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The squaring operation on generators of the Dickson algebra.

S0305004109990405_char1-NGUYỄN H V HƯNG and VÕ T N QUỲNH

Mathematical Proceedings of the Cambridge Philosophical Society / Volume 148 / Issue 02 / March 2010, pp

267 - 288

DOI: 10.1017/S0305004109990405, Published online: 03 December 2009

Link to this article: http://journals.cambridge.org/abstract_S0305004109990405

How to cite this article:

NGUYỄN H V HƯNG and VÕ T N QUỲNH (2010) The squaring operation on -generators of the Dickson algebra Mathematical Proceedings of the Cambridge Philosophical Society, 148, pp 267-288 doi:10.1017/S0305004109990405

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Math Proc Camb Phil Soc (2010), 148, 267  Cambridge Philosophical Society 2009c

doi:10.1017/S0305004109990405

First published online 3 December 2009

267

BYNGUY ˆE˜ N H V HƯNGANDV ˜O T N QU `YNH

Department of Mathematics, Vietnam National University, Hanoi 334 Nguyˆe˜n Tr˜ai Street, Hanoi, Vietnam e-mail: nhvhung@vnu.edu.vn e-mail: quynhvtn@vnu.edu.vn (Received 10 March 2009; revised 8 September 2009)

Abstract

We study the squaring operation Sq0 on the dual of the minimal A-generators of the

Dickson algebra We show that this squaring operation is isomorphic on its image We alsogive vanishing results for this operation in some cases As a consequence, we prove that

the Lannes–Zarati homomorphism vanishes (1) on every element in any finite Sq0-family in

E xt A∗(F2, F2) except possibly the family initial element, and (2) on almost all known

ele-ments in the Ext group This verifies a part of the algebraic version of the classical conjecture

on spherical classes

Dedicated to William Singer on the occasion of his 65th birthday

1 Introduction and statement of results

Throughout the paper, the coefficient ring for homology and cohomology is alwaysF2, the

field of two elements LetVs be an s-dimensionalF2-vector space The general linear group

G L s := GL(V s ) acts regularly on V s and therefore on H∗(BV s ) Let P(F2 ⊗

G L s

H∗(BV s )) be

the submodule ofF2 ⊗

G L s

H∗(BV s ) consisting of all elements, which are annihilated by every

positive-degree operation in the mod 2 Steenrod algebra,A.

The subject of this paper is the squaring operation

which is defined by the first named author in [12] as an analogue of the classical squaring

operation on the cohomology of the Steenrod algebra, E xt A∗(F2, F2).

The most important property of the squaring operation is that it commutes with the

clas-sical squaring operation Sq0on E xt A∗(F2, F2) through the Lannes–Zarati homomorphism

ϕ s : Ext s ,s+δ

A (F2, F2) −→ P(F2 ⊗

G L s

H∗(BV s )) δ ,

for any s (see [15]) Therefore the investigation of the squaring operation is useful to the

study of the Lannes–Zarati homomorphism

† Supported in part by a grant of the NAFOSTED.

268 NGUYEˆ ˜N H V HƯNG ANDV ˜O T N QUYNH`

The Lannes–Zarati homomorphism, defined in [23], is the one corresponding to an

associ-ated graded of the Hurewicz map H : π s

∗(S0) % π∗(Q0S0) → H∗(Q0S0) So, the following

is an algebraic version of the conjecture on spherical classes

CONJECTURE 1·1 ([12]) ϕ s = 0 in any positive stem for s > 2.

That the conjecture is no longer valid for s = 1 and 2 is respectively an exposition of

the existence of Hopf invariant one and Kervaire invariant one classes (See Adams [1], Browder [4], Curtis [7], Snaith and Tornehave [31], Wellington [34] for a discussion on spherical classes; and see Lannes–Zarati [23], Goerss [10], Hưng [12]–[15] for a discussion

P H∗(BV s ) in such a way that these two squaring operations commute with

each other through the canonical homomorphism

induced by the identity map onVs (see [12]) The first named author also showed in [12]

that j s∗ = ϕ s ◦ T r s Here T r s is the algebraic transfer, which was defined by Singer [30] and was shown to be highly nontrivial by Singer [30], Boardman [2], Bruner–H`a–Hưng [6], Hưng [16], H`a [11], Nam [28] and the authors [21] Further, Hưng and Nam proved in [17]

that j s∗ = 0 in positive degree for s > 2, or equivalently that the Lannes–Zarati

homomorph-ism vanishes on the positive stem part of the algebraic transfer’s image for the homological

degree s > 2.

A basis of the F2-vector space P (F2 ⊗

G L s

H∗(BV s )) was determined by Singer [30] for

s = 1, 2, by Hưng and Peterson [18] for s = 3, 4, and by Giambalvo and Peterson [9]

for s = 5 It is still unknown for s > 5 The squaring operation on P(F2 ⊗

G L s

H∗(BV s )) is

explicitly computed in [12] for s  4 This result shows that Sq0 is an isomorphism for

s = 1, 2 and is no longer an isomorphism for s = 3, 4.

The Dickson algebra of all G L s-invariants was determined in [8] as follows

s ,s−1 with respect to the basis of

D s consisting of all monomials in the Dickson invariants

The following theorem, which claims that the squaring operation is “eventually

iso-morphic” on P (F2 ⊗

G L s

H∗(BV s )), is the first main result of this paper.

THEOREM1·2 The squaring operation

Squaring operation on the Dickson algebra 269

For s = 1 or 2, the squaring operation is given on a basis of P(F2 ⊗

H∗(BV s )) or of its dual space F2⊗

A D s The theorem is an analogue of the

res-ult by the first name author [16, theorem 1·1] stating that Sq0: F2 ⊗

The following is an immediate consequence of the above theorem

COROLLARY1·3 Any Sq0-family in P (F2 ⊗

P H∗(BV s ) is either infinite or finite with length at most s − 2.

Letα(δ) be the number of ones in the dyadic expansion of δ, and ν(δ) the exponent of the

highest power of 2 dividingδ, with convention 2 ν(0)= 0

Following Giambalvo and Peterson [9], the function κ s is defined by setting κ s (r) =

r + 2ν(s−2−r) For convenience, setκ0

s (r) = r Finally, let κ 

s (r) = κ s (κ −1

s (r)) for   1.

Actually, Giambalvo and Peterson denoted the functionκ s by x s However, the letter x s will

be used in this paper to name an another object, so we denote it byκ s A discussion on an

earlier version of this function defined by Hưng and Peterson [18] is given in Section 4.

The following is the second main result of the paper

THEOREM1·4 The squaring operation Sq0 on P (F2 ⊗

G L s

H∗(BV s )) vanishes in any gree δ, which:

de-(i) either satisfies ν(δ + s)  [log2(s − 2)] + 1 for s  3 or

(ii) is not of the form δ s defined inductively for s  3 as follows

δ s = δ s−1− 1 + 2s−1[κ j s−1

1 κ j s−2

2 · · · κ j1

s−1(s − 2) + 1], for arbitrary non-decreasing sequence[log2(s − 2)] < j1  j2  · · ·  j s−1, where

δ2= 2j1 +1− 2.

The theorem does not seem to be possibly improved in the meaning that, Sq0 acts trivially in every degreeδ s given in the theorem at least for s = 3, 4 and 5 One can verify this claim by combining the explicit formulas for the action of Sq0 on P (F2 ⊗

non-G L s

H∗(BV s )) when s = 3, 4 (in [12]) and s = 5 (in Section 4 below) with Lemmas 6·1-6·3.

In Section 5 we give an inductive formula that is convenient for computingδ s By means

of this formula, we find explicitly the list of all degreesδ s for s  7 in Lemmas 6·1-6·5 In

principle, this procedure of computing can inductively be extended for any bigger value of s.

270 NGUYEˆ ˜N H V HƯNG ANDV ˜O T N QUYNH`

(See Section 5 for details.) In particular, the following is an immediate consequence of theabove theorem

COROLLARY1·5 Sq0 on P (F2 ⊗

G L s

H∗(BV s )) vanishes in any degree δ, which satisfies one of the two conditions:

(i) ν(δ + s)  [log2(s − 2)] + 1 for s  3;

(ii) δ + s is not of the forms listed respectively in Lemmas 6·3 – 6·5 for 5  s  7.

The remaining part of the introduction deals with some applications of the above results

to the study of Conjecture 1·1

The group E xt s

A (F2, F2) was determined for s = 1, 2 by Adams [1], for s = 3 by Wang [33] and for s = 4 by Lin [24] (see also [25]) It is unknown for s > 4 Based on these

results, Conjecture 1·1 was proved by the first named author in [12, 15] for s = 3, 4.

Hưng and Peterson showed in [19] thatϕ = ⊕ϕ s is a homomorphism of algebras and itvanishes on decomposable elements So, in order to prove Conjecture 1·1, it suffices to studythe Lannes–Zarati homomorphism on indecomposable elements

Our first result on the Lannes–Zarati homomorphism is the following consequence ofTheorem 1·2

COROLLARY1·6 If {a i |i  0} is a finite Sq0-family in E xt s

(i) ν(δ + s)  [log2(s − 2)] + 1 for s  3;

(ii) δ is not of the form δ s given in Theorem 1 ·4 In particular, δ +s is not of the forms listed respectively in Lemmas 6 ·3 – 6·5 for 5  s  7.

Then ϕ s (a i ) = 0 for any i > 0.

We note that every Sq0-family listed in the paper by Tangora [32], as well as in that by Bruner [5], satisfies either the hypothesis of Corollary 1·6 or the one of Proposition 1·7.Therefore, if{a i |i  0} denotes such a family in Ext s

A (F2, F2), then ϕ s (a i ) = 0 for any

i > 0 It should be noted that the above results do not conclude whether the Lannes–Zarati homomorphism vanishes on the initial element a0 of the Sq0-family in question The fol-

lowing proposition gives an answer to this problem in the case where Stem (a0) is rather

small

PROPOSITION1·8 If {a i |i  0} is an Sq0-family in E xt A s (F2, F2) with Stem(a0) < 2 s−1,

then ϕ s (a i ) = 0 for any i  0.

The paper is divided into seven sections and organized as follows Section 2 is a inary on the modular invariant theory and the squaring operation In Section 3, we show achain-level representation of the dual of the squaring operation on the Dickson invariants.This is a key tool in the study of the behavior of the squaring operation We prove The-orem 1·2 and Theorem 1·4 in Section 4 and Section 5 respectively Section 6 then deals with

prelim-Squaring operation on the Dickson algebra 271

an explicit expression for the degrees, in which the squaring operation vanishes at small

rank s  7 Finally, we show in Section 7 some applications in the investigation of theLannes–Zarati homomorphism

The main results of the present paper have already been announced in [20].

2 Preliminary The regular action of the general linear group G L s := GL(V s ) on V s induces its regularaction on the cohomology

H∗(BV s ) % P s := F2[x1, , x s ], which is the polynomial algebra on s generators x1, , x s, each of degree 1 Recall that the

Dickson algebra of the G L s-invariants was computed in [8]:

D s := F2[x1, , x s]G L s = F2[Qs ,0 , Q s ,1 , , Q s ,s−1 ].

Here the Dickson invariant Q s ,i of degree 2s− 2i

can inductively be defined by

Q s ,i = Q2

s −1,i−1 + V s Q s −1,i , where, by convention, Q s ,s = 1, Q s ,i = 0 for i < 0, and

V i = 

c j∈F 2

(c1x1+ · · · + c i−1x i−1+ x i )

is the M`ui invariant under the Sylow 2-subgroup T s of G L sconsisting of all upper triangular

s × s-matrices with 1 on the main diagonal (See M`ui [27].)

Let S (s) ⊂ P s be the multiplicative subset generated by all the non-zero linear forms in

P s Let sbe the localization: s = (P s ) S (s) Using the results of Dickson [8] and M`ui [27],

Singer noted in [29] that

272 NGUYEˆ ˜N H V HƯNG ANDV ˜O T N QUYNH`

which can be expressed in terms of∧as follows (see [15]):

Given a module M over the mod 2 Steenrod algebraA, let P(M) denote the submodule of

M consisting of all elements, which are annihilated by any positive-degree operation inA.

the Kameko squaring operation This was shown by Boardman [2] and Minami [26]

to commute with the classical squaring operation through the algebraic transfer T r s:F2 ⊗

A (F2, F2) through the Lannes–Zarati ism ϕ s : E xt A s (F2, F2) → P(F2 ⊗

homomorph-G L s

H∗(BV s )).

Squaring operation on the Dickson algebra 273

3 A chain-level representation for the dual of the squaring operation

The goal of this section is to describe a chain-level representation for the dual of thesquaring operation on the Dickson invariants It will be applied to investigate the behavior

of the squaring operation

For abbreviation, we denote by Q (I) the monomial Q i0

s ,0 · · · Q i s−1

s ,s−1 for I = (i0 , , i s−1)

an s-tuple of non-negative integers.

PROPOSITION3·1 A chain-level representation of the homomorphism Sq0

∗: F2⊗

A D s →F2⊗

v given in Section 2 Note that, Sq v0 depends on the rank s and, when

necessary, will be denoted by Sq0

v,s−1 (A) for any A ∈ F2[V1, , V s−1].

Proof of Proposition 3 ·1 Lemma 2·1 plays a key role in this proof More precisely, we use the chain-level representation Sq0

v = Sq0

x of the homomorphism Sq0

∗.

The proof proceeds by induction on s From Lemma 3·2(i), it suffices to show the

propos-ition in the case, where I is an s-tuple of numbers 0 and 1 only.

The conclusion is trivial for s = 1, as Q1 ,0 = V1 = v1 Suppose inductively that the

proposition holds for s Let I = (i0 , i1, , i s ) be an (s + 1)-tuple of numbers 0 and 1 Using the inductive formula Q s +1,i = Q2

Q s +1,i (for 1  i  s − 1) So, in order to get at least one J with Sq0

v,s (Q(J))  0, the necessary condition is i1= · · · = i s−1 = 1

If i s = 0, then the exponent of Q s ,0 in the monomial Q (J), which contains

Q s ,1 · · · Q s ,s−1 , is either Q s s ,0−1 or Q s s+1,0 In these two possibilities, by the inductive

hypo-thesis, Sq0

v,s (Q(J)) = 0, as (s − 1) + s and (s + 1) + s are both odd.

274 NGUYEˆ ˜N H V HƯNG ANDV ˜O T N QUYNH`

So, it suffices to consider the case of i1= · · · = i s−1= i s = 1 If in addition i0 = 0, then

4 The squaring operation is isomorphic on its image

The aim of this section is to prove the following theorem, whish is also numbered asTheorem 1·2 in the introduction

THEOREM4·1 The squaring operation

The following is also numbered as Corollary 1·3 in the introduction

COROLLARY4·2 Any Sq0-family in P (F2 ⊗

H∗(BV s )) Then, a0 and a1 are non-zero Let n be the biggest number with

a n  0 Thus, n  1 and a n = Sq0(a n−1) ∈ Im(Sq0) By Theorem 4·1, the two theses a n  0 and a n ∈ Im(Sq0) imply a n+1 = Sq0(a n )  0 This is a contradiction to the definition of n The corollary is proved.

Squaring operation on the Dickson algebra 275

only if< d, Q >= 0 for any Q ∈ AD s In this case, d is non-zero if and only if there exists

an indecomposable element Q ∈ D s such that< d, Q >= 1, where < ·, · > denotes the

dual pairing betweenF2 ⊗

G L s

H∗(BV s ) and D s.Using the functionκ s given in the introduction, Giambalvo and Peterson showed in [9] a

sufficient condition for a monomial in D sto be decomposable as follows

THEOREM4·3 ([9, theorem 2·1]) Let I = (i0 , i1, , i s−1) be an s-tuple of non-negative integers, and Q (I) = Q i0

Suppose inductively thatκ 

s (0) = 2 s1+ · · · + 2s  (For = 0, we mean the right-hand side is 0.) Since s − 2 − κ 

The lemma is proved

Actually, the central role of the numbersκ 

s (0) in the study of the problem was earlier

recognized by Hưng and Peterson in [18], where they found a minimal set ofA-generators for D s with s = 3, 4 They denoted by [s − 2 : 2 t] the non-negative integer less than 2twith

s − 2 ≡ [s − 2 : 2 t] mod 2t From Lemma 4·4, we have κ 

s (0) = [s − 2 : 2 s +1] for   0.

So, the following is a consequence of Theorem 4·3 and Lemma 4·4

COROLLARY4·5 If i0  [s − 2 : 2 t ] for every t  0, then Q(I) = Q(i0 , i1, , i s−1) is decomposable In other words, if Q (I) is indecomposable in D s , then i0 = [s − 2 : 2 t ] for some t  0.

For a monomial Q (I) ∈ D s, let us denote its class inF2⊗

Proof Let Q (I) = Q(i0, i1, , i s−1) be an indecomposable monomial in D s, that means

[Q(I)]  0 in D s /AD s By Proposition 3·1, we need only to consider the case i1 , , i s−1

276 NGUYEˆ ˜N H V HƯNG ANDV ˜O T N QUYNH`

odd and i0+ s even In this case, Proposition 3·1 claims

So, the proposition is valid for the case i0= s − 2.

In the case i0 s −2, let s −2 = 2 s1+· · ·+2s k be the dyadic expansion of s−2 Suppose

to the contrary that Sq∗0([Q(I)])  0, then by Corollary 4·5, we have (i0+ s − 2)/2 =

2s1+ · · · + 2s r for some r with 1  r  k From i0  s − 2, it implies r < k Then we get

i0= 2s1+ · · · + 2s r − 2s r+1− · · · − 2s k < 0.

This is a contradiction The proposition is proved

Proof of Theorem 4 ·1 It should be noted that the first part of the theorem is not a

con-sequence of the second part

The second part of the theorem is the dual statement of Proposition 4·6

Indeed, suppose d (I) = d(i0, , i s−1) ∈ P(F2 ⊗

Proof of Theorem ·1 It should be noted that the first part of the theorem is not a

con-sequence of the second part

The second... by Hưng and Peterson in [18], where they found a minimal set of< /b>A-generators for D s with s = 3, They denoted by [s − : t] the non-negative integer less... s.Using the functionκ s given in the introduction, Giambalvo and Peterson showed in [9] a

sufficient condition for a monomial in D sto