Báo cáo toán học: " On an Invariant-Theoretic Description of the Lambda Algebra" ppsx

Vietnam Journal of Mathematics 33:1 (2005) 19–32 On an Inva riant-Theoretic Description of the Lambda Algebra * Nguyen Sum Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Q uy Nhon, Binh Dinh, Vietnam Received May 12, 2003 Revised September 15, 2004 Dedicated to Professor Hu`ynh M`ui on the occasion of his sixtieth birthday Abstract The purpose of this paper is to give a mod-p analogue of the Lomonaco invariant-theoretic description of the lambda algebra for p an odd prime. More pre- cisely, using modular invariants of the general linear group GL n = GL(n, F p ) and its Borel subgroup B n , we construct a differential algebra Q − which is isomorphic to the lambda algebra Λ=Λ p . Introduction For the last few decades, the modular invariant theory has been playing an important role in stable homotopy theory. Singer [9] gave an interpretation for the dual of the lambda algebra Λ p , which was introduced by the six authors [1], in terms of modular invariant theory of the general linear group at the prime p = 2. In [8], Hung and the author gave a mod-p analogue of the Singer invariant-theoretic description of the dual of the lambda algebra for p an odd prime. Lomonaco [6] also gave an interpretation for the lambda algebra in terms of modular invariant theory of the Borel subgroup of the general linear group at p =2. ∗ This work was supported in part by the Vietnam National Research Program Grant 140801. 20 Nguyen Sum The purpose of this paper is to give a mod-p analogue of the Lomonaco invariant-theoretic description of the lambda algebra for p an odd prime. More precisely, using modular invariants of the general linear group GL n = GL(n, F p ) and its Borel subgroup B n , we construct a differential algebra Q − which is iso- morphic to the lambda algebra Λ = Λ p . Here and in what follows, F p denotes the prime field of p elements. Recall that, Λ p is the E 1 -term of the Adams spectral sequence of spheres for p an odd prime, whose E 2 -term is Ext ∗ A(p) (F p , F p )where A(p) denotes the mod p Steenrod algebra, and E ∞ -term is a graded algebra associated to the p-primary components of the stable homotopy of spheres. It should be noted that the idea for the invariant-theoretic description of the lambda algebra is due to Lomonaco, who realizes it for p = 2 in [6]. In this paper, we develope of his work for p any odd prime. Our main contributions are the computations at odd degrees, where the behavior of the lambda algebra is completely different from that for p =2. The paper contains 4 sections. Sec. 1 is a preliminary on the modular invari- ant theory and its localization. In Sec. 2 we construct the differential algebra Q by using modular invariant theory and show that Q can be presented by a set of generators and some relations on them. In Sec. 3 we recall some results on the lambda algebra and show that it is isomorphic to a differential subalgebra Q − of Q. Finally, in Sec. 4 we give an F p -vector space basis for Q. 1. Preliminaries on the Invariant Theory For an odd prime p,letE n be an elementary abelian p-group of rank n,andlet H ∗ (BE n )=E(x 1 ,x 2 , ,x n ) ⊗ F p (y 1 ,y 2 , ,y n ) be the mod-p cohomology ring of E n . It is a tensor product of an exterior algebra on generators x i of dimension 1 with a polynomial algebra on generators y i of dimension 2. Here and throughout the paper, the coefficients are taken over the prime field F p of p elements. Let GL n = GL(n, F p )andB n be its Borel subgroup consisting of all invert- ible upper triangular matrices. These groups act naturally on H ∗ (BE n ). Let S be the multiplicative subset of H ∗ (BE n ) generated by all elements of dimension 2andlet Φ n = H ∗ (BE n ) S be the localization of H ∗ (BE n ) obtained by inverting all elements of S.The action of GL n on H ∗ (BE n ) extends to an action of its on Φ n . We recall here some results on the invariant rings Γ n =Φ GL n n and Δ n =Φ B n n . Let L k,s and M k,s denote the following graded determinants (in the sense of Mui [3]) On an Invariant-Theoretic Description of the Lambda Algebra 21 L k,s =                 y 1 y 2 y k y p 1 y p 2 y p k . . . . . . . . . y p s−1 1 y p s−1 2 y p s−1 k y p s+1 1 y p s+1 2 y p s+1 k . . . . . . . . . y p k 1 y p k 2 y p k k                 , M k,s =                   x 1 x 2 x k y 1 y 2 y k y p 1 y p 2 y p k . . . . . . . . . y p s−1 1 y p s−1 2 y p s−1 k y p s+1 1 y p s+1 2 y p s+1 k . . . . . . . . . y p k−1 1 y p k−1 2 y p k−1 k                   . for 0 ≤ s ≤ k ≤ n and M k,k =0. WesetL k = L k,k , 1 ≤ k ≤ n, L 0 = 1. Recall that L k is invertible in Φ n . As is well known L k,s is divisible by L k . Dickson invariants Q k,s and Mui invariants R k,s ,V k , 0 ≤ s ≤ k, are defined by Q k,s = L k,s /L k ,R k,s = M k,s L p−2 k ,V k = L k /L k−1 . Note that dim Q k,s =2(p k − p s ), dim R k,s =2(p k − p s ) −1, dim V k =2p k−1 , Q k,0 = L p−1 k ,L k = V k V k−1 V 2 V 1 . From the results in Dickson [2] and Mui [3, 4.17] we observe Theorem 1.1. (see Singer [9]) Γ n = E(R n,0 ,R n,1 , ,R n,n−1 ) ⊗ F p (Q ±1 n,0 ,Q n,1 , ,Q n,n−1 ). Following Li–Singer [7], we set N k = M k,k−1 L p−2 k ,W k = V p−1 k , 1 ≤ k ≤ n. Then we have Theorem 1.2. (see Li–Singer [7]) Δ n = E(N 1 ,N 2 , ,N n ) ⊗ F p (W ±1 1 ,W ±1 2 , ,W ±1 n ). Forlatteruse,weset t k = N k /Q p−1 k−1,0 ,w k = W k /Q p−1 k−1,0 , 1 ≤ k ≤ n. Observe that dim t k =2p − 3, dim w k =2p − 2. From Theorem 1.2 we obtain 22 Nguyen Sum Corollary 1.3. Δ n = E(t 1 ,t 2 , ,t n ) ⊗ F p (w ±1 1 ,w ±1 2 , ,w ±1 n ). Moreover, from Dickson [2], Mui [3], we have Proposition 1.4. (i) Q n,s = Q p n−1,s−1 + Q p−1 n−1,0 Q n−1,s w n , (ii) R n,s = Q p−1 n−1,0 (R n−1,s w n + Q n−1,s t n ). 2. The Algebra Q In this section, we construct the differential algebra Q by using modular invari- ant theory. In Sec. 4, we will show that the lambda algeba is isomorphic to a subalgebra of Q. Definition 2.1. Let Δ n be as in Sec. 1.Set Δ= ⊕ n≥0 Δ n . Here, by convention, Δ 0 = F p . This is a direct sum of vector spaces over F p . Remark. For I =(ε 1 ,ε 2 , ,ε n ,i 1 ,i 2 , ,i n )withε j =0, 1,i j ∈ Z,set w I = t ε 1 1 t ε 2 2 t ε n n w i 1 +ε 1 1 w i 2 +ε 2 2 w i n +ε n n , even in the case when some of ε j or i j are zero. For example, the element t 1 ∈ Δ 2 will be written as t 1 t 0 2 w 0 1 w 0 2 , to be distinguished from t 1 ∈ Δ 1 ,since t 1 = t 1 t 0 2 w 0 1 w 0 2 . For any n>0wehaveamonomial t 0 1 t 0 2 t 0 n w 0 1 w 0 2 w 0 n ∈ Δ n which is the identity of Δ n . All these elements are distinct in Δ. Now we equip Δ with an algebra structure as follows. For any non-negative integers k,, we define an isomorphism of algebras μ k, :Δ k ⊗ Δ  → Δ k+ by setting μ k, (t ε 1 1 t ε 2 2 t ε k k w i 1 +ε 1 1 w i 2 +ε 2 2 w i k +ε k k ⊗ t σ 1 1 t σ 2 2 t σ   w j 1 +σ 1 1 w j 2 +σ 2 2 w j  +σ   ) = t ε 1 1 t ε 2 2 t ε k k t σ 1 k+1 t σ 2 k+2 t σ  k+ w i 1 +ε 1 1 w i 2 +ε 2 2 w i k +ε k k w j 1 +σ 1 k+1 w j 2 +σ 2 k+2 w j  +σ  k+ , for any i 1 ,i 2 , ,i k ,j 1 ,j 2 , ,j  ∈ Z,ε 1 ,ε 2 , ,ε k ,σ 1 ,σ 2 , ,σ  =0, 1. We assemble μ k, ,k,≥ 0, to obtain a multiplication μ :Δ⊗ Δ → Δ. This multiplication makes Δ into an algebra. For simplicity, we denote μ(x ⊗ y)=x ∗ y for any elements x, y ∈ Δ. On an Invariant-Theoretic Description of the Lambda Algebra 23 Definition 2.2. Let Γ denote the two-sides ideal of Δ generated by all elements of the forms t 0 1 t 0 2 w −1 1 w 0 2 Q a 2,0 Q b 2,1 , t 0 1 t 0 2 w −1 1 w 0 2 R 2,0 Q a 2,0 Q b 2,1 − R 2,1 Q a 2,0 Q b 2,1 , 2t 0 1 t 0 2 w 1 w 0 2 R 2,1 Q a 2,0 Q b 2,1 − R 2,0 Q a 2,0 Q b 2,1 , t 0 1 t 0 2 w 1 w 0 2 R 2,0 R 2,1 Q a 2,0 Q b 2,1 , where a, b ∈ Z,b≥ 0. We define Q =Δ/Γ to be the quotient of Δ by the ideal Γ. For any non-negative integer n, we define a homomorphism ¯ δ n :Δ n → Δ n+1 by setting ¯ δ n (x)=−t 1 w −1 1 ∗ x +(−1) dim x x ∗ t 1 w −1 1 , for any homogeneous element x ∈ Δ n . By assembling ¯ δ n ,n ≥ 0, we obtain an endomorphism ¯ δ :Δ→ Δ. Theorem 2.3. The endomorphism ¯ δ :Δ→ Δ induces an endomorphism δ : Q → Q which is a differential. Proof. Let u ∈ Δ n be a homogeneous element and suppose u ∈ Γ. From the definition of Γ we see that u is a sum of elements of the form u i ∗ s i ∗ z i , where u i ∈ Δ n i ,z i ∈ Δ n−n i −2 and s i is one of the elements given in Definition 2.2. Then ¯ δ(u)isasumofelementsoftheform −t 1 w −1 1 ∗ u i ∗ s i ∗ z i +(−1) dim u u i ∗ s i ∗ z i ∗ t 1 w −1 1 . Since t 1 w −1 1 ∗ u i ∈ Δ n i +1 ,z i ∗ t 1 w −1 1 ∈ Δ n−n i −1 , we obtain ¯ δ(u) ∈ Γ. So, ¯ δ induces an endomorphism δ : Q → Q. Now we prove that δδ = 0. It suffices to check that if x ∈ Δ n is a homogeneous element then ¯ δ ¯ δ(x) ∈ Γ. In fact, from the definition of ¯ δ we have ¯ δ ¯ δ(x)=t 1 t 2 w −1 1 w −1 2 ∗ x − x ∗ t 1 t 2 w −1 1 w −1 2 . A direct computation using Proposition 1.4 shows that R 2,0 Q −1 2,0 = t 1 t 0 2 w −1 1 w 0 2 + t 0 1 t 2 w 0 1 w −1 2 , R 2,1 Q −1 2,0 = t 0 1 t 2 w −1 1 w −1 2 . 24 Nguyen Sum From these, we have t 0 1 t 0 2 w 1 w 0 2 R 2,0 R 2,1 Q −2 2,0 = t 1 t 2 w −1 1 w −1 2 . Hence we obtain ¯ δ ¯ δ(x)=t 0 1 t 0 2 w 1 w 0 2 R 2,0 R 2,1 Q −2 2,0 ∗ x − x ∗ t 0 1 t 0 2 w 1 w 0 2 R 2,0 R 2,1 Q −2 2,0 ∈ Γ. The theorem is proved.  Now we give a new system of generators for Q. Let T be the free associative algebra over F p generated by x i+1 of degree 2(p − 1)i − 1andy i+1 of degree 2(p − 1)i, for any i ∈ Z. It is easy to see that there exists a unique derivation D : T → T satisfying D(x i )=x i−1 ,D(y i )=y i−1 ,i∈ Z. (Recall that D is called a derivation if D(uv)=D(u)v+uD(v), for any u, v ∈ T .) Denote by D n = D ◦ D ◦ ◦ D the composite of n-copies of D. For simplicity, we set x ε i =  x i ,ε=1 y i ,ε=0. By induction on n we easily obtain Lemma 2.4. Under the above notation, we have D n (x ε 1 q 1 x ε 2 q 2 )= n  k=0  n k  x ε 1 q 1 −k x ε 2 q 2 −n+k . Here  n k  denotes the binomial coefficient. We define a homomorphism of algebras π : T → Q by setting π(x i+1 )=t 1 w i−1 1 ,π(y i+1 )=t 0 1 w i 1 ,i∈ Z. That means π(x ε i+1 )=t ε 1 w i−ε 1 for any i ∈ Z,ε=0, 1. Proposition 2.5. The homomorphism π : T → Q is an epimorphism. Its kernel is the two-sides ideal of T generated by all elements of the forms D n (y pi y i+1 ), D n (x pi y i+1 ), D n (y pi+1 x i+1 − x pi+1 y i+1 ), D n (x pi+1 x i+1 ), with n ≥ 0,i∈ Z. Proof. It is easy to see that π is an epimorphim. Now we prove the remaining part of the proposition. On an Invariant-Theoretic Description of the Lambda Algebra 25 By a direct computation we obtain Q a 2,0 Q b 2,1 = b  k=0  b k  t 0 1 t 0 2 w p(a+b)−b+k 1 w a+b−k 2 R 2,0 Q a 2,0 Q b 2,1 = b  k=0  b k  t 1 t 0 2 w p(a+b+1)−b+k−1 1 w a+b+1−k 2 + b  k=0  b k  t 0 1 t 2 w p(a+b+1)−b+k 1 w a+b−k 2 R 2,1 Q a 2,0 Q b 2,1 = b  k=0  b k  t 0 1 t 2 w p(a+b+1)−b+k−1 1 w a+b−k 2 R 2,0 R 2,1 Q a 2,0 Q b 2,1 = b  k=0  b k  t 1 t 2 w p(a+b+2)−b+k−2 1 w a+b+1−k 2 . Using Lemma 2.4 and the definition of π we have π(D n (y pi y i+1 )) = π  n  k=0  n k  y pi−n+k y i+1−k  = n  k=0  n k  t 0 1 t 0 2 w pi−n+k−1 1 w i−k 2 = t 0 1 t 0 2 w −1 1 w 0 2 n  k=0  n k  t 0 1 t 0 2 w pi−n+k 1 w i−k 2 = t 0 1 t 0 2 w −1 1 w 0 2 Q i−n 2,0 Q n 2,1 =0 inQ. By an argument analogous to the previous one, we get π(D n (x pi y i+1 )) = t 0 1 t 0 2 w −1 1 w 0 2 R 2,0 Q i−n−1 2,0 Q n 2,1 − R 2,1 Q i−n−1 2,0 Q n 2,1 =0 inQ π(D n (y pi+1 x i+1 − x pi+1 y i+1 )) = (2t 0 1 t 0 2 w 1 w 0 2 R 2,1 − R 2,0 )Q i−n−1 2,0 Q n 2,1 =0 inQ π(D n (x pi+1 x i+1 )) = −t 0 1 t 0 2 w 1 w 0 2 R 2,0 R 2,1 Q i−n−2 2,0 Q n 2,1 =0 inQ. From these and the definition of Γ we obtain the proposition.  3. The Lambda Algebra and the Modular Invariant Theory In this section, we show that the lambda algebra, which is introduced by the six authors of [1], is isomorphic to a subalgebra of Q. Let ¯ Λ denote the graded free associative algebra over F p with generators λ i−1 of dimension −2(p − 1)i +1andμ i−1 of dimension −2(p − i),i≥ 0, subject to 26 Nguyen Sum the relations: n  k=0  n k  λ k+pi−1 λ i+n−k−1 =0 (1) n  k=0  n k   μ k+pi−1 λ i+n−k−1 − λ k+pi−1 μ i+n−k−1  =0 (2) n  k=0  n k  λ k+pi μ i+n−k−1 =0 (3) n  k=0  n k  μ k+pi μ i+n−k−1 =0 (4) for i, n ≥ 0. By Λ we mean the subalgebra of ¯ Λ generated by λ i−1 ,i > 0and μ i−1 ,i≥ 0. We note that this definition is the same as that given in [1], but we are writing the product in the order opposite to that used in [1]. For simplicity, we denote λ ε i =  λ i ,ε=1 μ i ,ε=0, for any i ≥−1. We set λ(ε 1 ,ε 2 ,i,n)= n  k=0  n k   λ ε 1 k+pi−ε 2 λ ε 2 i+n−k−1 − ε 2 (1 − ε 1 )λ ε 2 k+pi−ε 2 λ ε 1 i+n−k−1  , for any ε 1 ,ε 2 ,i,n with ε 1 ,ε 2 =0, 1andi, n ≥ 0. Then the defining relations (1) - (4) become λ(ε 1 ,ε 2 ,i,n)=0. (5) Then we can consider Λ as the free graded associative algebra over F p with generators λ ε i−1 ,i≥ ε, subject to the relation (5) with i ≥−ε 1 . Definition 3.1. AsequenceI =(ε 1 ,ε 2 , ,ε n ,i 1 ,i 2 , ,i n ),ε j =0, 1,i j ≥ 0, is said to be admissible if pi j ≥ i j+1 + ε j , 1 ≤ j<n, and i n ≥ ε n . In this case, the associated monomial λ I = λ ε 1 i 1 −1 λ ε 2 i 2 −1 λ ε n i n −1 is also said to be admissible Theorem 3.2. (Bousfield et al. [1]) The admissible monomials form an additive basis for Λ. Definition 3.3. The homomorphism ¯ d : ¯ Λ → ¯ Λ is defined by ¯ d(x)=−λ −1 x +(−1) dim x xλ −1 , On an Invariant-Theoretic Description of the Lambda Algebra 27 for any homogeneous element x ∈ ¯ Λ. In ¯ Λ, we have λ −1 λ −1 = 0, hence ¯ d ¯ d =0. So ¯ d is a differential on ¯ Λ. From the defining relations (1)-(4) we obtain ¯ d(λ 0 )=0, ¯ d(μ −1 )=0, ¯ d(μ 0 )=λ 0 μ −1 − μ −1 λ 0 , ¯ d(λ n−1 )= n−1  k=1  n k  λ k−1 λ n−k−1 , ¯ d(μ n−1 )=λ n−1 μ −1 + n−1  k=1  n k   λ k−1 μ n−k−1 − μ k−1 λ n−k−1  − μ −1 λ n−1 , for any n ≥ 2. From these, we obtain ¯ d(λ ε n−1 ) ∈ Λ,n ≥ ε, so ¯ d passes to a differential d on Λ. Now we describe the algebra Λ in terms of modular invariants. Definition 3.4. We define Q − to be the subalgebra of Q generated by all ele- ments x ε i+1 with i ≤−ε. For any ε 1 ,ε 2 =0, 1,n≥ 0,i∈ Z,weset x(ε 1 ,ε 2 ,i,n)=D n  x ε 1 pi+ε 2 x ε 2 i+1 − ε 2 (1 − ε 1 )x ε 2 pi+ε 2 x ε 1 i+1  . Then the defining relations of Q become x(ε 1 ,ε 2 ,i,n)=0. (6) So we can consider Q − as the free graded associative algebra over F p with generators x ε i+1 ,i≤−ε, subject to the relation (6) with i ≤−ε 1 . Theorem 3.5. As a graded differential algebra, Λ is isomorphic to Q − . Proof. We define a homomorphism of algebras Φ:Λ→ Q − by setting Φ(λ ε i−1 )=x ε −i+1 , for any i ≥−ε. From the definition of Q − we easily obtain Φ  λ(ε 1 ,ε 2 ,i,n)  = x(ε 1 ,ε 2 , −i, n) for any ε 1 ,ε 2 =0, 1,i,n≥ 0,i≥ ε 1 . Hence, the homomorphism Φ is well defined. Now we define a homomorphism of algebras Ψ:Q − → Λ, by setting Ψ(x ε i+1 )=λ ε −i−1 , for any i ≤−ε. It is easy to check that Ψ  x(ε 1 ,ε 2 ,i,n)  = λ(ε 1 ,ε 2 , −i, n), 28 Nguyen Sum for any ε 1 ,ε 2 =0, 1,n ≥ 0,i ≤−ε 1 . So, the homomorphism Ψ is well defined. Obviously, we have Φ ◦ Ψ=1 Q − , Ψ ◦ Φ=1 Λ . Hence Φ is an isomorphism of algebras. Finally we prove that Φ preserves the differential structure. We have Φ(δ(λ n−1 )) = Φ  n−1  k=1  n k  λ k−1 λ n−k−1  = n−1  k=1  n k  x −k+1 x k−n+1 = d(x −n+1 ) = dΦ(λ n−1 ), for any n ≥ 1. Similarly, we obtain Φ(δ(μ n−1 )) = dΦ(μ n−1 ), for any n ≥ 0. So Φ is an isomorphism of differential algebras. The theorem is proved.  4. An Additive Basis for Q For J =(ε 1 ,ε 2 , ,ε n ,j 1 ,j 2 , ,j n ), with ε k =0, 1,j k ∈ Z,k =1, 2, ,n, we set x J = x ε 1 j 1 +1 x ε 2 j 2 +1 x ε n j n +1 . Definition 4.1. The monomial x J is said to be admissible if j k ≥ pj k+1 + ε k+1 ,k =1, 2, ,n. Denote by J n the set of all sequences J such that x J is admissible. We note that if j k ≤−ε k ,k=1, 2, ,n,thenx J is admissible if and only if λ −J is admissible in Λ. Here −J =(ε 1 ,ε 2 , ,ε n , −j 1 , −j 2 , ,−j n ). From the relation D n (x pi+1 x i+1 )=0inQ we have x pi−n+1 x i+1 = − n−1  k=0  n k  x pi−k+1 x i+1−n+k . (7) Applying relations of the same form to those terms of the right hand side of (7) which are not admissible, after finitely many steps we obtain an expression of the form x pi−n+1 x i+1 =  a n,k x pi−k+1 x i+1−n+k , (8) [...].. .On an Invariant-Theoretic Description of the Lambda Algebra 29 where an, k ∈ Fp and all the monomials appearing on the right hand side are admissible (see the proof of Lemma 4.2) That means an, k = 0 if (p + 1)k ≥ pn By an argument analogous to the previous one, we get xpi−n yi+1 = bn,k xpi−k yi+1−n+k , xpi−n+1 yi+1 = (9) cn,k ypi−k+1... operations derived from modular invariants, Math Z 193 (1986) 151–163 32 Nguyen Sum 6 L A Lomonaco, Invariant theory and the total squaring operation, Ph D Thesis, Univ of Warwich, September, 1986 7 H H Li and W M Singer, Resolutions of modules over the Steenrod algebra and the classical theory of invariants, Math Z 18 (1982) 269–286 8 Nguyen H V Hung and Nguyen Sum, On Singer’s invariant-theoretic description. .. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans Amer Math Soc 12 (1911) 75–98 3 Huynh Mui, Modular invariant theory and the cohomology algebras of symmetric groups, J Fac Sci Univ Tokyo Sec IA Math 22 (1975) 319–369 4 Huynh Mui, Dickson invariants and Milnor basis of the Steenrod algebra, Colloq Math Soc Janos Bolyai, Topology and... + pn = p(n − ) > 0 Therefore in (8) the coefficient an, k such that an, k = 0, with the lowest possible k is an, 0 Hence an, k = 0 if k < 0 The main result of this section is Theorem 4.3 The set X = xJ : J ∈ Jn n≥0 is an Fp -vector space basis for Q Proof We first prove that X spans Q Let xJ be a monomial in Q We apply the relations (8)-(11) and Lemma 4.2 to the inadmissible pairs in xJ and after Nguyen Sum... number of steps we can write xJ as a linear combination of monomials of the form xJ xJ , where xJ is an admissible monomial involving generators xε with i > −ε and i+1 xJ is a monomial involving generators xε with i ≤ −ε Then xJ ∈ Q− i+1 Using Theorem 3.5 we get Ψ(xJ ) = αu λ−Ju , where αu ∈ Fp and λ−Ju is an admissible monomial in Λ From this we obtain xJ = αu xJu , where xJu is an admissible monomial... Nguyen H V Hung and Nguyen Sum, On Singer’s invariant-theoretic description of the lambda algebra: A mod p analogue, J Pure and Appl Algebra 99 (1995) 297–329 9 W M Singer, Invariant theory and the lambda algebra, Trans Amer Math Soc 280 (1983) 673–693 10 N E Steenrod and D B A Epstein, Cohomology operations, Ann of Math No 50, Princeton University Press, 1962 ... Definition 4.1) It is easy to see that the monomial xJ xJu is admissible in Q Therefore X spans Q We now prove that the set X is linearly independent Suppose that m au xJu = 0 in Q, u=1 with au ∈ Fp , Ju ∈ Jn , u = 1, 2, , m Then we have m au wJu ∈ Γ u=1 We order the set {wJ : J ∈ Jn } by agreeing that wJ1 > wJ2 if and only if J1 > J2 Here the order in Z2n is the antilexicographical one Suppose that there... there is an index u such that au = 0 Let wJ be the greatest monomial of all monomials wJu such that au = 0 and assume that J = n (ε1 , ε2 , , εn , j1 , j2 , , jn ) Since u=1 au wJu ∈ Γ, wJ is a term in the expression of elements of the form ε j j k−1 k−1 tε1 tk−1 w11 +ε1 wk−1 1 +εk−1 ε j k+2 ∗ z ∗ t1k+2 tεn n−k−2 w1 +εk+2 jn +εn wn−k−2 , where 1 ≤ k ≤ n − 2 and z is one of the elements... given in Definition 2.3 −1 0 If z = t0 t0 w1 w2 Qa Qb then 1 2 2,0 2,1 b z= j=0 b 0 0 p(a+b)−b+j−1 a+b−j w2 t t w j 1 2 1 Since wJ is the greatest monomial, from this we get jk = p(a + b) − b − 1, jk+1 = a + b, εk+1 = 0 Hence pjk+1 + εk+1 = p(a + b) > p(a + b) − b − 1 = jk On an Invariant-Theoretic Description of the Lambda Algebra 31 0 If z = t0 t0 w−1 w2 R2,0 Qa Qb − R2,1 Qa Qb then 1 2 2,0 2,1... 2) − b = jk Therefore xJ is inadmissible This contradicts the fact that xJ is admissible Hence, the theorem is proved ˜ Acknowledgements The author expresses his warmest thank to Professor Nguyˆn H V e ng for helpful suggestions which lead him to this paper Hu References 1 A K Bousfield, E B Curtis, D M Kan, D G Quillen, D L Rector, and J W Schlesinger, The mod-p lower central series and the Adams spectral . mod-p analogue of the Singer invariant-theoretic description of the dual of the lambda algebra for p an odd prime. Lomonaco [6] also gave an interpretation for the lambda algebra in terms of modular. (8) On an Invariant-Theoretic Description of the Lambda Algebra 29 where a n,k ∈ F p and all the monomials appearing on the right hand side are admissible (see the proof of Lemma 4.2). That means a n,k =0. Journal of Mathematics 33:1 (2005) 19–32 On an Inva riant-Theoretic Description of the Lambda Algebra * Nguyen Sum Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Q uy Nhon,