Block Basis for Coinvariants of Modular Pseudo-reflection Groups
aa r X i v : . [ m a t h . R T ] J u l BLOCK BASIS FOR COINVARIANTS OF MODULAR PSEUDO-REFLECTIONGROUPS
KE OU
Abstract.
As a sequel of [10], in this shot note, we investigate block basis for coinvariants of finitemodular pseudo-reflection groups. We are particularly interested in the case where G is a subgroupof the parabolic subgroups of GL n ( q ) which generalizes the Weyl groups of restricted Cartan typeLie 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 ) where V = F nq is thenatural GL n ( q ) -module. If G is any subgroup of GL n ( q ) , then we denote by P G the ring of G -invariant polynomials. The ring of coinvariants, which we denote by P G , is the quotient of P bythe ideal generated by the homogeneous elements in P G with positive degree.A block basis for P G (see 2.3 for definition) is a nice basis consisting of the monomial factors ofa single monomial. Such basis simplify computations in many ways. Generally, the combinatoricsinvolved in writing a polynomial or the products of elements in the block basis in terms of such abasis is less complicated since we are dealing with monomials. As an application, the authors of [1]use the block basis to determine the image of the transfer over some specific group G.The GL n ( q ) invariants in P are determined by Dickson [2]. The block basis for P GL n ( q ) isinvestigated by Steinberg [11] and Campell-Hughes-Shank-Wehlau [1]. For a composition I =( n , · · · , n l ) of n, let GL I be the parabolic subgroup associated to I . Generalizing [2], Kuhn andMitchell [7] showed that the algebra P GL I is a polynomial algebra in n explicit generators.Let G I and U I be a subgroup of GL I with forms(1.1) G I = G ∗ · · · ∗ G · · · ∗ ... ... ... ... · · · G l and U I = I n ∗ · · · ∗ I n · · · ∗ ... ... ... ... · · · I n l respectively,such that G i < GL n i ( q ) for all i where I j is the identity matrix of GL j ( q ) . Note that G I is a generalization of GL I as well as the Weyl groups of Cartan type Lie algebras.Precisely, G I = GL I if G i = GL n i ( q ) for all i. And G I is a Weyl group of Cartan type Lie algebras if l = 2 , q = p, G = GL n ( q ) , G = S n or G (2 , , n ) (cf. [5]). From the viewpoint of representationtheory, the coinvariants of Weyl group of Lie algebra g are providing very interesting yet limitedanswers to the problem of understanding g modules. For example, Jantzen describes the basicalgebra of block with regular weight by using coinvariants of Weyl group in [4, Proposition 10.12]. Date : July 21, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Modular invariant theory, Pseudo-reflection group, block basis, coinvariants.
Meanwhile, G I is a modular finite pseudo-reflection group if l ≥ and all G i are pseudo-reflectiongroups since p | | U I | (subsection 2.2). The invariants and coinvariants for a modular pseudo-reflection group can be quite complicate (see [9] for example). Our investigation generalizes theresults for modular coinvariants by Steinberg [11] and Campell-Hughes-Shank-Wehlau [1].In [10], we study P U I and P G I which are in turns out to be polynomial algebras. In this shortnote, we are interested in the block basis for the coinvariants P U I (Theorem 5.2) and P G I (Theorem6.3). As an application, the block basis for P G I are described precisely when G i is GL n i ( q ) and G ( m, a, n i ) in Theorem 4.6, Proposition 6.4 and Proposition 6.6. Our approach is in turn built on[1] where the authors describe sufficient conditions for the existence of a block basis for P G . The paper is organized as follows. In Section 2 and 3, we review some needed results from[1, 2, 3, 6, 7, 8, 10]. Section 4 deals with the block basis for coinvariants of G ( m, a, n ) . A blockbasis for P U I are given in Section 5 and section 6 describes the block basis for P 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 . Recall the definition of G I and U I in 1.1. Then G I = L I ⋉ U I , where L I = G · · · G · · · ... ... ... ... · · · G l , Suppose V = h x , · · · , x n i F q , the symmetric algebra S • ( V ) will be identified with F q [ x , · · · , x n ] .Namely, P = F q [ x , · · · , x n ] . Pseudo-reflection groups.
In this subsection, we will recall some basic facts for pseudo-reflection groups. More details refer to [6].For a finite dimensional vector space W over F q , a pseudo-reflection is a linear isomorphism s : W → W that is not the identity map, but leaves a hyperplane H ⊆ W pointwise invariant. G ⊆ GL( W ) is a pseudo-reflection group if G is generated by its pseudo-reflections. We call G isnon-modular if p G | while G is modular otherwise. Lemma 2.1. [10, Lemma 2.1] G I is a finite pseudo-reflection group if all G i are finite pseudo-reflection groups.Remark . As a corollary, the Weyl groups of restricted Cartan type Lie algebras with type
W, S and H are modular pseudo-reflection groups ([5]).2.3. block basis. In this subsection, we will list some results of block basis for coinvariants. Moredetails refer to [1].A homogeneous system of parameters for P is any collection of homogeneous elements e , · · · , e n with the property that they generate a polynomial subalgebra A = K [ f , · · · , f n ] over which P is afree A -module. Moreover, a basis for P as a free module over A is any set of elements of P thatprojects to a K -basis for P/ I where I = ( A + ) is the ideal generated by { f , · · · , f n } . Denote d i thedegree of f i for i = 1 , · · · , n. Definition 2.3. [1, Definition 1.1] Let f ∈ P be a monomial. We say f generates a block basis for P over A (or for P / I ) if the set of all monomial factors of f is a basis of vector space P / I . Such abasis consisting of all the monomial factors of a single monomial is called a block basis . LOCK BASIS 3
In [1], the authors provide sufficient conditions for the existence of a block basis for P/ I . Notethat the symmetric group S n acts on P by permuting the variables. Let Σ( A ) be the subgroup of S n fixing A pointwise. Denote a ≡ b if a − b ∈ I . Definition 2.4. [1, Definition 4.1] We will say that w = x m · · · x m n n is a critical monomial associatedto α = x t · · · x t n n , if:(1) for every g ∈ Σ( A ) , there is an i such that m i > t g ( i ) ;(2) no proper factor of w satisfies the first condition.Suppose Σ( A ) = S n and the t i form a decreasing sequence. In this case, the critical monomialsassociated to α are ( x · · · x i ) t i +1 . Lemma 2.5. [1, Theorem 4.1]
Suppose α = x t · · · x t n n generates a block basis. Then the sequence t i is a permutation of the sequence d i − and the degree of α is P ni =1 d i − n. Furthermore, if g ∈ Σ( A ) , then g ( α ) also generates a block basis. Lemma 2.6. [1, Theorem 4.4]
Suppose α = x t · · · x t n n has trivial critical monomials and the se-quence t i is a permutation of d i − . Then α generates a block basis for P/ I . invariants of P In this section, we will first recall the works by Dickson [2] and Kuhn-Mitchell [7] on invariantsin P. And then tcretirierhe G I invariants in P will be investigated.3.1. 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 ) ,F n ( X ) = Y λ , ··· ,λ i − ∈ F q ( X + λ x + · · · λ n x n ) = X q n + n − X k =0 Q n,k X q k . According to [2], the algebra of invariants over GL n ( q ) in P are polynomial algebras. Moreover,(3.1) P GL n ( q ) = F q [ Q n, , · · · , Q n,n − ] . For ≤ i ≤ l, ≤ j ≤ n i , define(3.2) v i,j = F m i − ( x m i − + j ) , (3.3) 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 the proof of [8, Lemma 1],(3.4) P U I = F q [ x , · · · , x n , v , , · · · , v ,n , · · · , v l, , · · · , v l,n l ] . Moreover, by [7, Theorem 2.2] and [3, Theorem 1.4],(3.5) P GL I = F q [ q i,j | ≤ i ≤ l, ≤ j ≤ n i ] , KE OU
The invariants of G I .Lemma 3.1. [10, Proposition 3.2] For ≤ i ≤ l, assume that F q [ x , · · · , x n i ] G i = F q [ e i, , · · · , e i,n i ] is a polynomial algebra. For ≤ j ≤ n i , define u i,j = e i,j ( v i, , · · · , v i,n i ) . The subalgebra P G I of G I -invariants in P is a polynomial ring on the generators u i,j with ≤ i ≤ l, ≤ j ≤ n i . Namely, P G = F q [ u i,j | ≤ i ≤ l, ≤ j ≤ n i ] . block basis for the coinvariants of G ( m, a, n ) Recall that G ( m, a, n ) ≃ S n ⋉ A ( m, a, n ) , which is called imprimitive reflection group, where A ( m, a, n ) = { diag( w , · · · , w n ) | w mj = ( w · · · w n ) m/a = 1 } . Denote G = G ( m, a, n ) , it is well-known that P G = K [ e , · · · , e n ] , where e i = (cid:26)P ≤ j < ··· For all ≤ i ≤ n and ≤ r ≤ n + 1 − i, one have ( x n · · · x n +1 − i ) mr e n +1 − i − r,n − i ∈ I . Proof. The proof is by induction on i. (1) When i = 1 , one need to verify that x mrn e n − r,n − ∈ I . We will use induction on r to proveit.(a) When r = 1 , x mn e n − ,n − = e n,n ∈ I . (b) Suppose r > . Note that(4.1) e k,i = e k,i − + x mi e k − .i − . Therefore, x mrn e n − r,n − + x m ( r − n e n − r +1 ,n − = x m ( r − n e n − r +1 ∈ I . By induction assump-tion on r, x m ( r − n e n − r +1 ,n − ∈ I . Thus x mrn e n − r,n − ∈ I . (2) Suppose i > . Again we will use induction on r to prove ( x n · · · x n +1 − i ) mr e n +1 − i − r,n − i ∈ I . (a) When r = 1 , ( x n · · · x n +1 − i ) m e n − i,n − i = e n,n ∈ I . (b) Suppose r > . By 4.1 ( x n · · · x n +1 − i ) mr e n +1 − i − r,n − i + ( x n · · · x n +2 − i ) mr x m ( r − n +1 − i e n +2 − i − r,n − i = ( x n · · · x n +2 − i ) mr x m ( r − n +1 − i e n +2 − i − r,n − i +1 . Using the primary induction hypothesis, ( x n · · · x n +2 − i ) mr x m ( r − n +1 − i e n +2 − i − r,n − i +1 ∈ I . Us-ing the secondary induction hypothesis, ( x n · · · x n +2 − i ) mr x m ( r − n +1 − i e n +2 − i − r,n − i ∈ I . Thus, ( x n · · · x n +1 − i ) mr e n +1 − i − r,n − i ∈ I . (cid:3) Set r = n + 1 − i, one have ( x n · · · x r ) mr e ,n − i = ( x n · · · x r ) mr ∈ I . Corollary 4.2. The critical monomial ( x n · · · x r ) mr , r = 1 , · · · , n, are trivial. LOCK BASIS 5 By Lemma 2.6, one have Corollary 4.3. The monomial α = x m − x m − · · · x mn − n generates a block basis for the coinvari-ants of G ( m, , n ) . By Lemma 2.5, it’s convenient to deal with the block basis given by x mn − x m ( n − − · · · x m − n todescribe the product in P G . In fact, we need to write x mrn +1 − r in terms of our block basis.Let s k,i = P j + ··· + j i = k x mj j · · · x mj i j i . Observe that s k,i = x mi s k − ,i + s k,i − . The proof of [1,Theorem 8.4] still works. Namely, if k + i ≥ n + 1 , s k,n +1 − i ∈ I . As Corollary, one have Proposition 4.4. Keep notations as above, x mrn +1 − r ≡ − X j ··· + jn +1 − r = r j n +1 − r = r x mj j · · · x mj n +1 − r j n +1 − r . Note that the summands on the right hand side form a subset of our block basis. This propositionprovides a recursive algorithm for computing products in P G . a = 1 case. Suppose a = 1 now. By Lemma 2.5, all candidates of block generators forcoinvariants of G ( m, a, n ) are α σ = x m − σ (1) x m − σ (2) · · · x mn/a − σ ( n ) for σ ∈ S n . Lemma 4.5. Keep notations as above. α σ ∈ I . In particular, there is no block basis for thecoinvariants of G ( m, a, n ) . Proof. Note that mn/a − ≥ m/a unless n = 1 which we omit. Since a = 1 by hypothesis, hence ( m − a − ≥ and m − ≥ m/a. Therefore, there exist f n ∈ P such that α σ = ( x · · · x n ) m/a f n = e n f n ∈ I . (cid:3) general case. Combining Corollary 4.3, Proposition 4.4 and Lemma 4.5, we have one of ourmain results. Theorem 4.6. The coinvariants of G ( m, a, n ) has block basis if and only if a = 1 . Moreover, if a = 1 , the monomial x mn − x m ( n − − · · · x m − n generates a block basis with products x mrn +1 − r = − X j ··· + jn = r j n = r x mj j · · · x mj n +1 − r j n +1 − r . block basis for P U I Denote H n = F q [ x , · · · , x n ] U I . In this section, we will prove that there is a block basis for P U I := P / ( H n, + ) . θ : F q [ x , · · · , x n ] → F q [ x , · · · , x m l − ] by θ ( x i ) = x i for ≤ i ≤ m l − and θ ( x i ) = 0 otherwise. Then ker( θ ) is the ideal generated by x m l − +1 , · · · , x n . KE OU Ψ n = { x i · · · , x i n n | ∀ k, ≤ i k < τ ( k ) } . We will prove that Ψ n is a basis for P over P U I . Namely,(5.1) f I := n Y i =1 x q τ ( i ) − i = x q m − m +1 · · · x q m − m x q m − m +1 · · · x q ml − − m l generates a block basis for P U I . Lemma 5.1. If α ∈ Ψ n , then x m l − + j α is in the P U I -module spanned by Ψ n for all j = 1 , · · · , n l . Proof. Suppose α = x i · · · , x i n n ∈ Ψ n . Without loss of generality, one can assume j = n l , and hence m l − + j = n. If i n < q m l − − , then x n α ∈ Ψ n and Lemma holds. Now, if i n = q m l − − , then x n α = x i · · · x i n − n − x q ml − n . By 3.2, v l,n l = F m l − ( x n ) = x q ml − n + m l − X i =1 Q m l − ,i x q ml − − i n . Therefore, x q ml − n = v l,n l − m l − X i =1 Q m l − ,i x q ml − − i n , and x n α = v l,n l x i · · · x i n − n − − m l − X i =1 Q m l − ,i x i · · · x i n − n − x q ml − − i n . Since Q m l − ,i ∈ P G I ⊆ P U I and v l,n l ∈ P U I . Lemma holds. (cid:3) Theorem 5.2. Ψ n is a basis for P over P U I . Namely, f I generates a block basis for P U I . Proof. Since the cardinality of Ψ n equals the dimension of P U I , it is sufficient to prove that Ψ n isa spanning set. We use induction on n. For n = 1 , the result is trivial.For n > , we use induction on the degree. The result is clear for polynomials of degree zero.Now, suppose f ∈ P with deg( f ) > . By the primary induction hypothesis θ ( f ) = X β ∈ Ψ ml − c β β for some c β ∈ H m l − . The kernel of θ is the ideal generated by x m l − +1 , · · · , x n and therefore f = θ ( f ) + x m l − + j f ′ for some m l − < j ≤ n and f ′ ∈ P . Since deg( f ′ ) < deg( f ) , by the secondaryinduction hypothesis, f = θ ( f ) + x m l − + j f ′ = X β ∈ Ψ ml − c β β + X γ ∈ Ψ n d γ x m l − + j γ. By Lemma 5.1, x m l − + j γ is in the span of Ψ n . Note that Ψ m l − ⊆ Ψ n and H m l − ⊆ H n . Therefore, f is in the H n span of Ψ n . (cid:3) products of block basis.Lemma 5.3. For each ≤ i ≤ l − , ≤ j ≤ n i , x q mi m i + j ≡ in P U I . Proof. By 3.2, v i +1 ,j = F m i ( x m i + j ) = x q mi m i + j + P m i k =1 Q m i ,k x q mi − k m i + j . All v i +1 ,j and Q m i ,k lie in I . Hence lemma holds. (cid:3) LOCK BASIS 7 Corollary 5.4. P U I ≃ N ni =1 K [ x i ] / ( x q τ ( i ) i ) . Remark . Note that the hypothesis of Lemma 2.6 are satisfied by monomial f I . This gives anotherproof of the existence of a block basis for P U I . In fact, Σ( H n ) = { } . And the critical monomialsassociated to f I are x q τ ( j ) j . By 5.3, x q τ ( j ) j ≡ . Moreover, by [1, Theorem 4.7], one get Corollary 5.4.6. Block basis for P G I ≤ k ≤ l, denote ˜ G k = G ∗ · · · ∗ G · · · ∗ ... ... ... ... · · · G k , Q k = F q [ x , · · · , x m k ] , I k = (cid:16) Q ˜ G k k, + (cid:17) and P k = F q [ x m k − +1 , · · · , x m k − + n k ] . For each ≤ i ≤ l, suppose x α i m i − +1 · · · x α i,ni m i generates a block basis for P i over P G i i . Lemma 6.1. Keep notations as above. P is a free P G I module with a basis consisting of n Y k =1 x a k k l Y i =1 n i Y j =1 v b ij ij ≤ a k < τ ( k ) , ≤ b ij < α ij . Proof. Let R i = F q [ v i, , · · · , v i,n i ] be a polynomial ring with generators v ij , ≤ j ≤ n i . Then R i = n i M j =1 α ij M b ij =1 v b i i, · · · v b i,ni i,n i ( R i ) G i . By Theorem 5.2, f I = Q ni =1 x q τ ( i ) − i generates a block basis for P U I . Equivalently, P = n M i =1 τ ( i ) − M a i =0 x a · · · x a n n P U I . Since P U I = R ⊗ · · · ⊗ R l , lemma holds. (cid:3) Lemma 6.2. For all ≤ i ≤ l, ≤ j ≤ n i , v ij ≡ x q mi − m i − + j . Proof. By 3.2, v i,j = x q mi − m i − + j + P m i − k =1 Q m i − ,k x q mi − − k m i − + j . Since Q m i − ,k ∈ I i − ⊆ I , lemma holds. (cid:3) Theorem 6.3. Suppose x α i m i − +1 · · · x α i,ni m i generates a block basis for P i over P G i i . Then l Y i =1 n i Y j =1 x ( α ij +1) q mi − − m i − + j generates a block basis of P over P G I . Block basis for P GL I . If G I = GL I , then G i = GL n i ( q ) . By [11, Theorem B] (a generalizationrefers to [1, Theorem 3.2]), the monomial Q n i j =1 x q ni − q ni − j − m i − + j generates a block basis for P i over P GL ni ( p ) i . The following proposition is a corollary of Theorem 6.3. Proposition 6.4. The monomial Q li =1 Q n i j =1 x q mi − q mi − j − m i − + j generates a block basis for P GL I . KE OU Remark . (1) Take I = ( n ) , then l = 1 , n = m = n. Proposition provides a block basis for P GL n ( q ) which coincide with [11, Theorem B] and [1, Theorem 3.2].(2) Take I = (1 , · · · , , then l = n, m i = i and the monomial Q li =1 x q i − q i − − i generates a blockbasis for P B where B is a Borel subgroup of GL n ( q ) consisting of all upper triangle matrices.6.3. Weyl groups of Cartan type Lie algebras. By [5], G I is a Weyl group of restricted Cartantype Lie algebra g of type W, S or H where I = ( n , n ) , G I = (cid:26)(cid:18) A B C (cid:19) A ∈ GL n ( p ) , C ∈ G (cid:27) ,G = G ( m, , n ) and m = (cid:26) if g is of type W or S, if g is of type H. By Theorem 4.6, the monomial Q ni = n +1 x m ( n − i +1) − i generates a block basis for P over P G . The following is a corollary of Theorem 6.3. Proposition 6.6. The monomial x p n − p n − − · · · x p n − n x m ( n +1) p n − n +1 x mn p n − n +2 · · · x mp n − n gen-erates a block basis for P over P G I . Acknowledgments. This work is supported by Fundamental Research Funds of Yunnan Province(No. 2020J0375), the Fundamental Research Funds of YNUFE (No. 80059900196). References [1] H. E. A. Campbell, I. P. Hughes, R. J. Shank and D. L. Wehlau, Bases for rings of coinvariants, TransformationGroups, Vol.1, No.4(1996), 307-336.[2] 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.[3] 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. 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Steinberg, On Dickson’s theorem on invariants, J. Fac. Sci. Univ. Tokyo. 34 (1987), 699-707.(Ke Ou) School of Statistics and Mathematics, Yunnan University of Finance and Economics,Kunming 650221, China E-mail address ::