aa r X i v : . [ m a t h . R T ] M a r MODULAR INVARIANTS OF SOME FINITE PSEUDO-REFLECTIONGROUPS
KE OU
Abstract.
We determine the modular invariants of finite modular pseudo-reflection subgroups ofthe finite general linear group GL n ( q ) acting on the tensor product of the symmetric algebra S • ( V ) and the exterior algebra ∧ • ( V ) of the natural GL n ( q ) -module V . We are particularly interested inthe case where G is a subgroup of the parabolic subgroups of GL n ( q ) which is a generalization ofWeyl group of Cartan type Lie algebra. Introduction
Let p be a fixed prime and F q be the finite field with q = p r for some r ≥ . The finite generallinear group GL n ( q ) acts naturally on the symmetric algebra P := S • ( V ) and the tensor product A := S • ( V ) ⊗ ∧ • ( V ) , where V = F nq is the standard GL n ( q ) -module and ∧ • ( V ) denotes the exterioralgebra of V . The GL n ( q ) invariants in P (resp. A ) are determined by Dickson [3] (resp. Mui [10]).For a composition I = ( n , · · · , n l ) of n, let GL I be the parabolic subgroup associated to I .Generalizing [3], Kuhn and Mitchell [8] showed that the algebra P GL I is a polynomial algebra in n explicit generators. Minh and Tùng [9] determined the GL I invariants in A in the case q = p , asthey used some Steenrod algebra arguments. Wan and Wang [12] generalized to relative invariantsof GL I in A in general q. Let G I and U I be a subgroup of GL I which have forms(1.1) G I = G ∗ · · · ∗ G · · · ∗ ... ... ... ... · · · G l and U I = I n ∗ · · · ∗ I n · · · ∗ ... ... ... ... · · · I n l . such that G i < GL n i ( q ) for all i where I j is the identity matrix of GL j ( q ) . In this paper, we studythe G I and U I invariants in A when G i is GL n i ( q ) , SL n i ( q ) , and G ( m, a, n i ) . One motivation is that G I is a generalization of GL I as well as the Weyl groups of Cartan typeLie algebras. Precisely, G I = GL I if G i = GL n i ( q ) for all i. And G I becomes a Weyl group ofCartan type Lie algebras if l = 2 , q = p, G = GL n ( q ) , G = S n or S n ⋉ Z n (cf. [6]). Fromthe viewpoint of representation theory, the invariants of Weyl group of Lie algebra g are providingvery interesting yet limited answers to the problem of understanding g modules, such as Chevalley’srestriction theorem in classical type Lie algebras (cf. [5]).Another motivation is that G I is a modular finite pseudo-reflection group if l ≥ and all G i are pseudo-reflection groups since p | | U I | . It’s well-known that if G is a nonmodular subgroup of GL n ( q ) , then G is a pseudo-reflection group if and only if P G is a polynomial algebra (this goes backto Chevalley, Shephard, Todd and Bourbaki, see [7, Theorem 18-1]). However, the invariants of a Date : March 10, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Modular invariant theory, Pseudo-reflection group, positive characteristic, Weyl group,Cartan type Lie algebra. modular pseudo-reflection group can be quite complicate (see [11] for example). Our investigationgeneralizes the results of modular invariants in A by Mui [10] and Minh-Túng [9].Our first main result is the following. Theorem 1.1.
Let I = ( n , · · · , n l ) be a composition of n . Then A U I is a free module of rank n over the algebra P U I . We refer to Theorem 5.13 for a more precise version of Theorem 1.1 where an explicit basis forthe free module is given. Theorem 1.1 is a generalization of [10], and our approach is in turn builtheavily on [10]. Since A G I = ( A U I ) G ×···× G l , we will then discuss ( A U I ) G i in section 6 case by casewhere G i = GL n i ( q ) or G ( m, a, n i ) . As applications, we have the following. Theorem 1.2.
Let I = ( n , · · · , n l ) be a composition of n . Suppose p > n i if G i = G ( r i , a i , n i ) . (1) If G i = G ( r i , a i , n i ) such that r i | q − for all i = 1 , · · · , l. Then A G I is a free module ofrank n a · · · a l over the algebra P G I where G I = ( G ( r , , n ) × · · · × G ( r l , , n l )) ⋉ U I . (2) If there is ≤ a ≤ l such that G i = (cid:26) GL n i ( q ) i = 1 , · · · , aG ( r i , , n i ) i = a + 1 , · · · , l. Then Then A G I is a free module of rank n over the algebra P G I . For more details and explicit basis of these free modules, we refer to Theorem 7.1 for the case a = 0 in (2), Theorem 7.3 for (1) and Theorem 7.4 for the case ≤ a ≤ l in (2).The paper is organized as follows. In Section 2, 3 and 4, we review some needed results from[3, 8, 9, 10, 12] and deal with P G I which overlaps with parts of [4] and [2]. The invariants of A aregiven in Section 5,6 and 7. Precisely, section 5 deals with A U I and section 6, 7 describe A G I forconcrete G i . Preliminary m = 0 and m k = P ki =1 n i , k = 1 , · · · , l. For each ≤ s ≤ n, define τ ( s ) = m j if m j < s ≤ m j +1 . Then τ ( n ) = m l − . Let L I = G · · · G · · · ... ... ... ... · · · G l , then G I = L I ⋉ U I . The definition of G I and U I refer to 1.1. Lemma 2.1. G I is a finite pseudo-reflection group if all G i are finite pseudo-reflection groups.Proof. Let J (resp. K ) be the set consisting of all pseudo-reflections of G × · · · × G l (resp. allelementary matrices of U I ). One can check that G I can be generated by J ∪ K. (cid:3) Suppose V = h x , · · · , x n i F q , the symmetric algebra S • ( V ) and the exterior algebra ∧ • ( V ) willbe identified with F q [ x , · · · , x n ] and E [ y , · · · , y n ] , respectively. Namely, P = F q [ x , · · · , x n ] and A = F q [ x , · · · , x n ] ⊗ E [ y , · · · , y n ] . Then A is an associative superalgebra with a Z -gradationinduced by the trivial Z -gradation of F q [ x , · · · , x n ] and the natural Z -gradation of E [ y , · · · , y n ] . Denote d ( f ) the parity of f ∈ A . Set B ( n ) = P nk =0 B k where B = ∅ and B k = { ( i , · · · , i k ) | ≤ i < · · · < i k ≤ n } . Then E [ y , · · · , y n ] has a basis { y J | J ∈ B ( n ) } where y J = y j · · · y j t if J = ( j , · · · , j t ) . For every
I, J ∈ B ( n ) , we say that I < J if ODULAR INVARIANTS 3 (1)
I, J ∈ B k for some ≤ k ≤ n ; (2) there is ≤ l ≤ k such that i l < j l and i s = j s , for all l < s ≤ k. Moreover, I ≤ J if I = J or I < J.
One can check that ( B k , ≤ ) is a total order on B k for all ≤ k ≤ n. For K = ( k , · · · , k t ) ∈ B ( n ) , define • K ∪ { a } := ( · · · , k s , a, k s +1 , · · · ) if k s < a < k s +1 , • K ∪ { a , · · · , a s } := ( · · · ( K ∪ { a } ) ∪ { a } · · · ) , • K \{ k j } := ( k , · · · , b k j , · · · , k t ) , • K \{ k j , · · · k j s } := ( · · · ( K \{ k j } ) \{ k j } · · · ) , • τ ( K ) := τ ( k t ) if K = ∅ and τ ( ∅ ) := 0 , • hd ( K ) := K \{ k j | k j ≤ τ ( K ) } . Namely, hd ( K ) = ( k i +1 , · · · , k t ) if k i ≤ τ ( K ) < k i +1 . H and W are non-modular pseudo-reflection groups and H is a subgroup of W, i.e. p > | W | . It’s well known that all S ( V ) , S ( V ) H and S ( V ) W are polynomial algebras. The following propo-sition is well-known. For convenient, we prove it independently. Proposition 2.2. S ( V ) H is a free S ( V ) W module of rank | W || H | . Proof.
Denote S = S ( V ) , S ′ = S ( V ) H and R = S ( V ) W . Let T := S/SR (resp. T ′ := S ′ /S ′ R ) bethe coinvariant algebra related to S (resp. S ′ ).Note that S is a free R module of rank | W | . For each homogeneous basis { ¯ e k } of T, let { e k } bethe homogeneous elements in S associated to { ¯ e k } . Then { e k } forms a basis of S as R module (cf.[7, Section 18-3]).Since S ′ ⊆ S and S ′ R ⊆ SR, we can induce a morphism i : T ′ → T such that i ( x + S ′ R ) = x + SR where x ∈ S ′ . We claim that i is injective. In fact, if i ( x + S ′ R ) ∈ SR for any x ∈ S ′ , then(2.1) x = X j s j r j , where s j ∈ S, r j ∈ R. Define
Av : S ( V ) → S ( V ) by letting Av( a ) = | H | P h ∈ H h · a for all a ∈ S ( V ) . Then
Av( x ) = x forall x ∈ S ′ and Im(Av) = S ′ . Applying Av on 2.1, we have x = P j Av( s j ) r j ∈ S ′ R. Therefore, i isinjective.Now, take a homogeneous basis { ¯ f q } of T ′ , and { f q } is associated homogeneous elements in S ′ . Then S ′ is generated, as an R -module, by f q (cf. [7, Lemma 17-5]), i.e. S ′ = P q Rf q . Since i is injective, { i ( ¯ f q ) } are linearly independent in T. Moreover, { f q } are linearly independentas R -module. Therefore, S ′ is a free R -module with basis { f q } . Namely, S ′ = ⊕ q Rf q . Note that S is a free S ′ (resp. R ) module of rank | H | (resp. | W | ). Hence, S ′ is a free R moduleof rank | W | / | H | . (cid:3) invariants of P In this section, we will first recall the works by Dickson [3] and Kuhn-Mitchell [8] on invariantsin P. And then the G I invariants in P will be investigated. KE OU
The invariants of Dickson and Kuhn-Mitchell.
For ≤ k ≤ n, define homogeneouspolynomials V k , L n , Q n,k as follows: V k = Y λ , ··· ,λ k − ∈ F q ( λ x + · · · λ k − x k − + x k ) ,L k = k Y i =1 V i = k Y i =1 Y λ , ··· ,λ i − ∈ F q ( λ x + · · · λ i − x i − + x i ) , Y λ , ··· ,λ i − ∈ F q ( X + λ x + · · · λ n x n ) = X q n + n − X k =0 Q n,n − k X q k . For ≤ i ≤ k, by [3], we have L k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x · · · x k x q x q · · · x qk ... ... . . . ... x q k − x q k − · · · x q k − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,L k,i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x · · · x k x q x q · · · x qk ... ... ... ... c x q i c x q i · · · c x q i k ... ... ... ... x q k x q k · · · x q k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where the hat b means the omission of the given term as usual. Moreover, Q k,i = L k,i /L k . According to [3], both subalgebras of invariants over SL n ( q ) and over GL n ( q ) in F q [ x , · · · , x n ] are polynomial algebras. Moreover,(3.1) P SL n ( q ) = F q [ L n , Q n, , · · · , Q n,n − ] , (3.2) P GL n ( q ) = F q [ Q n, , · · · , Q n,n − ] . For ≤ i ≤ l, ≤ j ≤ n i , define(3.3) v i,j = Y λ , ··· ,λ mi − ∈ F q ( λ x + · · · λ m i − x m i − + x m i − + j ) , (3.4) q i,j = Q n i ,j ( v i, , · · · , v i,n i ) . Then deg( v i,j ) = q m i − and deg( q i,j ) = q m i − q m i − j . By definition, v i,j = L i +1 ( x , · · · , x ) Recall the Hilbert series of a graded space W • = ⊕ i W i is by definition the generating function H ( W • , t ) := P i t i dim W i . By the proof of [9, Lemma 1],(3.5) P U I = F q [ x , · · · , x n , v , , · · · , v ,n , · · · , v l, , · · · , v l,n l ] . Moreover, by [8, Theorem 2.2] and [4, Theorem 1.4],(3.6) P GL I = F q [ q i,j | ≤ i ≤ l, ≤ j ≤ n i ] ,H (cid:0) P GL I , t (cid:1) = 1 Q li =1 Q n i j =1 (1 − t q mi − q mi − j ) . ODULAR INVARIANTS 5
The invariants of G I .Lemma 3.1. Keep notations as above. Then P G I = ⊗ li =1 P G i i where P i = F q [ v i, , · · · , v i,n i ] . Proof.
It comes from the fact that P U I = ⊗ li =1 P i , and G i acts on P j trivially whence i = j. (cid:3) As a corollary, the following proposition holds.
Proposition 3.2.
For ≤ i ≤ l, ≤ j ≤ n i , assume that F q [ x , · · · , x n i ] G i = F q [ e i, , · · · , e i,n i ] is a polynomial algebra such that deg( e i,j ) = α ij . Define u i,j = e i,j ( v i, , · · · , v i,n i ) . the subalgebra P G of G -invariants in P is a polynomial ring on the generators u i,j of degree α ij · q m i − with ≤ i ≤ n, ≤ j ≤ n i . Namely, P G = F q [ u i,j | ≤ i ≤ l, ≤ j ≤ n i ] . Moreover, the Hilbert series of P G is H (cid:0) P G , t (cid:1) = 1 Q li =1 Q n i j =1 (1 − t α ij · q mi − ) . Remark . (1) When l = 2 , [4] and [2] generalize Lemma 3.1 and their arguments indeed workin our case.(2) For non-modular finite group, the assumption holds, i.e. F q [ x , · · · , x n i ] G i is a polynomialalgebra, if and only if G i is generated by pseudo-reflections.(3) For modular finite group, the case will be complex. There are examples to satisfy theassumption, such as GL n i , SL n i ([3]), U n i , B n i ([1]), transitive imprimitive group generatedby pseudo-reflections ([11]) and etc. Meanwhile, there are pseudo-reflection groups such thatthe ring of invariants is not a polynomial ring (see [11] for concrete examples).4. Mui, Ming-Tùng and Wan-Wang Invariants of A In this section, we will recall the work of Mui, Ming-Tùng and Wan-Wang invariants in A . Mui invariants in A . Let A = ( a ij ) be a n × n matrix with entries in a possibly noncommu-tative ring R . Define the (row) determinant of A : | A | = det( A ) = X σ ∈ S n sgn ( σ ) a σ (1) · · · a nσ ( n ) . Recall that ab = ( − d( a ) d( b ) ba for all a, b ∈ A . By [10, equation 1.4], n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y y · · · y n y y · · · y n ... ... . . . ... y y · · · y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = y · · · y n and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y y · · · y y y · · · y ... ... . . . ... y n y n · · · y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . KE OU
Suppose ≤ j ≤ m ≤ n, and let ( b , · · · , b j ) be a sequence of integers such that ≤ b < · · ·
Keep notations as above. A GL I is a free P GL I module of rank n , with a basisconsisting of and M m i ; b , ··· ,b j θ q − · · · θ q − i for ≤ i ≤ l, ≤ j ≤ m i and ≤ b < · · · < b j ≤ m i − , b j ≥ m i − . Namely, A GL I = P GL I ⊕ n X j =1 X m i ≥ j X ≤ b < ···
Keep notations as above. Suppose S = ( s , · · · , s k , a , · · · , a t ) ∈ B k + t such that s k ≤ b < a . (1) If b + 1 < a , then N s : b ; a · V b +1 = ( − t N s : b +1; a + t X i =1 ( − i +1 N s ∪{ b +1 } : b +1; a \{ a i } V b,a i + k X j =1 ( − k + t + j N s ∪{ b +1 }\{ s j } : b +1; a Q b,s j . (2) If b + 1 = a , i.e. b = a − , then we have N s : b ; a · V a +1 = ( − t − N s ∪{ a } : a +1; a \{ a } + t X i =2 ( − i N s ∪{ a ,a +1 } : a +1; a \{ a ,a i } V a ,a i + k X j =1 ( − k + t + j N s ∪{ a ,a +1 }\{ s j } : a +1; a \{ a } Q a ,s j . KE OU
Proof. (1) We consider the following determinant: k + t )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · · · x b x · · · x b x b +1 x a x a · · · x a t ... ... ... ... ... ... ... ... ... ... ... x q b · · · x q b b x q b · · · x q b b x q b b +1 x q b a x q b a · · · x q b a t y · · · y b y b +1 y a y a · · · y a t ... ... ... ... ... ... ... ... y · · · y b y b +1 y a y a · · · y a t x · · · x b x b +1 x a x a · · · x a t ... ... ... ... ... ... ... ... \ x q s − · · · \ x q s − b \ x q s − b +1 \ x q s − a \ x q s − a · · · \ x q s − a t ... ... ... ... ... ... ... ... \ x q sk − · · · \ x q sk − b \ x q sk − b +1 \ x q sk − a \ x q sk − a · · · \ x q sk − a t ... ... ... ... ... ... ... ... x q b − · · · x q b − b x q b − b +1 x q b − a x q b − a · · · x q b − a t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x b + 1 rows yx k + t rows yx(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b − k rows (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)y By the Laplace’s development, we have ( − b L b +1 N s : b ; a + t X i =1 ( − b + i L b +1 ( x , · · · , x b , x a i ) N s ∪{ b +1 } : b +1; a \{ a i } = ( − b + t L b N s : b +1; a + k X i =1 ( − b +1 − s i L b,s i · ( − k + t + s i − ( i − N s ∪{ b +1 }\{ s j } : b +1; a . Divide ( − b L b ( x , · · · , x b ) on both side. Statement (1) holds.(2) Now we consider the following determinant: k + t )! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x · · · x a x · · · x a x a +1 x a · · · x a t ... ... ... ... ... ... ... ... ... ... x q a · · · x q a a x q a · · · x q a a x q a a +1 x q a a · · · x q a a t y · · · y a y a +1 y a · · · y a t ... ... ... ... ... ... ... y · · · y a y a +1 y a · · · y a t x · · · x a x a +1 x a · · · x a t ... ... ... ... ... ... ... d x q s · · · d x q s a \ x q s a +1 d x q s a · · · d x q s a t ... ... ... ... ... ... ... d x q sk · · · d x q sk a \ x q sk a +1 d x q sk a · · · d x q sk a t ... ... ... ... ... ... ... x q a · · · x q a − a x q a − a +1 x q a − a · · · x q a − a t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x a + 1 rows yx k + t rows yx(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − − k rows (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)y ODULAR INVARIANTS 9
By the Laplace’s development, we have ( − a L a +1 N s : b ; a + t X i =2 ( − a + i +1 L a +1 ( x , · · · , x a , x a i ) N s ∪{ a ,a +1 } : a +1; a \{ a ,a i } = ( − a − t L a N s ∪{ a } : a +1; a \{ a } + k X j =1 ( − a +1+ k + t − ( j − L a ,s j N s ∪{ a ,a +1 }\{ s j } : a +1; a \{ a } . Divide ( − a L a on both side. Statement (2) holds. (cid:3) Corollary 5.3.
Keep notations as above. For J = ( j , · · · , j t ) ∈ B t and ≤ b < j t , we have (1) N τ ( J ) ,J is U I -invariant. (2) If b = j s − , for all s = 1 , · · · , t, then N b,J · V b +1 = ǫN b +1 ,J + t X i =1 g i N b +1 ,J ∪{ b +1 }\{ j i } where ǫ ∈ {± } and g i ∈ P U I . If b = j s − , for some s = 1 , · · · , t, then N b,J · V b +1 = ǫN b +1 ,J + t X i =1 g i N b +1 ,J ∪{ j s +1 }\{ j i } where ǫ ∈ {± } and g i ∈ P U I . Remark . For arbitrary b and J , N b,J may not be U I -invariant. Corollary 5.5.
Keep notations as above. Let ≤ b ≤ c ≤ n and J = ( j , · · · , j t ) ∈ B t . If j i ≤ b < j i +1 ≤ j l ≤ c < j l +1 , then N b,J · V b +1 · · · [ V j i +1 · · · c V j l · · · V c = ǫN c,J + X J ′ N c,J ′ f J ′ where ǫ ∈ {± } , J ′ ≤ (1 , · · · , j i , c − l + i + 1 , · · · , c, j l +1 , · · · , j t ) and f J ′ ∈ P U I . Proof.
For any K ∈ B ( n ) and d ∈ K, it is a direct computation that N d − ,K = N d,K . Thanks to Lemma 5.2, one can check this corollary by induction. (cid:3)
Remark . Note that J ∗ i < ( τ ( n ) − i, · · · , τ ( c ) − , j i +1 , · · · , j t ) . We may denote N b,s = N b,S if S = ( s ) ∈ B . Lemma 5.7. If S = ( s , · · · , s k ) ∈ B k , and s j ≤ b < s j +1 , then N b,S = ( − jk − ( j +1) j/ N b,s · · · N b,s k /L k − b . In particular, if s i ≤ τ ( S ) < s i +1 , then N τ ( S ) ,S = ( − ik − ( i +1) i/ N τ ( S ) ,s · · · N τ ( S ) ,s k /L k − τ ( S ) . Proof.
The relation holds trivially for k = 1 . Let us suppose k > and that it is true for all N a,J where ≤ a ≤ n and J ∈ B k − . Now we consider the following determinant: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y · · · y b y · · · y b y s j +1 · · · y s k x · · · x b x · · · x b x s j +1 · · · x s k ... ... ... ... ... ... ... ... ... x q b − · · · x q b − b x q b − · · · x q b − b x q b − s j +1 · · · x q b − s k y · · · y b y s j +1 · · · y s k ... ... ... ... ... ... y · · · y b y s j +1 · · · y s k x · · · x b x s j +1 · · · x s k ... ... ... ... ... ... \ x q s − · · · \ x q s − b \ x q s − s j +1 · · · \ x q s − s k ... ... ... ... ... ... \ x q sj − · · · \ x q sj − b \ x q sj − s j +1 · · · \ x q sj − s k ... ... ... ... ... ... x q b − · · · x q b − b x q b − s j +1 · · · x q b − s k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x b + 1 rows yx k − rows yx(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b − j rows (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)y By the Laplace’s development, we have ( − b k ! L b N b,S + j X i =1 ( − b + k − i +1 N b,s i ( k − N b,S \{ s i } = k X i = j +1 ( − b +1+ i − j N b,s i ( k − N b,S \{ s i } . Therefore, we obtain: kL b N b,S = j X i =1 ( − k − i N b,s i N b,S \{ s i } + k X i = j +1 ( − i − j +1 N b,s i N b,S \{ s i } . From the induction hypothesis, we have kL k − b N b,S = P ji =1 ( − k − i N b,s i · ( − ( k − j − − j ( j − / N b,s · · · [ N b,s i · · · N b,s k + P ki = j +1 ( − i − j +1 N b,s i · ( − ( k − j − j ( j +1) / N b,s · · · [ N b,s i · · · N b,s k = ( − jk − ( j +1) j/ kN b,s · · · N b,s k . Consequently, L k − b N b,S = ( − jk − ( j +1) j/ N b,s · · · N b,s k . Lemma holds. (cid:3)
Corollary 5.8. If S = ( s , · · · , s k ) ∈ B k and b < s , then N b, · · · N b,b N b,s · · · N b,s k = ( − bk − b ( b +1) / L b + k − b y · · · y b y s · · · y s k . Proof.
Thanks to above lemma and equation 5.1, we have ( − bk − b ( b +1) / N b, · · · N b,b N b,S = L b + k − b N b,J = L b + k − b y · · · y b y s · · · y s k , where J = (1 , · · · , b, s , · · · , s k ) ∈ B b + k . (cid:3) Corollary 5.9.
For all ≤ b, s ≤ n, N b,s = 0 . ODULAR INVARIANTS 11
Proof. If b ≥ s, then N b, · · · d N b,s · · · N b,b N b,s = ± L b − b y · · · y b N b,s = 0 . If b > s, then N b, · · · N b,b N b,s = ± L bb y · · · y b y s N b,s = 0 . Note that N b, · · · N b,b = 0 . Corollary holds. (cid:3)
Similar arguments with [10, Lemma 5.2], by Corollary 5.5, Lemma 5.7, Corollary 5.8 and 5.9, thefollowing proposition holds.
Proposition 5.10.
Let f = P J ∈ B ( n ) N τ ( J ) ,J h J where h J ∈ P. Then f = 0 if and only if all h J = 0 . Lemma 5.11.
Suppose J ∗ = ( j , · · · , j k ) ∈ B ( n ) , and f = X J ≤ J ∗ y J f J ( x , · · · x n ) ∈ A is U I -invariant, then f J ∗ ∈ P is U I -invariant. Moreover, f J ∗ has factors (cid:8) V i | i ∈ { , · · · , τ ( j k ) }\{ j , · · · , j k } (cid:9) . Proof.
For all w = ( w ij ) ∈ U I , wy i = y i + w i − ,i y i − + · · · + w i y . Therefore, wf = X J Keep notatinos as above. Suppose S ∗ = ( s ∗ , · · · , s ∗ k ) ∈ B k , with s ∗ k = b and s ∗ j ≤ τ ( b ) < s ∗ j +1 . Let f = X S ≤ S ∗ y S f S ( x , · · · x n ) ∈ A be U I -invariant. Then f = X L ≤ hd ( S ∗ ) X S =( s , ··· ,s k )( s j +1 , ··· ,s k )= L N τ ( s k ) ,S h S ( x , · · · , x n ) , where h S ∈ P is U I -invariant.Proof. We will use double induction on both k and S ∗ . (1) Suppose k = 1 and S ∗ = ( b ) , ≤ b ≤ n. (i)If b = 1 , τ ( b ) = 0 . Moreover, N τ (1) , = y and f = y f . By Lemma 5.11, f ∈ P U I andproposition holds. (ii) For arbitrary b, denote c = τ ( b ) . Suppose f = y f + · · · y b f b . By Lemma 5.11, f b is U I -invariant and has factors { V i | ≤ i ≤ c } . Therefore, f b = ( − c +1 y b L c h b where h b ∈ P U I . Theexpension of N c,b along row 1 implies that N c,b = ( − c +1 y b L c + c X i =1 ( − i +1 y i N i where N i ∈ P is the minor of N c,b at position (1 , i ) . Hence f = N c,b h b + P b − i =1 y i f ′ i . Note that f − N c,b h b = P b − i =1 y i f ′ i is U I -invariant. By induction, there are h i ∈ P U I such that f − N c,b h b = b − X i =1 N τ ( i ) ,i h i and f = b X i =1 N τ ( i ) ,i h i . (2) For arbitrary k > , suppose s ∗ k − = l < b, and s ∗ i ≤ τ ( l ) < s ∗ i +1 . (i) If b = k, i.e. S ∗ = (1 , , · · · , k ) , then f = y S ∗ f S ∗ . Note that y S ∗ = N τ ( k ) ,S ∗ is U I -invariant.For all w ∈ U I , wf = y S ∗ ( w · f S ∗ ) = y S ∗ f S ∗ , and hence f S ∗ is U I -invariant.Proposition holds in this case.(ii) Let us suppose b > k and that it is true for all S < S ∗ . One can rewrite f as(5.2) f = X K ≤ K ∗ y K F K y b + X b SS ≤ S ∗ y S f S , where K ∗ = ( s ∗ , · · · , s ∗ k − ) ∈ B k − and F K = f K ∪{ b } . Now, set F = P K ≤ K ∗ y K F K . Define T ( K ∗ ) = { ( α , · · · , α i , s ∗ i +1 , · · · , s ∗ k − ) } ⊆ B k − . Similar to the proof of Lemma 5.11, one can prove that F is U I -invariant. Then by induction, F can be decomposed into(5.3) F = X L ≤ hd ( K ∗ ) X K =( s , ··· ,s k − )( s i +1 , ··· ,s k − )= L N τ ( s k − ) ,K h K ( x , · · · , x n ) where all h K are U I -invariant.Note that y S ∗ f S ∗ = y K ∗ y b F K ∗ . As a component of F, N τ ( s ) ,K has factor y K ∗ if and only if K ∈ T ( K ∗ ) which equivalent to L = hd ( K ∗ ) . Thanks to Lemma 5.11, f S ∗ has factors V τ ( l )+1 · · · [ V s ∗ i +1 · · · c V s ∗ j · · · V τ ( b ) . It is a direct computationthat N τ ( l ) ,K has no such factors if K ∈ T ( K ∗ ) . As a consequence, h K = V τ ( l )+1 · · · [ V s ∗ i +1 · · · c V s ∗ j · · · V τ ( b ) h ′ K where h ′ K ∈ P for all K ∈ T ( K ∗ ) . Since all of h K and V i ( τ ( l ) + 1 ≤ i ≤ τ ( b )) are U I -invariant, h ′ K is also U I -invariant.Denote ˜ K ∗ = ( τ ( b ) − j, · · · , τ ( b ) − , s ∗ j +1 , · · · , s ∗ k − ) . Thanks to Corollary 5.5, X K ∈ T ( K ∗ ) N τ ( l ) ,K V τ ( l )+1 · · · [ V s ∗ i +1 · · · c V s ∗ j · · · V τ ( b ) = X S ≤ ˜ K ∗ N τ ( b ) ,S f S where f S ∈ P U I . ODULAR INVARIANTS 13 Then(5.4) F = X L< hd ( K ∗ ) X K =( s , ··· ,s k − )( s i +1 , ··· ,s k − )= L N τ ( s k − ) ,K h K + X S ≤ ˜ K ∗ N τ ( b ) ,S h S where h S ∈ P U I since all f S ( S ≤ ˜ K ∗ ) and h ′ K ( K ∈ T ( K ∗ )) are U I -invariant.For each S = ( s , · · · , s j , s ∗ j +1 , · · · , s ∗ k − ) ≤ ˜ K ∗ , note that hd( S ∪ { b } ) = hd( S ∗ ) = ( s ∗ j +1 , · · · , s ∗ k − , b ) . By Laplace’s development, N τ ( b ) ,S y b = ( − u · τ ( b ) y hd( S ∗ ) N τ ( b ) , ( s , ··· ,s j ) + X S ′ Theorem 5.13. (1) P U I = F q [ x , · · · , x n , v , , · · · , v ,n , · · · , v l, , · · · , v l,n l ] , (2) A U I is a free P U I module of rank n with a basis consisting of all elements of { N τ ( S ) ,S | S ∈ B ( n ) } . In other words, there exists a decomposition A U I = X S ∈ B ( n ) N τ ( S ) ,S P U I . Remark . If I = (1 , · · · , , i.e. U I = U n ( q ) , then τ ( j ) = j − , j = 1 , · · · , n. Suppose ≤ j ≤ m ≤ n, and ≤ b < · · · < b j = m − . Then M m ; b , ··· b j = N m − ,B = N τ ( B ) ,B where B = ( b + 1 , · · · , b j + 1) ∈ B j . Therefore, formula 4.2 holds by above theorem.6. G I -invariant of A ≤ i ≤ l, recall that G i < GL n i ( q ) . Hence, G i acts on x j and y j trivially unless m i − ODULAR INVARIANTS 15 Lemma 6.2. If τ ( S ) ≥ m i , then f S is G i skew-invariant, i.e. g · f S = det( g ) − f S for all g ∈ G i . Moreover, f = X S ∈ B k τ ( S ) ≥ m i N τ ( S ) ,S f S , where f S ∈ P U I is G i skew-invariant . Proof. If τ ( S ) ≥ m i , one can check that g · N τ ( S ) ,S = det( g ) N τ ( S ) ,S .g · f = X S ∈ B k τ ( S ) ≥ m i ( g · N τ ( S ) ,S )( g · f S ) = X S ∈ B k τ ( S ) ≥ m i det( g ) N τ ( S ) ,S ( g · f S ) = X S ∈ B k τ ( S ) ≥ m i N τ ( S ) ,S f S . Therefore, g · f S = det( g ) − f S . Lemma holds. (cid:3) τ ( S ) = m i − , we will discuss case by case.6.2.1. G i = G ( m, a, n i ) < GL n i . Suppose p > n i . Therefore, G i is a nonmodular group.Recall that G ( m, a, n i ) ≃ S n i ⋉ A ( m, a, n i ) where A ( m, a, n i ) = { diag( w , · · · , w n i ) | w mj = ( w · · · w n i ) m/a = 1 } . Since G i < GL n i ( q ) , one can check directly that G ( m, a, n i ) = G ( m ′ , a ′ , n i ) , where m ′ = ( q − , m ) , a ′ = m ′ / ( q − , m/a ) . Moreover, assume that m | ( q − and m = ab. For each ≤ i ≤ l and ≤ k ≤ n i , we need the following notations. • σ i,S = ( m i − + 1 , s ) · · · ( m i − + k, s k ) ∈ G ( m, a, n i ) , where S := ( s , · · · , s k ) ∈ B ( n ) suchthat m i − < s < · · · < s k ≤ m i ; • c i,k := P S =( s , ··· ,s k ) ∈ B k m i − Keep notations as above. (1) G i,k := Stab G i ( h x m i − +1 , · · · , x m i − + k i ) ≃ ( S k × S n i − k ) ⋉ A ( m, a, n i ) . (2) For each ≤ k ≤ n i , G i is generated by G i,k and all σ i,S where S := ( s , · · · , s k ) ∈ B ( n ) such that m i − < s < · · · < s k ≤ m i . Lemma 6.4. Keep notations as above. Then f is G i invariant if and only if the following conditionshold for all T ∈ B ( m i − ) and S = ( s , · · · , s k ) ∈ B ( m i ) \ B ( m i − ) : (1) f T ∪ S ( x ) = f T i,k ( σ S ( x )) = σ S · f T i,k (( x )) . Moreover, N m i − ,T ∪ S f T ∪ S = σ S ( N m i − ,T i,k f T i,k ) . (2) N m i − ,T i,k f T i,k is G i,k invariant. Proof. One can check directly that f is G i invariant if the two conditions hold for all T and S. Conversely, suppose f is G i invariant. Then(1) σ S · N m i − ,T i,k = N m i − ,T ∪ S and σ S ( R ) = T ∪ S if and onyl if R = T i,k ; (2) σN m i ,T i,k = χ ( σ ) N m i ,T i,k for some χ ( σ ) ∈ F q and σ ( R ) = T i,k if and only if R = T i,k for each σ ∈ G i,k . Lemma holds. (cid:3) Proposition 6.5. Keep notations as above. ( A U I ) G ( m,a,n i ) is a free ( P U I ) G ( m, ,n i ) module with abasis consisting of { β i,n i ,r | r = 1 , · · · a } and c i,k ( N m i − ,T i,k ∆ i,k β i,k,r α i,k,j ) , where T ∈ B ( m i − ) , ≤ k ≤ n i , ≤ j ≤ C kn i , ≤ r ≤ a. Proof. By above lemmas, f = n i X k =1 X T ∈ B ( m i − ) X S =( s , ··· ,s k ) ∈ B k m i − We will use induction on S ∗ . For some S appears in f , denote S ′ = { , · · · , n }\ S. For each a ∈ S ′ ∩ { m i − + 1 , · · · , m i } . Suppose s b < a < s b +1 , ≤ b ≤ k. Let r = (cid:26) s b s b > m i − s b +1 s b = m i − . Take w = E + E a,r ∈ G. Then(6.3) w · f = f . (i) Suppose m i − < s b = r. By comparing the coefficient of y K on both side of 6.3 where K = S ∪ { a }\{ s b } , we have(6.4) N m i − ,K ( w · f S ) + N m i − ,K ( w · f K ) = N m i − ,K f K . In fact, w ( N m i − ,J f J ) = N m i − ,J ( w · f J ) + N m i − ,E a,r · J ( w · f J ) has factor y K if and only if J = K with term N m i − ,K ( w · f K ) or E a,r · J = K, i.e. J = S, with term N m i − ,K ( w · f S ) . By Proposition 5.10, equation 6.4 implies that f S ( x , · · · , x r + x a , · · · , x a , · · · ) = f K ( x , · · · , x n ) − f K ( x , · · · , x r + x a , · · · , x a , · · · ) . Setting x a = 0 yields f S ( · · · , x a − , , x a +1 , · · · ) = 0 , which implies that x a | f S . (ii) Suppose m i − = r, i.e. s b ≤ m i − = r < s l +1 . Similar to (i), by comparing the coefficient of y K ′ on both side of 6.3 where K ′ = S ∪ { a }\{ s l +1 } , we have x a | f S . In particular, x a | f S ∗ for all a ∈ ( S ∗ ) ′ ∩ { m i − + 1 , · · · , m i } . Thanks to Lemma 6.8, V a | f S ∗ . By Corollary 5.5, we have f = N m i − ,S ∗ V m i − +1 · · · [ V S ∗ j +1 · · · d V S ∗ k · · · V m i h S ∗ + X S Corollary 6.10. Keep notations as above. (1) ( A U I ) SL ni is a free ( P U I ) SL ni module with a basis consisting of { N m i ,S | S ∈ B ( m i ) \ B ( m i − ) } . (2) ( A U I ) GL ni is a free ( P U I ) GL ni module with a basis consisting of { N m i ,S θ q − i | S ∈ B ( m i ) \ B ( m i − ) } . Applications In this section, we will apply above results and describe A G I for some concrete groups G I asexamples.7.1. G i = G ( r i , , n i ) for all i = 1 , · · · , l such that r i | q − . Suppose p > n i for all i. Hence, all G i ’s are non-modular.For each ≤ i ≤ l, T ∈ B ( m i − ) , ≤ k ≤ n i and ≤ j ≤ C kn i , we keep the notations c i,k , T i,k , ∆ i,k , H i,k , H ′ i,k and α i,k,j as subsection 6.2.1. Furthermore, define • u i,k := e i,k ( v i, , · · · , v i,n i ) , where F q [ x m i − +1 , · · · , x m i ] G ( r i , ,n i ) = F q [ e i, , · · · , e i,n i ] , namely, e i,j = P m i − +1 ≤ t < ··· Keep notations as above. Suppose p > n i for all i. (1) P G I = F q [ u , , · · · u ,n , · · · u l,n l ] . ODULAR INVARIANTS 19 (2) A G I is a free P G I module of rank n with a basis consisting of and c i,k ( N m i − ,T i,k Ω i,k α i,k,j ) , where ≤ i ≤ l, T ∈ B ( m i − ) , ≤ k ≤ n i and ≤ j ≤ C kn i . Proof. (1) By Proposition 3.2, statement holds.(2) For each f ∈ A G I , by Proposition 5.12, suppose f = X S ∈ B ( n ) N τ ( S ) ,S h S = h + l X i =1 f i , where h ∈ P U I , f i = X ∅6 = S ∈ B ( n ) τ ( S )= m i − N τ ( S ) ,S h S , and h S ∈ P U I for all S ∈ B ( n ) . Now, for each ≤ i ≤ l, by Lemma 6.1, 6.2 and Proposition 6.5, f i = n i X k =1 X T ∈ B ( m i − ) C kni X j =1 c i,k (cid:0) N m i − ,T i,k Ω i,k α i,k,j (cid:1) f T,i,k,r,j where f T,i,k,r,j ∈ ( P U I ) G ( m, ,n i ) . Consequently, A G I is generated, as P G I module, by and n c i,k (cid:0) N m i − ,T i,k Ω i,k α i,k,j (cid:1) | ≤ i ≤ l, ≤ k ≤ n i , T ∈ B ( m i − ) , ≤ j ≤ C kn i o . Thanks to Proposition 5.10 and the definition of { α i,k,j } , these generators are linear independentas P G I module. The rank is l X i =1 n i X k =1 m i − C kn i = 1 + l X i =1 m i − (2 n i − 1) = 1 + l X i =1 (2 m i − m i − ) = 2 n . (cid:3) G i = G ( r i , a i , n i ) for all i such that r i = a i b i and r i | q − . Suppose p > n i for all i. For each ≤ i ≤ l, T ∈ B ( m i − ) , ≤ k ≤ n i , ≤ r ≤ a i and ≤ j ≤ C kn i , we keep the notations c i,k , T i,k , Ω i,k , β i,k,r and α i,k,j as above subsection.Suppose F q [ x m i − +1 , · · · , x m i ] G ( r i ,a i ,n i ) = F q [ e i, , · · · , e i,n i ] , define u i,k := e i,k ( v i, , · · · , v i,n i ) , where v i,k refers to 3.3.Denote G I := ( G ( r , , n ) ×· · ·× G ( r l , , n l )) ⋉ U I . For our convenience, denote β s := β ,n ,s · · · β l,n l ,s l where s = ( s , · · · s l ) such that ≤ s i ≤ a i for all i. By Proposition 2.2, we have Lemma 7.2. P G I is a free G I module of rank ( a · · · a l ) with a basis consisting of β s for all s. Similar to above arguments, we can prove the following result. Theorem 7.3. Keep notations as above. Suppose p > n i for all i. (1) P G I = F q [ u , , · · · u ,n , · · · u l,n l ] . (2) A G I is a free P G I module of rank (2 n a · · · a l ) with a basis consisting of β s and c i,k (cid:0) N m i − ,T i,k Ω i,k α i,k,j β s (cid:1) , where ≤ i, i ′ ≤ l, T ∈ B ( m i − ) , ≤ k ≤ n i , ≤ j ≤ C kn i , ≤ r ≤ a i and s = ( s , · · · , s l ) . ≤ a ≤ l such that G i = (cid:26) GL n i ( q ) i = 1 , · · · , aG ( r i , , n i ) i = a + 1 , · · · , l and p > n i for i = a + 1 , · · · , l. For each ≤ i ≤ l, ≤ k ≤ n i and ≤ j ≤ C kn i , recall q i,k is defined as 3.4. And we keep thenotations c i,k , u i,k , T i,k , H i,k , H ′ i,k and α i,k,j as subsection 6.2.1.Moreover, if a < i ≤ l, by [7, section 20-2], all skew-invariants of G a +1 × · · · × G i − × H i,k form afree F q [ x , · · · , x m i − + k ] G a +1 ×···× G i − × H i,k module with one generator, which is denoted by Ω ( a ) i,k . Infact, Ω ( a ) i,k := i − Y t = a +1 Y m t − Keep notations as above. Suppose p > n i for i = a + 1 , · · · , l. (1) P G I = F q [ q , , · · · q ,n , · · · q a,n a , u a +1 , , · · · u a +1 ,n a +1 , · · · u l,n l ] . (2) A G I is a free P G I module of rank n with a basis consisting of , n N m i ,S θ q − · · · θ q − i | ≤ i ≤ a, S = ( s , · · · , s k ) ∈ B ( m i ) \ B ( m i − ) o and n c i,k (cid:16) N m i − ,T i,k Ω ( a ) i,k α i,k,j (cid:17) θ q − · · · θ q − a | a + 1 ≤ i ≤ l, ≤ k ≤ n i , T ∈ B ( m i − ) , ≤ j ≤ C kn i o . Proof. (1) By Proposition 3.2, statement holds.(2) Suppose f = f + f + f ∈ A G I , where f ∈ P G I ,f = X ∅6 = S ∈ B ( m a ) N τ ( S ) ,S h S and f = X S ∈ B ( n ) τ ( S ) ≥ m a − N τ ( S ) ,S h S . Note that f is G × · · · × G l invariant. By Proposition 6.9, f = a X i =1 X ∅6 = S ∈ B ( m a ) τ ( S )= m i − N m i , Sθ q − · · · θ q − i h ′ S where h ′ S ∈ P G I . By Lemma 6.2 and Proposition 6.5, f = l X i = a +1 n i X k =1 X T ∈ B ( m i − ) C kni X j =1 c i,k (cid:16) N m i − ,T i,k Ω ( a ) i,k · α i,k,j (cid:17) h T,i,k,j where h T,i,k,j ∈ P U I is G × · · · × G a skew-invariant and G a +1 × · · · × G l invariant.Similar to the proof of [12, Theorem 3.1], h T,i,k,j = θ q − · · · θ q − a h ′ T,i,k,j , where h ′ T,i,k,j ∈ P G I . Theorem holds. (cid:3) Weyl groups of Cartan type Lie algebras. As a corollary, suppose G I is a Weyl group ofCartan type Lie algebra g of type W, S or H. Precisely, by [6], G I = (cid:26)(cid:18) A B C (cid:19) | A ∈ GL n ( p ) , C ∈ G (cid:27) < GL n ( p ) , where G = (cid:26) S n if g is of type W or S,G (2 , , n ) if g is of type H. ODULAR INVARIANTS 21 Recall that P U I = F q [ x , · · · , x n , v , , · · · v ,n ] and F q [ x n +1 , · · · , x n ] S n = F q [ e , · · · , e n ] where e j = P n +1 ≤ i < ···
Corollary 7.5. Keep notations as above. (1) P G I = F q [ Q n , , · · · , Q n ,n − , u , , · · · , u ,n ] . (2) A G I is a free P G I module of rank n with a basis consisting of , { N n ,S L q − n | ∅ 6 = S ∈ B ( n ) } and n c k (cid:16) N n ,T ,k Ω (1)1 ,k α ,k,j (cid:17) L q − n | ≤ k ≤ n , T ∈ B ( n ) , ≤ j ≤ C kn i o . References [1] M.-J. Bertin, Sous-anneaux d’invariants d’anneaux de polynomes, C. R. Acacl. Sci. Paris. 260 (1965), 5655-5658.[2] Y. Chen, R. Shark and D. Wehlau, Modular invariants of finite gluing groups, arxiv: 1910.02659v2 (2019).[3] L. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the formproblem, Trans. Amer. Math. Soc. 12 (1911), 75-98.[4] T. Hewett, Modular invariant theory of parabolic subgroups of GL n ( F q ) and the associated steenrod modules, DukeMath. J. 82 (1996), 91-102; Erratum, Duke Math. J. 97 (1999), 217.[5] J. Humphreys, Lie Algebras and their representations, Graduate Texts in Math. Vol. 9, Springer, 1972.[6] M. Jensen, Invariant Theory of Restricted Cartan Type Lie Algebras , PhD-Thesis, Aarhus University, 2015.[7] R. Kane, Reflection Group and Invariant Theory, CMS Books in Mathematics 5, Springer-Verlag, New York,1997.[8] N. Kuhn, S. Mitchell, The multiplicity of the Steinberg representation of GL n ( F q ) in the symmetric algebra, Proc.Amer. Math. Soc. 96 (1986), 1-6.[9] P. Minh, V. Tùng, Modular invariants of parabolic subgroups of general linear groups, J. Algebra 232 (2000),197-208.[10] H. Mui, Modular invariant theory and chomomogy algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo, Sect.1A Math. 22 (1975), 319-369.[11] H.Nakajima, Invariants of finite groups generated by pseudo-reflections in positive characteristic, Tsukuba J.Math, Vol. 3, No. 1 (1979), 109-122.[12] J. Wan, W. Wang, Twisted Dickson-Mui invariants and the Steinberg module multiplicity , Mathematical Pro-ceedings of the Cambridge Philosophical Society 151, issue 01 (2011), 43-57.(Ke Ou) School of Statistics and Mathematics, Yunnan University of Finance and Economics,Kunming 650221, China E-mail address ::
n i , then A G ( m, ,n i ) is a free P G ( m, ,n i ) module with a basis consisting of and c i,k ( N m i − ,T i,k ∆ i,k α i,k,j ) , where T ∈ B ( m i − ) , ≤ k ≤ n i , ≤ j ≤ C kn i , . G i = SL n i ( q ) or GL n i ( q ) . Suppose f = P S ≤ S ∗ N m i − ,S f S , where S ∗ = ( s ∗ , · · · , s ∗ k ) and s ∗ j < m i − ≤ s ∗ j +1 . Let U i be the subgroup of G i consisting of all upper triangular matrices of theform ∗ · · · ∗ · · · ∗ ... ... ... ... · · · . Lemma 6.8. f S ∗ is U i -invariant.Proof. ∀ u ∈ U i , u · N m i − ,S = N m i − ,S + P L