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9LHWQDP -RXUQDOR I 0 $ 7 + 0 $ 7 , & 6 ‹ 9$67 On an Invariant-Theoretic Description Nguyen Sum Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Binh Din

9LHWQDP -RXUQDO

R I

0 $ 7 + ( 0 $ 7 , & 6

‹ 9$67 

On an Invariant-Theoretic Description

Nguyen Sum

Department of Mathematics, University of Quy Nhon,

170 An Duong Vuong, Quy 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-panalogue of the Lomonaco invariant-theoretic description of the lambda algebra for pan odd prime More pre-cisely, using modular invariants of the general linear groupGL n = GL(n, F p and its Borel subgroupB n, we construct a differential algebraQ −which is isomorphic to the lambda algebraΛ = Λp

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 Λp, 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.

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 = Λp Here and in what follows,Fpdenotes the

prime field of p elements Recall that, Λ p is the E1-term of the Adams spectral

sequence of spheres for p an odd prime, whose E2-term is Ext∗ A(p)(Fp , 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, let E n be an elementary abelian p-group of rank n, and let

H ∗ (BE n ) = E(x1, x2, , x n)⊗ F p (y1, y2, , 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 fieldFp of p elements.

Let GL n = GL(n, F p ) and B 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

2 and let

Φn = 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 Φn We recall here some results on the invariant rings Γn= ΦGL n

n and Δn= ΦB n n. Let L k,s and M k,sdenote the following graded determinants (in the sense of

Mui [3])

L k,s=

















y1 y2 y k

y p1 y2p y p k

.

. .

y1p s−1 y2p s−1 y p k s−1

y p1s+1 y2p s+1 y p k s+1

.

. .

y p1k y2p k y p k k

















,

M k,s=



















x1 x2 x k

y1 y2 y k

y1p y p2 y k p

.

. .

y p1s−1 y2p s−1 y k p s−1

y p1s+1 y2p s+1 y p k s+1

. . .

y1p k−1 y p2k−1 y k p k−1



















.

for 0≤ s ≤ k ≤ n and M k,k = 0 We set L k = L k,k , 1 ≤ k ≤ n, L0= 1 Recall

that L k is invertible in Φn.

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 V2V1.

From the results in Dickson [2] and Mui [3, 4.17] we observe

Theorem 1.1 (see Singer [9])

Γn = 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 k p−1 , 1 ≤ k ≤ n.

Then we have

Theorem 1.2 (see Li–Singer [7])

Δn = E(N1, N2, , N n)⊗ F p (W1±1 , W2±1 , , W n ±1 ).

For latter use, we set

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

Corollary 1.3.

Δn = E(t1, t2, , t n)⊗ F p (w ±11 , w ±12 , , w n ±1 ).

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

n≥0Δn Here, by convention, Δ0=Fp This is a direct sum of vector spaces over F p .

Remark For I = (ε1, ε2, , ε n , i1, i2, , i n ) with ε j = 0, 1, i j ∈ Z, set

w I = t ε1

1 t ε2

2 t ε n

n w i1+ε1

1 w i2+ε2

2 w i n +ε n

even in the case when some of ε j or i j are zero For example, the element

t1 ∈ Δ2 will be written as t1t0w0w0, to be distinguished from t1 ∈ Δ1, since

t1= t1t0w0w0 For any n > 0 we have a monomial

t01t02 t0n w01w02 w0n ∈ Δ n

which is the identity of Δn 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, : Δk ⊗ Δ  → Δ k+

by setting

μ k, (t ε1

1 t ε2

2 t ε k

k w i1+ε1

1 w i2+ε2

2 w i k +ε k

k ⊗ t σ1

1 t σ2

2 t σ 

 w j1+σ1

1 w j2+σ2

2 w j  +σ 

= t ε1

1 t ε2

2 t ε k

k t σ1

k+1 t σ2

k+2 t σ 

k+ w i1+ε1

1 w i2+ε2

2 w i k +ε k

k w k+1 j1+σ1w j k+22+σ2 w j  +σ 

k+ ,

for any i1, i2, , i k , j1, j2, , j  ∈ Z, ε1, ε2, , ε k , σ1, σ2, , 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.

Definition 2.2 Let Γ denote the two-sides ideal of Δ generated by all elements

of the forms

t01t02w1−1 w02Q a 2,0 Q b 2,1 ,

t01t02w1−1 w02R 2,0 Q a 2,0 Q b 2,1 − R 2,1 Q a 2,0 Q b 2,1 ,

2t01t02w1w20R 2,1 Q a 2,0 Q b 2,1 − R 2,0 Q a 2,0 Q b 2,1 ,

t0t0w1w0R 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: Δn → Δ n+1

by setting

¯

δ n (x) = −t1w −11 ∗ x + (−1) dim x x ∗ t1w −11 ,

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) is a sum of elements of the form

−t1w −11 ∗ u i ∗ s i ∗ z i+ (−1) dim u u i ∗ s i ∗ z i ∗ t1w1−1

Since t1w1−1 ∗ u i ∈ Δ n i+1, z i ∗ t1w −11 ∈ 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) = t1t2w −11 w −12 ∗ x − x ∗ t1t2w −11 w −12 .

A direct computation using Proposition 1.4 shows that

R 2,0 Q −1 2,0 = t1t02w −11 w20+ t01t2w01w2−1 ,

R 2,1 Q −1 2,0 = t01t2w −11 w2−1

From these, we have

t01t02w1w20R 2,0 R 2,1 Q −2 2,0 = t1t2w −11 w −12 .

Hence we obtain

¯

δ¯ δ(x) = t0t0w1w0R 2,0 R 2,1 Q −2 2,0 ∗ x − x ∗ t0t0w1w0R 2,0 R 2,1 Q −2 2,0 ∈ Γ.

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 − 1 and y 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 ε1

q1x ε2

q2) =

n



k=0



n k



x ε1

q1−k x ε2

q2−n+k Here n

k



denotes the binomial coefficient.

We define a homomorphism of algebras π : T → Q by setting

π(x i+1 ) = t1w1i−1 , π(y i+1 ) = t01w i1, i ∈ Z.

That means π(x ε i+1 ) = t ε1w i−ε1 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

By a direct computation we obtain

Q a 2,0 Q b 2,1=

b



k=0



b k



t01t02w p(a+b)−b+k1 w a+b−k2

R 2,0 Q a 2,0 Q b 2,1=

b



k=0



b k



t1t02w p(a+b+1)−b+k−11 w a+b+1−k2

+

b



k=0



b k



t01t2w1p(a+b+1)−b+k w a+b−k2

R 2,1 Q a 2,0 Q b 2,1=

b



k=0



b k



t01t2w p(a+b+1)−b+k−11 w a+b−k2

R 2,0 R 2,1 Q a 2,0 Q b 2,1=

b



k=0



b k



t1t2w p(a+b+2)−b+k−21 w a+b+1−k2 .

Using Lemma 2.4 and the definition of π we have

π(D n (y pi y i+1 )) = πn

k=0



n k



y pi−n+k y i+1−k

=

n



k=0



n k



t01t02w1pi−n+k−1 w2i−k

= t01t02w1−1 w02

n



k=0



n k



t01t02w pi−n+k1 w i−k2

= t01t02w1−1 w02Q i−n 2,0 Q n 2,1

= 0 in Q.

By an argument analogous to the previous one, we get

π(D n (x pi y i+1 )) = t01t02w1−1 w20R 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 in Q

π(D n (y pi+1 x i+1 − x pi+1 y i+1 )) = (2t01t02w1w02R 2,1 − R 2,0 )Q i−n−1 2,0 Q n 2,1 = 0 in Q

π(D n (x pi+1 x i+1)) =−t0

1t02w1w02R 2,0 R 2,1 Q i−n−2 2,0 Q n 2,1 = 0 in Q.

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 overFp with generators λ i−1

of dimension −2(p − 1)i + 1 and μ i−1 of dimension−2(p − i), i ≥ 0, subject to

the relations:

n



k=0



n k



n



k=0



n k



μ k+pi−1 λ i+n−k−1 − λ k+pi−1 μ i+n−k−1

n



k=0



n k



n



k=0



n k



for i, n ≥ 0 By Λ we mean the subalgebra of ¯ Λ generated by λ i−1 , i > 0 and

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(ε1, ε2, i, n) =

n



k=0



n k



λ ε1

k+pi−ε2λ ε2

i+n−k−1 − ε2(1− ε1)λ ε2

k+pi−ε2λ ε1

i+n−k−1



,

for any ε1, ε2, i, n with ε1, ε2 = 0, 1 and i, n ≥ 0 Then the defining relations

(1) - (4) become

Then we can consider Λ as the free graded associative algebra overFp with

generators λ ε i−1 , i ≥ ε, subject to the relation (5) with i ≥ −ε1

Definition 3.1 A sequence I = (ε1, ε2, , ε n , i1, i2, , 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 ε1

i1−1 λ ε2

i2−1 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 ,

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(λ0) = 0, ¯ d(μ −1 ) = 0, ¯ d(μ0) = λ0μ −1 − μ −1 λ0,

¯

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 ε1, ε2= 0, 1, n ≥ 0, i ∈ Z, we set

x(ε1, ε2, i, n) = D n

x ε1

pi+ε2x ε2

i+1 − ε2(1− ε1)x ε2

pi+ε2x ε1

i+1



.

Then the defining relations of Q become

So we can consider Q − as the free graded associative algebra overFp with

generators x ε i+1 , i ≤ −ε, subject to the relation (6) with i ≤ −ε1

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(ε1, ε2, i, n)

= x(ε1, ε2, −i, n)

for any ε1, ε2= 0, 1, i, n ≥ 0, i ≥ ε1 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(ε1, ε2, i, n)

= λ(ε1, ε2, −i, n),

for any ε1, ε2 = 0, 1, n ≥ 0, i ≤ −ε1 So, the homomorphism Ψ is well defined Obviously, we have

Φ◦ Ψ = 1 Q − , Ψ ◦ Φ = 1Λ.

Hence Φ is an isomorphism of algebras

Finally we prove that Φ preserves the differential structure We have

Φ(δ(λ n−1)) = Φn−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

4 An Additive Basis for Q

For J = (ε1, ε2, , ε n , j1, j2, , j n ), with ε k = 0, 1, j k ∈ Z, k = 1, 2, , n, we

set

x J = x ε1

j1 +1x ε2

j2 +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, then x J is admissible if and only

if λ −J is admissible in Λ Here

−J = (ε1, ε2, , ε n , −j1, −j2, , −j n ).

From the relation D n (x pi+1 x i+1 ) = 0 in Q we have

x pi−n+1 x i+1=− n−1

k=0



n k



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)