On the transfer between the Dickson algebras as modules over the Steenrod algebra

On the transfer between the Dickson algebras as modules over the Steenrod algebra tài liệu, giáo án, bài giảng , luận vă...

J Homotopy Relat Struct DOI 10.1007/s40062-014-0097-0 On the transfer between the Dickson algebras as modules over the Steenrod algebra Võ T N Quỳnh · Lưu X Trường Received: 25 June 2014 / Accepted: November 2014 © Tbilisi Centre for Mathematical Sciences 2014 Abstract Let D := D(W) be the Dickson–Mùi algebra of W for an elementary abelian p-group W of rank , which consists of all invariants in the mod p cohomology of W under the general linear group G L(W) Hưng (Math Ann 353:827–866, 2012) determined explicitly all the homomorphisms between the Dickson–Mùi algebras (regarded as modules over the Steenrod algebra, A) He showed that the compositions of the restrictions r es ,m : D → Dm and the transfers trm,n : Dm → Dn for m ≤ min{ , n} form a basis of HomA (D , Dn ) The restriction r es ,m has explicitly been known (from Lemmas 3.3 and 3.4 for p = and Lemma 9.1 for p > of the cited article), while the transfer trm,n has only been computed for m = in some degrees (see Lemma 9.2 of the article) In this paper, we study trm,n for p = in general We determine completely tr1,n for any n, and compute the image of trm,n for arbitrary m, n on some powers of multilinear and alternating invariants Then, we recognize some families of invariants in Dm on which the transfer trm,n vanishes Keywords Steenrod algebra · Modular representations · Invariant theory · Dickson algebra Communicated by Lionel Schwartz ˜ H V Hưng on the occasion of his 60th birthday Dedicated to Professor Nguyên The work was supported in part by a grant of the NAFOSTED V T N Quỳnh (B) · L X Trường Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Vietnam e-mail: quynhvtn@vnu.edu.vn L X Trường e-mail: lxtruong.lt@gmail.com 123 V T N Quỳnh, L X Trường Mathematics Subject Classification Primary 55S10 · 55S05 · 20G10 · 20G05 Introduction Let V be an elementary abelian 2-group rank n Then V can also be regarded as an n-dimensional vector space over F2 , the field of two elements Let H ∗ (V) denote the cohomology of the group V As it is well-known, H ∗ (V) ∼ = S(V∗ ) where S(V∗ ) ∗ denote the symmetric algebra over the space V Throughout the paper, the coefficient ring for homology and cohomology is always F2 Let x1 , , xn be a basis of V∗ , we have H ∗ (V) ∼ = F2 [x1 , , xn ] The general linear group G L(V) ∼ = G L(n, F2 ) acts regularly on V and therefore on H ∗ (V) The Dickson algebra is the algebra of all invariants of H ∗ (V) under the action of G L(V) It is explicitly determined by Dickson in [2] as follows: D(V) := H ∗ (V)G L(V) ∼ = F2 [x1 , , xn ]G L(n,F2 ) = F2 [Q n,0 , Q n,1 , , Q n,n−1 ], where Q n,i denotes the Dickson invariant of degree 2n − 2i (A precise definition for the Dickson invariants will be given in Sect 2.) Being the cohomology of the classifying space BV, the group H ∗ (V) is equipped with a structure of module over the mod Steenrod algebra, A Each γ ∈ G L(V) induces an A-isomorphism γ ∗ on H ∗ (V) The map γ → γ ∗ gives rise to the regular action of G L(V) on H ∗ (V) So, the actions of G L(V) and A on H ∗ (V) commute with each other Hence, the Dickson algebra D(V) inherits a structure of module over the Steenrod algebra A from H ∗ (V) Let U be a F2 -vector space of dimension m with m ≤ n The subject of the present paper is the transfer trU,V : D(U) → D(V), which is also denoted by trm,n : Dm → Dn defined in [3] For fixed n and , the set {trm,n | m < min{ , n}} is known as a key component forming all A-homomorphisms from D to Dn in the sense as follows Let W be a vector space of dimension The restriction r es ,m : D → Dm is the homomorphism induced from an inclusion U → W (in [3, Lemma 3.1], Hưng showed that any inclusion U → W induces the same homomorphism on ¯ or D¯ n be the augmentation ideal of all positive degree H ∗ (W)G K (W) ) Let D(V) elements in the Dickson algebra Dn Theorem 1.1 [3, Theorem 1.1] The A-module homomorphisms {trm,n r es ,m : ≤ m ≤ min{ , n}} form a basis of the vector space HomA ( D¯ , D¯ n ) of A-module homomorphisms from D¯ to D¯ n In particular, dim HomA ( D¯ , D¯ n ) = min{ , n} The restrictions r es ,m are explicitly determined on each Dickson monomial in [3, Lemma 3.4]; so, in order to compute HomA ( D¯ , D¯ n ) we need to compute the transfers trm,n First, the transfer tr1,n is concretely computed in [3, Lemma 4.2] Then, it is further studied in the undergraduate thesis of Phạm H -Dăng under the guidance of Nguyễn H V Hưng (see [1]) In the case m = 1, we have Dm = F2 [x] in which deg x = The following theorem is one of main results of the paper 123 On the transfer between the Dickson algebras Theorem 1.2 For k > 0, n−1 i=0 si tr1,n (x ) = k n−1 n i i=0 (2 −2 )si =k Here n−1 ( i=0 si −1)! (s0 −1)!s1 ! sn−1 ! −1 ! s (s0 − 1)!s1 ! sn−1 ! n−1 Q sn,1 Q n,n−1 Q sn,0 = if s0 = Let x1 , , xm be a basis for U∗ Then each f ∈ Dm = D(U) is a polynomial in the variables xi (1 ≤ i ≤ m); so, we rewrite f as f (x1 , , xm ) We denote by f the map V∗ × · · · × V∗ → S(V∗ ), (t1 , , tm ) → f (t1 , , tm ) We m times say that f (x1 , , xm ) is multilinear on V∗ if f is m-multilinear The invariant f (x1 , , xm ) is called alternating on V∗ if the map f is alternating The set of invariants in Dm which are multilinear and alternating forms a subspace of Dm Furthermore, this subspace is subset of the ideal of Dm generated by Q m,0 (see Corollary 4.7) It is infinity dimensional and can be explicitly determined as follows For each set of m distinct non-negative numbers β = {β1 , , βm }, let m be the symmetric group on {1, , m} and let ωβ (x1 , , xm ) = x12 σ∈ βσ (1) βσ (m) xm2 m Proposition 1.3 The set {ωβ (x1 , , xm ) | β = (β1 , , βm ) ∈ Nm , βi = β j for all i = j} forms a basis for the subspace of Dm generated by multilinear and alternating invariants The following theorem shows the image of the transfer trm,m+1 on powers of multilinear and alternating invariants in Dm For each Q ∈ Dm and y1 , , ym ∈ F[x1 , , xm+1 ], we denote by Q(y1 , , ym ) the polynomial in F[x1 , , xm+1 ] obtained from Q by the substitution y j for x j ( j = 1, , m) Theorem 1.4 Suppose that Q is a multilinear and alternating invariant in Dm Let y j = Q(x1 , , x j−1 , x j+1 , , xm+1 ) for j = 1, , m + Then, for r > we have trm,m+1 (Q r ) = (s0 + · · · + sm − 1)! m Q sm+1,0 (y1 , , ym+1 ) Q sm+1,m (s0 − 1)!s1 ! sm ! summed over all sequences of non-negative integers (s0 , , sm ) such that m (2m+1 − 2i )si = r i=0 123 V T N Quỳnh, L X Trường Since Q m,0 , Q m,0 Q m,i are multilinear and alternating invariants (see Example 4.8), we get the two following explicit formulas for image of trm,m+1 on some powers of these invariants j Corollary 1.5 (i) trm,m+1 (Q m,0 ) = for < j < 2m+1 − 2m+1 −1 (ii) trm,m+1 (Q m,0 2m −1 ) = Q m+1,0 Corollary 1.6 For r > 0, tr2,3 (Q r2,0 Q r2,1 ) = 7s0 +6s1 +4s2 =r (s0 + s1 + s2 − 1)! Q 33,0 Q 3,1 Q 23,2 + Q 53,0 (s0 − 1)!s1 !s2 ! × Q 23,1 Q 23,2 + Q 23,0 Q 3,1 s2 s0 Q 23,0 Q 43,2 s1 The following proposition gives some families of invariants on which the transfer is zero Proposition 1.7 trm,n (Q) is zero for each of the following cases (i) Q ∈ Dm is a multilinear and alternating invariant; sm−1 s0 s1 Q 2m,1 Q 2m,m−1 where s0 , s1 , , sm−1 are nonnegative with m, n (ii) Q = Q 2m,0 such that n ≥ 3m 2+3m ; s s (iii) m = 2, n ≥ and Q = Q 22,00 Q 22,11 for any nonnegative integers s0 , s1 The statement in Proposition 1.7(iii) is no longer true for the case n = In fact, s s tr2,3 (Q 22,00 Q 22,11 ) = for s0 − s1 > or s0 − s1 < (see Remark 7.3) The paper is divided into seven sections The introduction in Sect is followed by the preliminary in Sect 2, where we recall definition of the transfer between the Dickson algebras Section is a proof of Theorem 1.2 The concepts of multilinear and alternating invariants are introduced in Sect 4, where we prove Proposition 1.3 In Sect 5, we study the transfer on powers of multilinear and alternating invariants, then prove Theorem 1.4 and Proposition 1.7(i) Section deals with a method of finding invariant monomials in the image of the transfer by finding “leading elements” of it By using this method, we prove Lemma 5.5, the most technical lemma of the paper that is used in the proofs of Corollaries 1.5 and 1.6 Finally, in Sect 7, we prove Proposition 1.7(ii) and (iii) The paper was in progress while the first named author was visiting to the Vietnam Institute for Advanced Study in Mathematics (VIASM) She would like to thank VIASM for the financial support and the warm hospitality Preliminary In this section, we exploit Hưng’s definition of the transfer trm,n Let U and V be a F2 -vector spaces of dimensions m and n with m ≤ n Let K be a subspace of V and πK : V → U be an epimorphism with ker πK = K Suppose that π : V → U is another epimorphism whose kernel is also K Then 123 On the transfer between the Dickson algebras ∗ = there is an isomorphism α: U → U such that πK = απ It follows that πK π ∗ α ∗ : H ∗ (U) → H ∗ (V) Since α ∗ acts identically on the invariants H ∗ (U)G L(U) , ∗ and π ∗ are the same on H ∗ (U)G L(U) Therefore, the induced homomorphism πK ∗ : H ∗ (U)G L(U) → H ∗ (V) does not depend on the choice of the epimorphism π πK K It only depends on the kernel of πK , K On the other hand, the group G L(V) permute ∗ maps H ∗ (U)G L(U) to the subspaces K of dimension n − m in V So the sum of πK the G L(V)-invariants ∗ : H ∗ (U)G L(U) → H ∗ (V)G L(V) , Definition 2.1 [3, Definition 4.1] trU,V = K πK where the sum runs over all the subspaces K of dimension (dim V − dim U) in V As it is well known that H (V) ∼ = V∗ and H ∗ (V) ∼ = S(V∗ ) where S(V∗ ) denote ∗ the symmetric algebra over the space V Let x1 , , xn be a basis of V∗ , we have H ∗ (V) ∼ = F2 [x1 , , xn ] Recall that the algebra of the G L(V)-invariants H ∗ (V)G L(V) was computed by Dickson [2] He showed in [2] that H ∗ (V)G L(V) is also a polynomial algebra and denoted it by Dn : Dn := H ∗ (V)G L(V) = F2 [Q n,0 , Q n,1 , , Q n,n−1 ] Here Q n,0 , Q n,1 , , Q n,n−1 are the Dickson invariants They are inductively defined by the formula Q n,i = Q 2n−1,i−1 + Vn Q n−1,i , where, by convention, Q n,n = 1, Q n,i = for i < 0, and Vn = (c1 x1 + · · · + cn−1 xn−1 + xn ) c j ∈F2 is the Mùi invariant under the Sylow 2-subgroup Tn of G L n consisting of all upper triangular n × n-matrices with on the main diagonal (see Mùi [5]) Since the transfer trU,V only depends on the dimensions of U and V, it is denoted by trm,n : Dm → Dn , where m = dim U and n = dim V For each epimorphism π : V → U, the induced homomorphism in cohomological degree 1, π ∗ : H (U) → H (V) is a monomorphism So W := π ∗ H (U) is a subspace of dimension m in H (V) Let x1 , , xm be a basis of H (U) Then π ∗ (x1 ), , π ∗ (xm ) a basis for π ∗ H (U) And for each Q ∈ Dm we have π ∗ (Q) = Q(π ∗ (x1 ), , π ∗ (xm )) Since Q is invariant under any invertible transformation on U, Q(π ∗ (x1 ), , π ∗ (xm )) = Q(y1 , , ym ) for any basis y1 , , ym of W Hence Q(π ∗ (x1 ), , π ∗ (xm )) only depends on W ; so, we denote it by Q(W ) for short Therefore, by Definition 2.1 we have trm,n (Q) = Q(W ) W 123 V T N Quỳnh, L X Trường where the sum runs over all m-dimensional subspaces W in H (V) For example, with m = 1, n = let x1 , x2 , x3 be a basis for V∗ we have tr1,3 (x k ) = x1k +x2k +x3k + (x1 + x2 )k + (x1 + x3 )k + (x2 + x3 )k + (x1 + x2 + x3 )k since all one-dimensional subspaces of V∗ are x1 , x2 , x3 , x1 + x2 , x1 + x3 , x2 + x3 , x1 + x2 + x3 The transfer from D1 to Dn The aim of this section is to prove the following theorem, which is also numbered as Theorem 1.2 in the introduction Theorem 3.1 For k > 0, n−1 i=0 si tr1,n (x ) = k n−1 n i i=0 (2 −2 )si =k Here n−1 ( i=0 si −1)! (s0 −1)!s1 ! sn−1 ! −1 ! (s0 − 1)!s1 ! sn−1 ! s n−1 Q sn,0 Q sn,1 Q n,n−1 = if s0 = At small degrees, tr1,n was explicitly determined by Hưng in [3] as follows Lemma 3.2 [3, Lemma 4.2] (i) tr1,n (x −1 ) = Q n,0 (ii) tr1,n (x j ), for < j < 2n − n In order to prove Theorem 3.1, we need Lemma 3.2 and the following inductive formula Lemma 3.3 tr1,n (x k ) = n−1 i=0 Q n,i tr1,n (x k−2 n +2i ) for k ≥ 2n − Proof The lemma is proved by induction on n As k ≥ 2n − 1, we rewrite k as the form k = + 2n − for ≥ Then the lemma is equivalent to tr1,n (x +2n −1 n−1 )= Q n,i tr1,n (x +2i −1 ), i=0 ≥ For n = 1, as tr1,1 is the identity map, we have tr1,1 (x +1 ) = x +1 = Q 1,0 tr1,1 (x ) So the statement holds for n = Suppose inductively that it is true for n Let for U = {c1 x1 + · · · + cn xn |ci ∈ F2 for i = 1, , n}, W = {c1 x1 + · · · + cn xn + xn+1 |ci ∈ F2 for i = 1, , n} 123 On the transfer between the Dickson algebras For s ≥ 0, we set RsU := xs and RsW := x∈U x s x∈W Then, by the definition of tr1, , we have tr1,n (x s ) = RsU and tr1,n+1 (x s ) = RsU +RsW Furthermore, we get the two following formulas They are proved latter n−1 i=0 (a) (b) Vn+1 Q 2n,i tr1,n+1 (x n i=0 +2i+1 −1 ) Q 2n,i tr1,n+1 (x n−1 i=0 = R U+2n+1 −1 + +2i −1 ) = R W+2n+1 −1 + Q 2n,i R W+2i+1 −1 , n−1 i=0 Q 2n,i R W+2i+1 −1 Then, we have n Q n+1,i tr1,n+1 (x +2i −1 ) i=0 n = (Q 2n,i−1 + Q n,i Vn+1 )tr1,n+1 (x +2i −1 ) i=0 (as Q n+1,i = Q 2n,i−1 + Q n,i Vn+1 for i ≥ and Q n,i = for i < 0) n = i Q 2n,i−1 tr1,n+1 (x +2 −1 ) + i=1 n−1 Q n,i tr1,n+1 (x Vn+1 +2i −1 ) i=0 +2i+1 −1 Q 2n,i tr1,n+1 (x = n n−1 ) + R W+2n+1 −1 + i=0 Q 2n,i R W+2i+1 −1 i=0 [applying (b)] = R U+2n+1 −1 + R W+2n+1 −1 = tr1,n+1 (x +2n+1 −1 [applying (a)] ) So, the lemma is proved Now, we prove the two formulas (a) and (b) above Case (a) tr1,n (x +2n+1 −1 ) = tr1,n (x ( +2n )+2n −1 n−1 Q n,i tr1,n (x = ) +2n +2i −1 ) i=0 (by the inductive hypothesis on tr1,n for + 2n ) n−1 n−1 = Q n,i Q n, j tr1,n (x ( +2i )+2 j −1 ) i=0 j=0 [applying the inductive hypothesis to each tr1,n (x +2n +2i −1 )] 123 V T N Quỳnh, L X Trường However, we have Q n,i Q n, j tr1,n (x ( +2i )+2 j −1 )= (Q n,i Q n, j + Q n, j Q n,i )tr1,n (x i=j +2i +2 j −1 ) i< j = Therefore tr1,n (x +2n+1 −1 n−1 )= Q 2n,i tr1,n (x ( +2i+1 −1 ) i=0 It is equivalent to n−1 R U +2n+1 −1 = x Q 2n,i tr1,n (x ( +2i+1 −1 ) i=0 Then, we get n−1 Q 2n,i tr1,n+1 (x +2i+1 −1 n−1 )= i=0 Q 2n,i R U+2i+1 −1 + R W+2i+1 −1 i=0 n−1 = n−1 Q 2n,i R U+2i+1 −1 + i=0 Q 2n,i R W+2i+1 −1 i=0 n−1 = R U+2n+1 −1 + Q 2n,i R W+2i+1 −1 i=0 So, we obtain (a) Case (b) We have n +2i −1 Q n,i tr1,n+1 (x i=0 n Q n,i R U+2i −1 + R W+2i −1 )= i=0 Since the inductive hypothesis tr1,n (x Q n,n = 1, it follows that +2n −1 ) n Q n,i tr1,n (x +2i −1 = ) = i=0 In the other word n Q n,i R U+2i −1 = i=0 123 n−1 i=0 Q n,i tr1,n (x +2i −1 ) and On the transfer between the Dickson algebras So n +2i −1 Q 2n,i tr1,n+1 (x n )= i=0 Q 2n,i R U+2i −1 + R W+2i −1 i=0 n n = Q 2n,i R U+2i −1 + Q 2n,i R W+2i −1 i=0 n i=0 = Q 2n,i R W+2i −1 i=0 n = Q 2n,i i=0 = x x∈W x x∈W n −1 +2i −1 as R W+2i −1 = x +2i −1 x∈W i Q 2n,i x i=0 It implies that n Q 2n,i tr1,n+1 (x Vn+1 +2i −1 )= i=0 x −1 n i Q 2n,i x Vn+1 x∈W i=0 On the other hand, Vn+1 (x1 , , xn , xn+1 ) = Vn+1 (x1 , , xn , x) for all x ∈ W ; so, by [5, Proposition 2.6], we get n Vn+1 = i Q 2n,i x for all x ∈ W i=0 As a consequence n Vn+1 i Q 2n,i tr1,n+1 (x +2 −1 ) = i=0 −1 x x∈W = −1 x x∈W n i Q 2n,i x i=0 n i=0 n = = = = i+1 Q 2n,i x Q 2n,i x +2i+1 −1 x∈W i=0 n i+1 Q 2n,i x +2 −1 i=0 x∈W n Q 2n,i R W+2i+1 −1 i=0 n−1 R W+2n+1 −1 + Q 2n,i R W+2i+1 −1 i=0 123 V T N Quỳnh, L X Trường Therefore, we get (b) Proof of Theorem 3.1 for convenience, we denote sn −1 Q (s0 ,s1 , ,sn−1 ) = Q sn,0 Q sn,1 Q n,n−1 , C(s0 , s1 , , sn−1 ) = n−1 i=0 si −1 ! (s0 − 1)!s1 ! sn−1 ! Here C(s0 , s1 , , sn−1 ) = if one of the numbers (s0 −1), s1 , , sn−1 is negative We see that n−1 i=0 si n−1 C(s0 , s1 , , sn−1 ) = si − i=0 = n−1 i=0 si −2 ! (s0 − 1)!s1 ! sn−1 ! −2 ! (s0 − 2)!s1 ! sn−1 ! n−1 i=0 si n−1 + i=1 −2 ! (s0 − 1)! (si − 1) sn−1 ! n−1 = C(s0 , , si − 1, , sn−1 ) i=0 For ≤ i ≤ n − 1, setting di = deg Q n,i , we get di = 2n − 2i So, the theorem is equivalent to tr1,n (x k ) = C(s0 , s1 , , sn−1 )Q (s0 ,s1 , ,sn−1 ), where the sum is over all sequences s0 , s1 , , sn−1 such that The proposition is proved by induction on k Case k < 2n − We have n−1 = k n−1 di si = k ⇐⇒ (2n − 1)s0 + i=0 n−1 i=0 di si (1) si = k ⇒ s0 = i=1 n−1 Then C(s0 , s1 , , sn−1 ) = for all sequences (s0 , , sn−1 ) such that i=0 di si = k As a consequence, the right hand side of (1) is equal to zero On the other hand, by Lemma 3.2, tr1,n (x k ) = as k < 2n − Therefore, we get (1) for this case 123 On the transfer between the Dickson algebras m+1 = j=1 m+1 = ⎛ ⎞ (i) ⎝ cσ (1),1 cσ (m+1),m−1 ⎠ Q m (X j , xi ) σ∈ {1, ,m+1}\{i, j} c j Q m (X (i) j , xi ) = j=1 m+1 c j Q m (X j ) j=1 (i) as Q m X j , xi = Q m (X j ) So we get the conclusion Case ci = for all ≤ i ≤ m +1 By the similar argumentation as in Case for (c1 , , cm ), there is one (m − 1)-dimensional subspace W (m+1) of F2 x1 , , xm such that (m+1) Q m−1 (W (m+1) ) = Q m−1 X j 1≤ j≤m for all multilinear alternating invariants Q m−1 of Dm−1 We see that x1 ∈ W (m+1) Indeed, if x1 ∈ W (m+1) , let (α1 , α2 , , αm−1 ) be a basis for W (m+1) such that α1 = x1 and α2 , , αm−1 ∈ F2 < x2 , , xm > Then, the coefficient of (m+1) ) in the expansion of Q m−1 (W (m+1) ) = Q(x1 , α2 , , αm−1 ) is equal Q m−1 (X to This contradicts to c1 = So x1 ∈ W (m+1) Similarly, we can prove that xi ∈ W (m+1) for all ≤ i ≤ m Now let W be the subspace of V∗ generated by W (m+1) and x1 + xm+1 Since xm+1 ∈ W (m+1) it follows that dim W = m Suppose that (α1 , , αm−1 ) is a basis of W (m+1) and α j = m k=1 ck j x k By the similar argumentation as in Case 1, we get cj = cσ (1),1 cσ (m),m−1 σ∈ for ≤ j ≤ m {1, ,m}\{ j} Hence, for any multilinear and alternating invariant Q m of Dm , we obtain Q(W ) = Q(α1 , , αm−1 , x1 + xm+1 ) = Q(α1 , , αm−1 , x1 ) + Q(α1 , , αm−1 , xm+1 ) m = (m+1) cj Q X j m , x1 + j=1 (m+1) cj Q X j , xm+1 j=1 = c1 Q X 1(m+1) , x1 + m c j Q X (m+1) , xm+1 j j=1 (m+1) as Q X j , x1 = for all j = 123 V T N Quỳnh, L X Trường m = c1 Q(X m+1 ) + (m+1) c j Q(X j ) as Q X , x1 = Q(X m+1 ) j=1 = Q(X ) + · · · + Q(X m+1 ) (as c j = for all j) The lemma follows The following theorem is also numbered as Theorem 1.4 in the introduction Theorem 5.2 Suppose that Q is a multilinear and alternating invariant in Dm Let y j = Q(X j ) for j = 1, , m + Then, for r > we have (s0 + · · · + sm − 1)! s0 m Q Q sm+1,m (y1 , , ym+1 ) (s0 − 1)!s1 ! sm ! m+1,0 trm,m+1 (Q r ) = summed over all sequences of non-negative intergers (s0 , , sm ) such that m (2m+1 − 2i )si = r i=0 Proof By the definition of trm,m+1 , we have trm,m+1 (Q r ) = Q r (W ) W ≤V∗ ,dim W =m Following Lemma 5.1 and the definition of tr1,m+1 , it follows that trm,m+1 (Q r ) = (c1 Q(X ) + · · · + cm+1 Q(X m+1 ))r c1 , ,cm+1 ∈F2 = (c1 y1 + · · · + cm+1 ym+1 )r c1 , ,cm+1 ∈F2 = tr1,m+1 y1r Now, applying Theorem 3.1, we get the theorem The following is numbered as Corollary 1.5 in the introduction j Corollary 5.3 (i) trm,m+1 (Q m,0 ) = for < j < 2m+1 − m+1 −1 (ii) trm,m+1 (Q m,0 −1 ) = Q m+1,0 m Proof By Example 4.8(i), Q m,0 is multilinear and alternating So, we apply Theorem 5.2 to compute trm,m+1 (Q rm,0 ) m (2m+1 − 2i )si = r (i) We consider the equation i=0 m m+1 − 2i )s ≥ 2m+1 − But If r is odd then s0 ≥ We have i i=0 (2 m+1 − It follows that the equation has no solution Therefore < r < 123 On the transfer between the Dickson algebras trm,m+1 (Q rm,0 ) = If r is even, we write r = 2k where k, ∈ N and (ii) odd k We have trm,m+1 (Q rm,0 ) = (trm,m+1 (Q m,0 ))2 Since < r < 2m+1 − we get < < 2m+1 − So, applying the above case, we obtain trm,m+1 (Q m,0 ) = Hence trm,m+1 (Q rm,0 ) = Part (i) of the corollary is proved m We have the equation i=0 (2m+1 − 2i )si = 2m+1 − has exactly one solution, namely s0 = 1, s j = for ≤ j ≤ m By Theorem 5.2, we get m+1 −1 trm,m+1 Q 2m,0 = Q m+1,0 (y1 , , ym+1 ), where y j = Q(x1 , , x j−1 , x j+1 , , xm+1 ) for ≤ j ≤ m +1 Furthermore, from [4, Theorem 2.2] we have m Q m+1,0 (y1 , , ym+1 ) = y12 y22 m−1 ym2 ym+1 + (symmetried) We will use Lemma 6.2 to find invariant monomials of Q m+1,0 (y1 , , ym+1 ) First, we show by induction on m that the maximal monomial in variables x1 , , xm+1 of Q m+1,0 (y1 , , ym+1 ), by Definition 6.1, is x12 m −1 2(2m −1) x2 m−1 (2m −1) xm2 2m (2m −1) xm+1 For m = 1, we have y1 = Q 1,0 (x2 ) = x2 , y2 = Q 1,0 (x1 ) = x2 So Q 2,0 (y1 , y2 ) = y12 y2 + y1 y22 = x1 x22 + x12 x2 Hence, the maximal monomial of Q 2,0 (y1 , y2 ) is x1 x22 Now we consider Q m+1,0 (y1 , , ym+1 ) From the inductive definitions of the Dickson invariants and the Mùi invariants in [2, 5], for i = 1, , m, we have yi = Q m,0 (x1 , , xi−1 , xi+1 , , xm+1 ) = Q m−1,0 (x1 , , xi−1 , xi+1 , , xm )Vm (x1 , , xi−1 , xi+1 , , xm+1 ) m−1 j (Q m−1,0 Q m−1, j )(x1 , , xi−1 , xi+1 , , xm )xm+1 = j=1 (applying [5, Proposition 2.6] ) Thus, the maximal monomial of each yi is a term of the polynomial m−1 Q m−1,0 (x1 , , xi−1 , xi+1 , , xm )xm+1 It should be noted that ym+1 = Q m,0 (x1 , , xm ); so, the power of xm+1 in ym+1 equals to zero For ≤ i ≤ m, denoting yi = Q m−1,0 (x1 , , xi−1 , xi+1 , , xm ) 123 V T N Quỳnh, L X Trường Then, we get Q m+1,0 (y1 , , ym+1 ) (2 = Q 2m,0 y1 , , ym ym+1 xm+1 m−1 m +2m−1 +···+2) 2m−1 = Q 2m,0 y1 , , ym x1 x22 xm m−1 as ym+1 = x1 x22 xm2 hypothesis, we have Q m,0 y1 , , ym = x12 2m (2m −1) xm+1 + (smaller terms) + (smaller terms), + (smaller terms) On the other hand, by the inductive m−1 −1 2(2m−1 −1) x2 m−1 (2m−1 −1) xm2 + (smaller terms) Therefore 2(2m −1) Q m+1,0 (y1 , , ym+1 ) = x12 −1 x2 xm2 +(smaller terms) m m−1 (2m −1) 2m (2m −1) xm+1 So, the maximal monomial of Q m+1,0 (y1 , , ym+1 ) is x12 m −1 2(2m −1) x2 m−1 (2m −1) xm2 2m (2m −1) xm+1 Hence, by Lemma 6.2, −1 Q m+1,0 (y1 , , ym+1 ) = Q m+1,0 + (others) m m Q im+1,1 Q im+1,m is a term of Q m+1,0 (y1 , , ym+1 ) We see that if Q im+1,0 then i (2m+1 − 1) + i (2m+1 − 2) + · · · + i m (2m+1 − 2m ) = (2m+1 − 1)(2m − 1) Since 2i ≥ for i = 0, , m, we get (2m+1 − 1)(2m − 1) ≤ (2m+1 − 1)(i + i + · · · + i m ) So 2m − ≤ (i + i + · · · + i m ) (2) On the other hand, for j = 1, , m we have m Q m+1, j = Q 2m, j−1 + s (Q m, j Q m,s )(x1 , , xm )xm+1 s=0 So 2m ( m m x Q im+1,0 Q im+1,1 Q im+1,m = Q im,0 Q im,m m+1 123 m j=0 i j ) + (smaller terms) On the transfer between the Dickson algebras From the above conclusion, the powers of xm+1 in the maximal monomial of Q m+1,0 (y1 , , ym+1 ) is 2m (2m − 1) It follows that 2m ( mj=0 i j ) ≤ 2m (2m − 1) Or equivalently mj=0 i j ≤ 2m − Combining this with (2), we get mj=0 i j = 2m − It follows that i = · · · = i m = and i = 2m − −1 m Q im+1,1 Q im+1,m = Q m+1,0 As a consequence Then Q im+1,0 m −1 Q m+1,0 (y1 , , ym+1 ) = Q m+1,0 m So, we get the corollary The following is numbered as Corollary 1.6 in the introduction Corollary 5.4 For r > 0, tr2,3 (Q r2,0 Q r2,1 ) = 7s0 +6s1 +4s2 =r (s0 + s1 + s2 − 1)! Q 33,0 Q 3,1 Q 23,2 + Q 53,0 (s0 − 1)!s1 !s2 ! × Q 23,1 Q 23,2 + Q 23,0 Q 3,1 s2 s0 Q 23,0 Q 43,2 s1 Proof Setting y1 = Q 2,0 Q 2,1 (x2 , x3 ), y2 = Q 2,0 Q 2,1 (x1 , x3 ), y3 = Q 2,0 Q 2,1 (x1 , x2 ) Since Q 2,0 Q 2,1 is multilinear and alternating by Example 4.8, applying Theorem 5.2 we get tr2,3 Q r2,0 Q r2,1 = 7s0 +6s1 +4s2 =r (s0 + s1 + s2 − 1)! Q s3,0 (y1 , y2 , y3 ) Q s3,1 Q s3,2 (s0 − 1)!s1 !s2 ! Now, applying Lemma 5.5, we get the corollary The following technical lemma will be proved in the next section Lemma 5.5 Let y1 = Q 2,0 Q 2,1 (x2 , x3 ), y2 = Q 2,0 Q 2,1 (x1 , x3 ), y3 = Q 2,0 Q 2,1 (x1 , x2 ) Then Q 3,0 (y1 , y2 , y3 ) = Q 33,0 Q 3,1 Q 23,2 + Q 53,0 , Q 3,1 (y1 , y2 , y3 ) = Q 23,0 Q 43,2 , Q 3,2 (y1 , y2 , y3 ) = Q 23,1 Q 23,2 + Q 23,0 Q 3,1 Next, we study the image of transfer on multilinear and alternating invariants in general The following is also numbered as Proposition 1.7(i) in the introduction Proposition 5.6 If Q ∈ Dm is multilinear and alternating then trm,n (Q) = 123 V T N Quỳnh, L X Trường Proof From Proposition 4.5, it is sufficient to prove trm,n ωβ = with β = (s0 , s1 , , sm−1 ) such that si = s j for all i = j By the definition of trm,n we have trm,n ωβ = ωβ (W ) W ≤V∗ ,dim W =m Since ωβ is multilinear and alternating, for each m-dimensional subspace W of V∗ , there exist scalars ci1 , ,im such that ωβ (W ) = ci1 , ,im ωβ (xi1 , , xim ) 1≤i < We have s0 −s1 Q = tr2,3 Q 22,0 = x12 x22 s0 −s1 x32 Q 2,1 s0 −s1 +1 2s1 + (symmetried) 2s1 [as shown in the proof of Remark 7.3(i)] = x12 s1 +1 s0 x22 x32 s0 +1 + (symmetried) From Proposition 4.5, we see that Q is the multilinear and alternating invariant ωβ with β = (s1 + 1, s0 , s0 + 1) So, by Proposition 5.6, we get tr3,n (Q) = Case s0 − s1 = or Following Remark 7.3(ii), we get Q = Thus tr3,n (Q) = 123 On the transfer between the Dickson algebras Case s0 − s1 < We have s1 −s0 Q = tr2,3 Q 2,0 Q 22,1 = x1 x22 x32 s1 −s0 +1 2s0 + (symmetried) 2s0 [as shown in the proof of Remark 7.3(ii)] s0 = x12 x22 s0 +1 x32 s1 +1 + (symmetried) Similarly, from Proposition 4.5, we obtain that Q is the multilinear and alternating invariant ωβ with β = (s0 , s0 +1, s1 +1) So, by Proposition 5.6, we get tr3,n (Q) = The proposition is proved Acknowledgments We would like to thank Phạm H -Dăng, who wrote an undergraduate thesis in 2011 under the guidance of Prof Nguyễn H V Hưng on transfers, and is no longer studying Maths, for valuable discussion on Theorem 1.2 We also thank to Prof Nguyễn H V Hưng for many helpful suggestions References - ăng, P.H.: The transfer between the Dickson algebras Undergraduate thesis, VNU University of D Science (2011) Dickson, L.E.: A fundamental system of invariants of the general modular linear group with a solution of the form problem Trans Amer Math Soc 12, 75–98 (1911) Hưng, N.H.V.: The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra Math Ann 353, 827–866 (2012) Hưng, N.H.V., Peterson, F.P.: Spherical classes and the Dickson algebra Math Proc Camb Philos Soc 124, 253–264 (1998) Mùi, Huỳnh: Modular invariant theory and cohomology algebras of symmetric groups J Fac Sci Univ Tokyo Sect IA Math 22, 319–369 (1975) 123 ... im,m m+1 123 m j=0 i j ) + (smaller terms) On the transfer between the Dickson algebras From the above conclusion, the powers of xm+1 in the maximal monomial of Q m+1,0 (y1 , , ym+1 ) is 2m... group with a solution of the form problem Trans Amer Math Soc 12, 75–98 (1911) Hưng, N.H.V.: The homomorphisms between the Dickson Mùi algebras as modules over the Steenrod algebra Math Ann 353,... As a consequence, the right hand side of (1) is equal to zero On the other hand, by Lemma 3.2, tr1,n (x k ) = as k < 2n − Therefore, we get (1) for this case 123 On the transfer between the Dickson