗ , u } ∪ { f j , u j : 2 j m } ∪ { f ∗ k , v k : 2 k d } . (3) For n > , F q ( mW ⊕ dW ∗ ) U ( n , F q ) is generated by D ∪ E ∪ F .Proof.
The proof of (1) is immediate since U ( n , F q ) is the trivial group.For (2) let L denote the field generated by { f , f ∗ , f ∗ , u } ∪ { f j , u j : 2 j m } ∪ { f ∗ k , v k :2 k d } over F q . By Corollary 3.2 (3), F q ( mW ⊕ dW ∗ ) U ( n , F q ) is generated by { f j , f j , u j :1 j m } ∪ { f ∗ k , f ∗ k : 1 k d } ∪ { v k : 2 k d } . It suffices to show that f j ∈ L for 1 j m (‡ ) f ∗ k ∈ L for 2 k d . (‡ )For any 1 j m , by the first equality in Proposition 2.5 (2), we see that F q ( W j ⊕ W ∗ ) U ( , F q ) = F q ( x j , x j , y , y ) U ( , F q ) = F q [ f j , f ∗ , f ∗ , u j ] . Thus f j ∈ F q ( W j ⊕ W ∗ ) U ( , F q ) ⊆ L , whichproves (‡ ). To show (‡ ), for any 2 k d , we consider F q ( W ⊕ W ∗ k ) U ( , F q ) = F q ( x , x , y k , y k ) U ( , F q ) = F q [ f , f ∗ k , f , v k ] , by the second equality in Proposition 2.5 (2). This together with (‡ ) implies that f ∗ k ∈ F q ( W ⊕ W ∗ k ) U ( , F q ) ⊆ L for all 2 ≤ k ≤ d .To prove (3) we first note that Corollary 3.2 (3) implies F q ( mW ⊕ dW ∗ ) U is generated by { f j , . . . , f jn , u j : 1 j m } ∪ { f ∗ k , . . . , f ∗ kn : 1 k d } ∪ { v k : 2 k d } . Let L be the subfieldgenerated by D ∪ E ∪ F over F q . Note that L ⊆ F q ( mW ⊕ dW ∗ ) U and | D | + | E | + | F | = ( m + d ) n .Thus it is sufficient to show: f , . . . , f n , f ∗ , n − , f ∗ n ∈ L , († ) f j , f j , . . . , f jn ∈ L , for 2 j m , († ) f ∗ k , f ∗ k , . . . , f ∗ kn ∈ L , for 2 k d . († )By the first equality of Proposition 2.5 (3), F q ( W ⊕ W ∗ ) U ( n , F q ) = F q ( D ) . Thus f , . . . , f n , f ∗ , n − , f ∗ , n ∈ F q ( D ) ⊆ L which proves († ).To show († ), consider the invariant field F q ( W j ⊕ W ∗ ) U ( n , F q ) for any 2 j m . By the firstequality of Proposition 2.5 (3) again, we have seen that F q ( W j ⊕ W ∗ ) U ( n , F q ) is generated by { f j , f j , f ∗ , f ∗ , . . . , f ∗ , n − , u j , u j , u j , − , . . . , u j , − n } . N INVARIANT FIELDS OF VECTORS AND COVECTORS 13
Thus { f j , . . . , f jn } ⊆ F q ( W j ⊕ W ∗ ) U ( n , F q ) ⊆ L ( f j , u j ) . Recall the relations ( ( R + ) j ) and ( ( R ) j )in F q [ W j ⊕ W ∗ ] U ( n , F q ) : f qj · f ∗ n = u qj , − n + α j , − n · u qj , − n + · · · + α j · u qj + α j · u j , ( ( R + ) j ) where all α jk ∈ F q [ f ∗ , . . . , f ∗ , n − ] are not zero . f j · f ∗ , n − = u j , − n + α j , − n · u j , − n + · · · + α j · u j ( ( R ) j ) + u qj , − n + β j , − n · u qj , − n + · · · + β j · u qj , + β j · u j , where all α jk , β jk ∈ F q [ f j , f ∗ j , . . . , f ∗ j , n − ] are not zero . Since α j , ( R + ) j ), we have u j ∈ L . This fact, together with ( ( R ) j ), implies that f j ∈ L .Therefore L ( f j , u j ) = L and { f j , . . . , f jn } ⊆ L .To show († ), we consider F q ( W ⊕ W ∗ k ) U ( n , F q ) for any 2 k d . By the second equality ofProposition 2.5 (3), it follows that F q ( W ⊕ W ∗ k ) U ( n , F q ) is generated by { f ∗ k , f ∗ k , f , f , . . . , f , n − , v k , − , v k , v k , , . . . , v k , n − } . Since f ∗ k , f ∗ k , . . . , f ∗ kn ∈ F q ( W ⊕ W ∗ k ) U ( n , F q ) , it suffices to show F q ( W ⊕ W ∗ k ) U ( n , F q ) ⊆ L . It followsfrom († ) that f , . . . , f , n − ∈ L . Since f ∗ k , f , f , v k , v k , , . . . , v k , n − ∈ L , we need only to showthat f ∗ k , v k , − ∈ L . As in the above proof of († ), we now use ∗ (( R + ) k ) and ∗ (( R ) k ) to deduce that ( v k , − ) and f ∗ k lie in L . This completes the proof. (cid:3) A CKNOWLEDGMENTS
We thank Gregor Kemper for helpful comments on an earlier draft of this article. The firstauthor thanks Queen’s University at Kingston for providing a comfortable working environ-ment during his visit in 2014–2016. The first author was supported by the Fundamental Re-search Funds for the Central Universities (2412017FZ001), NNSF of China (11401087) andCSC (201406625007). Both authors were partially supported by NSERC.R
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