DSpace at VNU: The squaring operation on A-generators of the Dickson algebra.

Mathematical Proceedings of the Cambridge Philosophical Society http://journals.cambridge.org/PSP Additional services for Mathematical Proceedings of the Cambridge Philosophical Society: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here The squaring operation on S0305004109990405_char1generators of the Dickson algebra 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 Request Permissions : Click here Downloaded from http://journals.cambridge.org/PSP, IP address: 169.230.243.252 on 22 Mar 2015 Math Proc Camb Phil Soc (2010), 148, 267 c Cambridge Philosophical Society 2009 267 doi:10.1017/S0305004109990405 First published online December 2009 The squaring operation on A-generators of the Dickson algebra† ˜ T N QUYNH ` B Y NGUYE˜ˆ N H V HƯNG AND VO 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 September 2009) Abstract We study the squaring operation Sq on the dual of the minimal A-generators of the Dickson algebra We show that this squaring operation is isomorphic on its image We also give 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 Sq -family in ∗ E xtA (F2 , F2 ) except possibly the family initial element, and (2) on almost all known elements 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 Introduction and statement of results Throughout the paper, the coefficient ring for homology and cohomology is always F2 , the field of two elements Let Vs be an s-dimensional F2 -vector space The general linear group G L s := G L(Vs ) acts regularly on Vs and therefore on H∗ (BVs ) Let P(F2 ⊗ H∗ (BVs )) be G Ls the submodule of F2 ⊗ H∗ (BVs ) consisting of all elements, which are annihilated by every G Ls positive-degree operation in the mod Steenrod algebra, A The subject of this paper is the squaring operation Sq : P(F2 ⊗ H∗ (BVs ))δ −→ P(F2 ⊗ H∗ (BVs ))s+2δ , G Ls G Ls which is defined by the first named author in [12] as an analogue of the classical squaring ∗ (F2 , F2 ) operation on the cohomology of the Steenrod algebra, E xtA The most important property of the squaring operation is that it commutes with the clas∗ sical squaring operation Sq on E xtA (F2 , F2 ) through the Lannes–Zarati homomorphism s,s+δ ϕs : E xtA (F2 , F2 ) −→ P(F2 ⊗ H∗ (BVs ))δ , G Ls 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 ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 268 The Lannes–Zarati homomorphism, defined in [23], is the one corresponding to an associated graded of the Hurewicz map H : π∗s (S ) % π∗ (Q S ) → H∗ (Q S ) So, the following is an algebraic version of the conjecture on spherical classes C ONJECTURE 1·1 ([12]) ϕs = in any positive stem for s > That the conjecture is no longer valid for s = and 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 on the homomorphism.) The squaring operation on P(F2 ⊗ H∗ (BVs )) is derived from the Kameko squaring opG Ls eration on F2 ⊗ P H∗ (BVs ) in such a way that these two squaring operations commute with G Ls each other through the canonical homomorphism js∗ : F2 ⊗ P H∗ (BVs ) −→ P(F2 ⊗ H∗ (BVs )) G Ls G Ls induced by the identity map on Vs (see [12]) The first named author also showed in [12] that js∗ = ϕs ◦ T rs Here T rs 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 js∗ = in positive degree for s > 2, or equivalently that the Lannes–Zarati homomorphism vanishes on the positive stem part of the algebraic transfer’s image for the homological degree s > A basis of the F2 -vector space P(F2 ⊗ H∗ (BVs )) was determined by Singer [30] for G Ls s = 1, 2, by Hưng and Peterson [18] for s = 3, 4, and by Giambalvo and Peterson [9] for s = It is still unknown for s > The squaring operation on P(F2 ⊗ H∗ (BVs )) is G Ls explicitly computed in [12] for s This result shows that Sq is an isomorphism for s = 1, and is no longer an isomorphism for s = 3, The Dickson algebra of all G L s -invariants was determined in [8] as follows Ds := H ∗ (BVs )G L s % F2 [x1 , , xs ]G L s = F2 [Q s,0 , Q s,1 , , Q s,s−1 ], where Q s,i denotes the Dickson invariant of degree 2s − 2i Let d(i , i , , i s−1 ) ∈ i s−1 · · · Q s,s−1 with respect to the basis of F2 ⊗ H∗ (BVs ) be the element that is dual to Q is,0 G Ls Ds consisting of all monomials in the Dickson invariants The following theorem, which claims that the squaring operation is “eventually isomorphic” on P(F2 ⊗ H∗ (BVs )), is the first main result of this paper G Ls T HEOREM 1·2 The squaring operation Sq : P(F2 ⊗ H∗ (BVs )) −→ P(F2 ⊗ H∗ (BVs )) G Ls is an isomorphism on its image For s P(F2 ⊗ H∗ (BVs )), then G Ls > 2, if d(i , , i s−1 ) is an element in G Ls Sq d(i , , i s−1 ) = d(s − 2, 2i + 1, , 2i s−1 + 1), 0, i = s − 2, otherwise Squaring operation on the Dickson algebra 269 For s = or 2, the squaring operation is given on a basis of P(F2 ⊗ H∗ (BVs )) as follows G Ls (see [12, page 3906]): Sq d(2k − 1) = d(2k+1 − 1), Sq d(2k − 1, 0) = d(2k+1 − 1, 0) Evidently, this theorem could be applied to investigate the structure of the space P(F2 ⊗ H∗ (BVs )) or of its dual space F2 ⊗ Ds The theorem is an analogue of the resA G Ls ult by the first name author [16, theorem 1·1] stating that Sq : F2 ⊗ P H∗ (BVs ) → G Ls F2 ⊗ P H∗ (BVs ) is an isomorphism on the image of (Sq )s−2 G Ls A sequence {ai | i s 0} of elements in P(F2 ⊗ H∗ (BVs )) (or in E xtA (F2 , F2 )) is called G Ls an Sq -family if = Sq (ai−1 ) for every i > It is called finite with length s if it has exactly s non-zero elements Otherwise, it is called infinite The following is an immediate consequence of the above theorem C OROLLARY 1·3 Any Sq -family in P(F2 ⊗ H∗ (BVs )) is either infinite or finite with G Ls length This is an analogue of the result by the first name author [16, corollary 1·7] stating that any Sq -family in F2 ⊗ P H∗ (BVs ) is either infinite or finite with length at most s − G Ls Let α(δ) be the number of ones in the dyadic expansion of δ, and ν(δ) the exponent of the highest power of dividing δ, with convention 2ν(0) = Following Giambalvo and Peterson [9], the function κs is defined by setting κs (r ) = r + 2ν(s−2−r ) For convenience, set κs0 (r ) = r Finally, let κs (r ) = κs (κs −1 (r )) for Actually, Giambalvo and Peterson denoted the function κs by xs However, the letter xs will be used in this paper to name an another object, so we denote it by κs A discussion on an earlier version of this function defined by Hưng and Peterson [18] is given in Section The following is the second main result of the paper T HEOREM 1·4 The squaring operation Sq on P(F2 ⊗ H∗ (BVs )) vanishes in any deG Ls gree δ, which: (i) either satisfies ν(δ + s) [log (s − 2)] + for s or (ii) is not of the form δs defined inductively for s as follows j j j δs = δs−1 − + 2s−1 [κ1 s−1 κ2 s−2 · · · κs−1 (s − 2) + 1], for arbitrary non-decreasing sequence [log (s − 2)] < j1 δ2 = j1 +1 − j2 ··· js−1 , where The theorem does not seem to be possibly improved in the meaning that, Sq acts nontrivially in every degree δs given in the theorem at least for s = 3, and One can verify this claim by combining the explicit formulas for the action of Sq on P(F2 ⊗ H∗ (BVs )) G Ls when s = 3, (in [12]) and s = (in Section below) with Lemmas 6·1-6·3 In Section we give an inductive formula that is convenient for computing δs By means of this formula, we find explicitly the list of all degrees δs for s in Lemmas 6·1-6·5 In principle, this procedure of computing can inductively be extended for any bigger value of s 270 ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH (See Section for details.) In particular, the following is an immediate consequence of the above theorem C OROLLARY 1·5 Sq on P(F2 ⊗ H∗ (BVs )) vanishes in any degree δ, which satisfies one of the two conditions: G Ls (i) ν(δ + s) [log (s − 2)] + for s 3; (ii) δ + s is not of the forms listed respectively in Lemmas 6·3 – 6·5 for s The remaining part of the introduction deals with some applications of the above results to the study of Conjecture 1·1 s (F2 , F2 ) was determined for s = 1, by Adams [1], for s = by The group E xtA Wang [33] and for s = by Lin [24] (see also [25]) It is unknown for s > Based on these results, Conjecture 1·1 was proved by the first named author in [12, 15] for s = 3, Hưng and Peterson showed in [19] that ϕ = ⊕ϕs is a homomorphism of algebras and it vanishes on decomposable elements So, in order to prove Conjecture 1·1, it suffices to study the Lannes–Zarati homomorphism on indecomposable elements Our first result on the Lannes–Zarati homomorphism is the following consequence of Theorem 1·2 C OROLLARY 1·6 If {ai |i s 0} is a finite Sq -family in E xtA (F2 , F2 ), then ϕs (ai ) = for i > Our second result on the Lannes–Zarati homomorphism is the following application of Theorem 1·4 and Corollary 1·5 s 0} be an Sq -family in E xtA (F2 , F2 ) Suppose δ = P ROPOSITION 1·7 Let {ai |i Stem(a0 ) satisfies one of the following conditions: (i) ν(δ + s) [log (s − 2)] + for s 3; (ii) δ is not of the form δs given in Theorem 1·4 In particular, δ + s is not of the forms listed respectively in Lemmas 6·3 – 6·5 for s Then ϕs (ai ) = for any i > We note that every Sq -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 s 0} denotes such a family in E xtA (F2 , F2 ), then ϕs (ai ) = for any Therefore, if {ai |i i > It should be noted that the above results not conclude whether the Lannes–Zarati homomorphism vanishes on the initial element a0 of the Sq -family in question The following proposition gives an answer to this problem in the case where Stem(a0 ) is rather small P ROPOSITION 1·8 If {ai |i then ϕs (ai ) = for any i s 0} is an Sq -family in E xtA (F2 , F2 ) with Stem(a0 ) < 2s−1 , The paper is divided into seven sections and organized as follows Section is a preliminary on the modular invariant theory and the squaring operation In Section 3, we show a chain-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 Theorem 1·2 and Theorem 1·4 in Section and Section respectively Section then deals with Squaring operation on the Dickson algebra 271 an explicit expression for the degrees, in which the squaring operation vanishes at small rank s Finally, we show in Section some applications in the investigation of the Lannes–Zarati homomorphism The main results of the present paper have already been announced in [20] Preliminary The regular action of the general linear group G L s := G L(Vs ) on Vs induces its regular action on the cohomology H ∗ (BVs ) % Ps := F2 [x1 , , xs ], which is the polynomial algebra on s generators x1 , , xs , each of degree Recall that the Dickson algebra of the G L s -invariants was computed in [8]: Ds := F2 [x1 , , xs ]G L s = F2 [Q s,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 = Q 2s−1,i−1 + Vs Q s−1,i , where, by convention, Q s,s = 1, Q s,i = for i < 0, and Vi = (c1 x1 + · · · + ci−1 xi−1 + xi ) c j ∈F2 is the M`ui invariant under the Sylow 2-subgroup Ts of G L s consisting of all upper triangular s × s-matrices with on the main diagonal (See M`ui [27].) Let S(s) ⊂ Ps be the multiplicative subset generated by all the non-zero linear forms in Ps Let s be the localization: s = (Ps ) S(s) Using the results of Dickson [8] and M`ui [27], Singer noted in [29] that s s := ( := ( s) s) G Ls Ts = F2 [V1±1 , , Vs±1 ], = F2 [Q ±1 s,0 , Q s,1 , , Q s,s−1 ] Further, he set v1 = V1 , vs = Vs /V1 · · · Vs−1 (s 2), so that s−2 s−3 Vs = v12 v22 · · · vs−1 vs (s 2) Then, he obtained s = F2 [v1±1 , , vs±1 ], with degree vi = for every i Singer defined s∧ to be the submodule of s = Ds [Q −1 s,0 ] spanned by all monomials i s−1 i0 0, i ∈ Z, and i + deg γ He also equipped γ = Q s,0 · · · Q s,s−1 with i , , i s−1 ∧ ∧ = with a differential coalgebra structure Then, he proved that the differential s s ∧ is dual to the (differential) lambda algebra of the six authors of [3] Thus, coalgebra Hs ( ∧ ) % T orsA (F2 , F2 ) (See [29].) s,t s,2t (F2 , F2 ) → E xtA (F2 , F2 ) be the classical squaring operation Its dual Let Sq : E xtA A homomorphism Sq∗ : T ors (F2 , F2 ) → T orsA (F2 , F2 ) has a chain-level representation, ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 272 which can be expressed in terms of ∧ s Sqv0 : ∧ −→ as follows (see [15]): ∧ s , j1 −1 j Sqv0 (v11 · · · vsjs ) = js −1 v1 · · · vs 0, , j1 , , js odd, otherwise Given a module M over the mod Steenrod algebra A, let P(M) denote the submodule of M consisting of all elements, which are annihilated by any positive-degree operation in A In particular, P(F2 ⊗ H∗ (BVs )) and F2 ⊗ P H∗ (BVs ) are dual to F2 ⊗ Ds and (F2 ⊗ Ps )G L s G Ls A G Ls respectively In [22], Kameko defined a homomorphism A Sq : H∗ (BVs ) −→ H∗ (BVs ), a1(i1 ) · · · as(is ) −→ a1(2i1 +1) · · · as(2is +1) , where a1(i1 ) · · · as(is ) is dual to x1i1 · · · xsis with respect to the basis of H ∗ (BVs ) consisting of all monomials in x1 , , xs He proved that this is a G L s -homomorphism and maps P H∗ (BVs ) to itself The induced homomorphism Sq :F2 ⊗ P H∗ (BVs ) → F2 ⊗ P H∗ (BVs ) is called G Ls G Ls 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 rs : s (F2 , F2 ) defined by Singer [30] F2 ⊗ P H∗ (BVs ) → E xtA G Ls On the other hand, the first named author considered in [12] the homomorphism Sq := ⊗ Sq : F2 ⊗ H∗ (BVs ) −→ F2 ⊗ H∗ (BVs ) G Ls G Ls G Ls and proved that it sends the primitive part P(F2 ⊗ H∗ (BVs )) to itself The resulting hoG Ls momorphism on P(F2 ⊗ H∗ (BVs )) is also denoted by Sq Its dual homomorphism Sq∗0 : G Ls F2 ⊗ Ds → F2 ⊗ Ds has a chain-level representation Sqx0 := Sq∗0 : Ps → Ps , which is given by A A j1 −1 j Sqx0 (x11 · · · xsjs ) = js −1 x1 · · · xs 0, , j1 , , js odd, otherwise L EMMA 2·1 ([15, proposition 3·2]) Sqx0 coincides with Sqv0 on the M`ui algebra F2 [x1 , , xs ]Ts = F2 [V1 , , Vs ], for any s The first named author also proved in [14] that the inclusion of the Dickson algebra Ds into s+ is a chain-level representation of the Lannes–Zarati dual homomorphism This result together with Lemma 2·1 led him to the following T HEOREM 2·2 ([15, theorem 3·1]) The squaring operation Sq on P(F2 ⊗ H∗ (BVs )) G Ls s (F2 , F2 ) through the Lannes–Zarati homomorphcommutes with the classical Sq on E xtA s ism ϕs : E xtA (F2 , F2 ) → P(F2 ⊗ H∗ (BVs )) G Ls Squaring operation on the Dickson algebra 273 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 the squaring operation on the Dickson invariants It will be applied to investigate the behavior of the squaring operation i s−1 For abbreviation, we denote by Q(I ) the monomial Q is,0 · · · Q s,s−1 for I = (i , , i s−1 ) an s-tuple of non-negative integers P ROPOSITION 3·1 A chain-level representation of the homomorphism Sq∗0 : F2 ⊗ Ds → A F2 ⊗ Ds is given by A i +s−2 Sq∗0 (Q(I )) = i s−1 −1 i −1 Q s,02 Q s,12 · · · Q s,s−1 , i + s even, i , , i s−1 odd, 0, otherwise The proposition will be proved by means of the following lemma, which comes from the definition of Sqv0 given in Section Note that, Sqv0 depends on the rank s and, when necessary, will be denoted by Sqv,s L EMMA 3·2 ( [15, lemmas 3·4-3·5]) 0 (AB ) = Sqv,s (A)B for any A, B ∈ F2 [V1 , , Vs ], (i) Sqv,s 0 (ii) Sqv,s (Avs ) = Sqv,s−1 (A) for any A ∈ F2 [V1 , , Vs−1 ] Proof of Proposition 3·1 Lemma 2·1 plays a key role in this proof More precisely, we use the chain-level representation Sqv0 = Sqx0 of the homomorphism Sq∗0 The proof proceeds by induction on s From Lemma 3·2(i), it suffices to show the proposition in the case, where I is an s-tuple of numbers and only The conclusion is trivial for s = 1, as Q 1,0 = V1 = v1 Suppose inductively that the proposition holds for s Let I = (i , i , , i s ) be an (s + 1)-tuple of numbers and Using the inductive formula Q s+1,i = Q 2s,i−1 + Q s,i Q s,0 vs+1 , we get j Q(J )vs+1 Q(I ) = Q(J )∈Ds , j Hence j−1 Sqv,s+1 (Q(I )) = Sqv,s (Q(J ))vs+1 Q(J )∈Ds , j odd Sqv,s (Q(J ))  By the inductive hypothesis, if and only if j0 + s even, while j1 , , js−1 odd To go further, we need the explicit expansions Q s+1,s = Q 2s,s−1 + Q s,0 vs+1 , Q s+1,i = Q 2s,i−1 + Q s,i Q s,0 vs+1 , Q s+1,0 = i s − 1, Q 2s,0 vs+1 Note that, among these formulas, the only odd exponent of Q s,i occurs in the expansion of i s − 1) So, in order to get at least one J with Sqv,s (Q(J ))  0, the Q s+1,i (for necessary condition is i = · · · = i s−1 = If i s = 0, then the exponent of Q s,0 in the monomial Q(J ), which contains s+1 Q s,1 · · · Q s,s−1 , is either Q s−1 s,0 or Q s,0 In these two possibilities, by the inductive hypo0 thesis, Sqv,s (Q(J )) = 0, as (s − 1) + s and (s + 1) + s are both odd ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 274 So, it suffices to consider the case of i = · · · = i s−1 = i s = If in addition i = 0, then 0 s (Q(I )) = Sqv,s+1 (Q ss,0 Q s,1 · · · Q s,s−1 vs+1 ) Sqv,s+1 It is non-zero if and only if s odd In this case, by Lemma 3·2, we have s−1 0 (Q(I )) = Sqv,s (Q ss,0 Q s,1 · · · Q s,s−1 )vs+1 Sqv,s+1 s−1 = Q s−1 s,0 vs+1 (by the inductive hypothesis) s−1 = Q s+1,0 Otherwise, if in addition i = 1, then s+1 0 Sqv,s+1 (Q(I )) = Sqv,s+1 (Q s+2 s,0 Q s,1 · · · Q s,s−1 vs+1 ) It is non-zero if and only if s even In this case, by Lemma 3·2, we get s (Q(I )) = Q ss,0 vs+1 Sqv,s+1 (by the inductive hypothesis) s = Q s+1,0 Then the proposition also holds for s + Therefore, it is completely proved The squaring operation is isomorphic on its image The aim of this section is to prove the following theorem, whish is also numbered as Theorem 1·2 in the introduction T HEOREM 4·1 The squaring operation Sq : P(F2 ⊗ H∗ (BVs )) −→ P(F2 ⊗ H∗ (BVs )) G Ls G Ls is an isomorphism on its image For s P(F2 ⊗ H∗ (BVs )), then > 2, if d(i , , i s−1 ) is an element in G Ls Sq d(i , , i s−1 ) = d(s − 2, 2i + 1, , 2i s−1 + 1), 0, i = s − 2, otherwise The following is also numbered as Corollary 1·3 in the introduction C OROLLARY 4·2 Any Sq -family in P(F2 ⊗ H∗ (BVs )) is either infinite or finite with G Ls length Proof Suppose to the contrary that {ai |i 0} is a finite Sq -family with length at least in P(F2 ⊗ H∗ (BVs )) Then, a0 and a1 are non-zero Let n be the biggest number with G Ls and an = Sq (an−1 ) ∈ I m(Sq ) By Theorem 4·1, the two hypoan  Thus, n theses an  and an ∈ I m(Sq ) imply an+1 = Sq (an )  This is a contradiction to the definition of n The corollary is proved Note that F2 ⊗ H∗ (BVs ) and P(F2 ⊗ H∗ (BVs )) are respectively dual to Ds and G Ls G Ls F2 ⊗ Ds = Ds /ADs , where A denotes the augmentation ideal of the Steenrod algebra A An A element in Ds is called decomposable if it is in ADs Otherwise, it is called indecomposable Squaring operation on the Dickson algebra 275 So, an element d ∈ F2 ⊗ H∗ (BVs ) is in the primitive submodule P(F2 ⊗ H∗ (BVs )) if and G Ls G Ls only if < d, Q >= for any Q ∈ ADs In this case, d is non-zero if and only if there exists an indecomposable element Q ∈ Ds such that < d, Q >= 1, where < ·, · > denotes the dual pairing between F2 ⊗ H∗ (BVs ) and Ds G Ls Using the function κs given in the introduction, Giambalvo and Peterson showed in [9] a sufficient condition for a monomial in Ds to be decomposable as follows T HEOREM 4·3 ([9, theorem 2·1]) Let I = (i , i , , i s−1 ) be an s-tuple of non-negative i s−1 integers, and Q(I ) = Q is,0 · · · Q s,s−1 ∈ Ds If i  κs (0) for any non-negative then Q(I ) is decomposable We explicitly determine the finite set {κs (0)| 0} as follows L EMMA 4·4 Let s − = 2s1 + · · · + 2sk be the dyadic expansion of s − with s1 < · · · < sk Then ⎧ = 0, ⎨ 0, k, κs (0) = 2s1 + · · · + 2s , ⎩ s − 2, k Proof The proof proceeds by induction on the number For = 0, by definition, we get κs0 (0) = Suppose inductively that κs (0) = 2s1 + · · · + 2s (For = 0, we mean the right-hand side is 0.) Since s − − κs (0) = 2s +1 + · · · + 2sk , we have κs +1 (0) = κs (0) + 2s +1 = 2s1 + · · · + 2s +1 In particular κsk (0) = s − So, it implies that κs (0) = s − for any The lemma is proved k Actually, the central role of the numbers κs (0) in the study of the problem was earlier recognized by Hưng and Peterson in [18], where they found a minimal set of A-generators for Ds with s = 3, They denoted by [s − : 2t ] the non-negative integer less than 2t with s − ≡ [s − : 2t ] mod 2t From Lemma 4·4, we have κs (0) = [s − : 2s +1 ] for So, the following is a consequence of Theorem 4·3 and Lemma 4·4 C OROLLARY 4·5 If i  [s − : 2t ] for every t 0, then Q(I ) = Q(i , i , , i s−1 ) is decomposable In other words, if Q(I ) is indecomposable in Ds , then i = [s − : 2t ] for some t For a monomial Q(I ) ∈ Ds , let us denote its class in F2 ⊗ Ds by [Q(I )] A P ROPOSITION 4·6 The dual of the squaring operation, Sq∗0 : F2 ⊗ Ds → F2 ⊗ Ds , is A given by Sq∗0 ([Q(I )]) = , , is−12−1 )], [Q(s − 2, i1 −1 0, A i = s − and i , , i s−1 odd, otherwise Proof Let Q(I ) = Q(i , i , , i s−1 ) be an indecomposable monomial in Ds , that means [Q(I )]  in Ds /ADs By Proposition 3·1, we need only to consider the case i , , i s−1 ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 276 odd and i + s even In this case, Proposition 3·1 claims Sq∗0 ([Q(I )]) = Q i s−1 − i0 + s − i1 − , , , 2 So, the proposition is valid for the case i = s − In the case i  s − 2, let s − = 2s1 + · · · + 2sk be the dyadic expansion of s − Suppose to the contrary that Sq∗0 ([Q(I )])  0, then by Corollary 4·5, we have (i + s − 2)/2 = 2s1 + · · · + 2sr for some r with r k From i  s − 2, it implies r < k Then we get i = 2s1 + · · · + 2sr − 2sr +1 − · · · − 2sk < 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 consequence of the second part Let d be non-zero in I m(Sq ) ⊂ P(F2 ⊗ H∗ (BVs )) We need only to show that G Ls Sq (d)  Write d = Sq (d0 ) for a certain non-zero element d0 ∈ P(F2 ⊗ H∗ (BVs )) As G Ls d  0, there exists an indecomposable monomial Q(J ) = Q( j0 , j1 , , js−1 ) such that < d, [Q(J )] >= Then < d, [Q(J ]) >=< Sq (d0 ), [Q(J )] >=< d0 , Sq∗0 [Q(J )] >= It implies that Sq∗0 [Q(J )] is non-zero So, by Proposition 4·6, we get j0 = s − and Sq∗0 [Q(s − 2, j1 + 1, , js−1 + 1)] = [Q(s − 2, j1 , , js−1 )] As Q(s − 2, j1 , , js−1 ) is indecomposable, its class is non-zero in F2 ⊗ Ds Then so is Q(s − 2, j1 + 1, , js−1 + 1) Further, we have A < Sq (d), [Q(s − 2, j1 + 1, , js−1 + 1)] > = < d, Sq∗0 [Q(s − 2, j1 + 1, , js−1 + 1)] > = < d, [Q(s − 2, j1 , , js−1 )] >= Since Q(s − 2, j1 + 1, , js−1 + 1) is indecomposable, we conclude that Sq (d)  in P(F2 ⊗ H∗ (BVs ) G Ls The second part of the theorem is the dual statement of Proposition 4·6 Indeed, suppose d(I ) = d(i , , i s−1 ) ∈ P(F2 ⊗ H∗ (BVs )), then G Ls < Sq d(i , , i s−1 ), Q( j0 , , js−1 ) >=< d(i , , i s−1 ), Sq∗0 Q( j0 , , js−1 ) > If either j0  s − or at least one of j1 , , js−1 is even, then by Proposition 4·6, Sq∗0 Q( j0 , , js−1 ) is decomposable So, the right-hand side is zero Otherwise, if j0 = s−2 and j1 , , js−1 are all odd, then < Sq d(I ), Q(s − 2, j1 , , js−1 ) > = < d(I ), Sq∗0 Q(s − 2, j1 , , js−1 ) > j1 − js−1 − = < d(I ), Q s − 2, , , 2 > Squaring operation on the Dickson algebra 277 The right-hand side is non-zero if and only if j0 = s − = i , j1 = 2i + 1, , js−1 = 2i s−1 + Therefore, for any J = ( j0 , , js−1 ), we get < Sq d(i , , i s−1 ), Q(J ) > = 0, if i  s − 2, < Sq d(s − 2, , i s−1 ), Q(J ) > = < d(s − 2, 2i + 1, , 2i s−1 + 1), Q(J ) > The theorem is completely proved An explicit formula for the action of Sq on P(F2 ⊗ H∗ (BVs )) was given by the first name author in [12] for s G Ls We are now giving such a formula for this action with s = Definition 4·7 A monomial basis {[Q(I )] | I ∈ I} of F2 ⊗ Ds is called closed if from the A hypothesis (s − 2, j1 , , js−1 ) ∈ I with j1 , , js−1 odd it implies either (s − 2, ( j1 − 1)/2, , ( js−1 − 1)/2) ∈ I or [Q(s − 2, ( j1 − 1)/2, , ( js−1 − 1)/2)] = The basis of F2 ⊗ Ds given by Hưng–Peterson [18] for s and by Giambalvo– A Peterson [9, theorem 5·7] for s = is closed We predict that there exists a closed basis of F2 ⊗ Ds for any s A Let d(I ) be the element in P(F2 ⊗ H∗ (BVs )) that is dual to [Q(I )] with respect to the G Ls basis {[Q(I )] | I ∈ I} of F2 ⊗ Ds In general, the following proposition is not a direct A consequence of Theorem 1·2, because d(I ) is always primitive while d(I ) may not be In fact, one has d(I ) = d(I ) + other terms However, in some special cases, for instance when the elements of the basis are in pairwise distinct degrees, then Proposition 4·8 follows immediately from Theorem 4·1 P ROPOSITION 4·8 If {[Q(I )] | I ∈ I} is a closed basis of F2 ⊗ Ds , then A Sq (d(i , i , , i s−1 )) = d(i , 2i + 1, , 2i s−1 + 1), 0, i = s − 2, otherwise, for arbitrary I = (i , , i s−1 ) ∈ I Proof For I = (i , , i s−1 ) and J = ( j0 , , js−1 ) both in I, we have < Sq d(I ), [Q(J )] >=< d(I ), Sq∗0 [Q(J )] > If either j0  s − or at least one of j1 , , js−1 is even, then Sq∗0 [Q(J )] = by Proposition 4·6 So < Sq d(I ), [Q(J )] >= Otherwise, if j0 = s − and j1 , , js−1 are all odd, then by Proposition 4·6 again < d(I ), Sq∗0 [Q(J )] >=< d(I ), Q s − 2, js−1 − j1 − , , 2 > Since {[Q(I )] | I ∈ I} is a closed basis, from the hypothesis J ∈ I it implies either [Q(s − 2, ( j1 − 1)/2, , ( js−1 − 1)/2)] = or (s − 2, ( j1 − 1)/2, , ( js−1 − 1)/2) ∈ I Hence, by the definition of d(I ) as the dual element of [Q(I )] with respect to the basis in question, we conclude that < Sq d(I ), [Q(J )] >= if and only if I = (s − 2, ( j1 − 1)/ 2, , ( js−1 − 1)/2), or equivalently i = s − 2, j1 = 2i + 1, , js−1 = 2i s−1 + The proposition follows ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 278 The following is the dual statement of [9, theorem 5·7] The basis elements in the theorem are not in pairwise distinct degrees However, the basis in question is closed So one can apply Proposition 4·8 but not Theorem 4·1 in order to get an explicit formula for the action of Sq on the following module T HEOREM 4·9 ([9, theorem 5·7]) P(F2 ⊗ H∗ (BV5 )) has a basis consisting of: G L5 (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) d(0, 0, 0, 0, 2d − 1), d; d(0, 0, 1, 2c − 1, 2d+1 − 2c − 1), c d; d(0, 2, 2b − 1, 2c+1 − 2b − 1, 2d − 1), b c = d; d(0, 2, 2b − 1, 2c+1 − 2b − 1, 2d+1 − 2c+1 − 1), b c < d; d(1, 1, 2b − 1, 2c+1 − 2b − 1, 2d − 1), b c = d; d(1, 1, 2b − 1, 2c+1 − 2b − 1, 2d+1 − 2c+1 − 1), b c < d; d(3, 2a − 1, 2b+1 − 2a − 1, 2c+1 − 2b+1 − 1, 2d+1 − 2c+1 − 1), a b < c < d; d(3, 2a − 1, 2b+1 − 2a − 1, 2c+1 − 2b+1 − 1, 2c − 1), a b < c = d; d(3, 2a − 1, 2b+1 − 2a − 1, 2b − 1, 2d+2 − 3.2b − 1), a b = c d Note that, in the expressions of items (vii)-(ix), if d(3, i , i , i , i ) is assigned (a, b, c, d) then d(3, 2i + 1, 2i + 1, 2i + 1, 2i + 1) is assigned (a + 1, b + 1, c + 1, d + 1) Vanishing degrees of the squaring operation In this section, we prove the following theorem, which is also numbered as Theorem 1·4 in the introduction, that gives some degrees in which the squaring operation vanishes T HEOREM 5·1 The squaring operation Sq on P(F2 ⊗ H∗ (BVs )) vanishes in any G Ls degree δ, which: (i) either satisfies ν(δ + s) [log (s − 2)] + for s 3; (ii) or is not of the form δs defined inductively for s as follows j j j δs = δs−1 − + 2s−1 [κ1 s−1 κ2 s−2 · · · κs−1 (s − 2) + 1], for arbitrary non-decreasing sequence [log (s − 2)] < j1 δ2 = j1 +1 − ··· j2 js−1 , where In [9] Giambalvo and Peterson showed that every indecomposable can be expressed in terms of the so-called “reducible” indecomposable monomials (modulo ADs ) i s−1 Definition 5·2 ([9]) A monomial Q(I ) = Q is,0 Q is,1 Q s,s−1 of Ds is called reducible if there exists an s-tuple J = [ j0 , j1 , , js−1 ] of non-negative integers such that i = κsj0 (0), j k (i + i + · · · + i k−1 ) − (i + i + · · · + i k−1 ), i k = κs−k for k s − Then J = [ j0 , j1 , , js−1 ] is called the reduced form of I The reduced form J = [ j0 , j1 , , js−1 ] is said to have non-decreasing terms if j for all j +1 T HEOREM 5·3 ([9, theorems 2·3 and 5·4]) The F2 -vector space F2 ⊗ Ds is generated A by classes of indecomposable monomials that are reducible and their reduced forms have non-decreasing terms Squaring operation on the Dickson algebra 279 Example 5·4 Following Hưng and Peterson [18], F2 ⊗ D3 has a basis consisting of the A classes of monomials Q(0, 0, 2t − 1), Q(1, 2t − 1, 2u − 2t − 1), t, < t < u By Giambalvo and Peterson [9], these monomials are reducible with respectively the reduced forms [0, 0, t] and [1, t, u − 1], which have non-decreasing terms Similar examples of Theorem 5·3 for s = and are available in [18] and [9] respectively According to Lemma 4·4, if i = κsj0 (0) = s −2 then j0 = α(s −2), the number of ones in the dyadic expansion of s − In this case, we rewrite the reduced form J by [ j1 , , js−1 ] for short, because j0 is known L EMMA 5·5 Q(s − 2, i , , i s−1 ) is reducible with the reduced form [ j1 , , js−1 ] if and only if Q(s − 2, 2i + 1, , 2i s−1 + 1) is also reducible with reduced form [ j1 + 1, , js−1 + 1] Proof It is easy to see that 2(κs (r ) − r ) = κs (2r − (s − 2)) − (2r − (s − 2)) for positive integers r, s with 2r s − Therefore 2(κsa (r ) − r ) = κsa (2r − (s − 2)) − (2r − (s − 2)) j1 for any a Applying this to i , we have i = κs−1 (s − 2) − (s − 2) if and only if j1 j1 2i + = κs−1 (s − + 1) − (s − 2) = κs−1 (κs−1 (s − 2)) − (s − 2) By the same argument, we get the statement for i , , i s−1 The lemma is proved i i1 s−1 L EMMA 5·6 Let Q(I ) = Q s−2 s,0 Q s,1 · · · Q s,s−1 be a reducible monomial with the reduced j1 form [ j1 , , js−1 ] Then i = − Additionally, if Q(I ) is indecomposable, then j1 > [log (s − 2)] Proof The first claim is proved by the definition of the reducible monomial We see that κs−1 (s − 2) = (s − 2) + So, by induction on , we get κs−1 (s − 2) = s − + − j for any Thus i = κs−1 − (s − 2) = j1 − The second claim is a consequence of case in the proof of [9, theorem 5·4] j Using the same argument in the proof of Lemma 4·4, we can show that κk (2 N + a) = j N +κk (a) for any positive integers k, a and a large enough integer N Then, by the definition of the reduced form, if [ j1 , , js−1 ] is the reduced form of Q(I ), then it is also the reduced N form of Q 2s,0 Q(I ) Suppose to the contrary that j1 [log (s − 2)], then by [9, theorem 5·4], we have N Q 2s,0 Q(I ) = Sq i (Q(J )) 0 and N large enough, so we can factor Q 2s,0 out, and get a decomposition of Q(I ) The lemma is proved N N L EMMA 5·7 Suppose that for d ∈ P(F2 ⊗ H∗ (BVs )) there is no reducible indecomposG Ls able monomial of the form Q(s − 2, i , , i s−1 ), whose reduced form has non-decreasing terms, such that < d, Q(s − 2, i , , i s−1 ) >= Then Sq (d) = ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 280 Proof To prove Sq (d) = we show that < Sq (d), Q(K ) >= for any reducible indecomposable Q(K ) = Q(k0 , , ks−1 ) ∈ Ds , whose reduced form has non-decreasing terms We have < Sq (d), Q(K ) >=< d, Sq∗0 (Q(K )) > If Sq∗0 (Q(K )) is indecomposable, then by Proposition 4·6, k0 = s − and k1 , , ks−1 are odd In this case, Sq∗0 (Q(K )) = Q(s − 2, (k1 − 1)/2, , (ks−1 − 1)/2) So, from Lemma 5·5 and the hypothesis of the lemma, we get < d, Sq∗0 (Q(K )) >= Otherwise, if Sq∗0 (Q(K )) is decomposable, then < d, Sq∗0 (Q(K )) >= as d ∈ P(F2 ⊗ H∗ (BVs )) G Ls The lemma is proved The following is an immediate consequence of Lemma 5·7 C OROLLARY 5·8 If there is no reducible indecomposable of the form Q(s − 2, i , , i s−1 ) in degree δ, whose reduced form has non-decreasing terms, then Sq : P(F2 ⊗ H∗ (BVs ))δ → P(F2 ⊗ H∗ (BVs ))s+2δ vanishes G Ls G Ls For abbreviation, we denote by Is the s-tuple (s − 2, i , , i s−1 ) L EMMA 5·9 If Q(Is ) is a reducible indecomposable in Ds , whose reduced form has nondecreasing terms, then δs = deg Q(Is ) satisfies the following: (i) ν(δs + s) > [log (s − 2)] + for s 3; (ii) δ is of the form δs given in Theorem 1·4 In particular, δ + s is of the forms listed respectively in Lemmas 6·3 – 6·5 for s i i1 s−1 L EMMA 5·10 Suppose Q(Is ) = Q s−2 s,0 Q s,1 · · · Q s,s−1 ∈ Ds is reducible with the reduced form Js = [ j1 , , js−1 ] Then: i i1 s−2 (i) ([18, theorem 4·2(2b)]) Q(Is−1 ) = Q s−3 s−1,0 Q s−1,1 · · · Q s−1,s−2 ∈ Ds−1 is also reducible with the reduced form Js−1 = [ j1 , , js−2 ] (ii) δs = deg Q(Is ) can inductively be computed as follows: j j j δs = δs−1 − + 2s−1 [κ1 s−1 κ2 s−2 · · · κs−1 (s − 2) + 1], where δ2 = j1 +1 − Proof In this proof, we need to use Lemma 5·11 below (i) To make the paper self-contained, we give here an alternative expression and proof of this part by using some terminologies that differ from the original ones in Hưng and Peterson [18, theorem 4·2] By Definition 5·2 of reduced form and Lemma 5·11, we have j j 1 i = κs−1 (s − 2) − (s − 2) = κs−2 (s − 3) − (s − 3) Similarly j k i k = κs−k (s − + i + · · · + i k−1 ) − (s − + i + · · · + i k−1 ) j k = κs−k−1 (s − + i + · · · + i k−1 ) − (s − + i + · · · + i k−1 ), for k s − From Definition 5·2, (i) follows Squaring operation on the Dickson algebra 281 (ii) As deg Q s,i = 2s − 2i , δs = deg Q(Is ) can be written as follows: δs = 2s−1 (s − + i + · · · + i s−1 ) + 2s−2 (s − + i + · · · + i s−2 ) + · · · +2(s − + i ) + s − j j j j j 1 = 2s−1 (κ1 s−1 κ2 s−2 κs−1 (s − 2)) + 2s−2 (κ2 s−2 κs−1 (s − 2)) + · · · j (s − 2) + s − +2κs−1 Applying Lemma 5·11 to each term of this sum, we get j j j (s − 2) + 1] + δs−1 − δs = 2s−1 [(κ1 s−1 κ2 s−2 · · · κs−1 Obviously, Q(I2 ) = Q 22,1−1 , so δ2 = j1 +1 − The lemma is proved j1 The following is an arithmetical property of the function κs L EMMA 5·11 Let r and i be positive integers Then i κs+1 (r + 1) = κsi (r ) + Proof We only consider the case r s − The case r < s − is similarly investigated By definition, κs+1 (r + 1) = r + + 2ν(r −(s−2)) It implies that κs+1 (r + 1) = κs (r ) + By induction on i, we get i−1 i (r + 1) = κs+1 (κs+1 (r + 1)) = κs+1 (κsi−1 (r ) + 1) = κsi (r ) + κs+1 The lemma is proved The degrees of indecomposable monomials Q(Is ), whose reduced forms have nondecreasing terms, can inductively be computed by means of the functions κs j j j1 (s − 2) From Lemmas 5·10 and 5·11, More precisely, set σs = κ1 s−1 κ2 s−2 · · · κs−1 δs = δs−1 − + 2s−1 (σs + 1), j σs = κ1 s−1 (σs−1 + 1), where s 3, σ2 = j1 − and δ2 = j1 +1 − j This is why we need an explicit formula for κ1 as given below Note that, by definition, κ1 (r ) = r + 2ν(r +1) fills in the first gap in the dyadic expansion of r L EMMA 5·12 Let a, b, t, j be any positive integers with b (a + t) + Then ⎧ j+t+2 − 1, j b, ⎨2 j b a t+1 b j+t+1 − 1, b > j a, κ1 (2 + (2 − 1)) = + ⎩ b + 2a (2t+1 − 1) + j − 1, a > j Proof The lemma is proved by induction on j It is easy to see that κ1 (2b + 2a (2t+1 − 1)) = 2b + 2a (2t+1 − 1) + Suppose inductively that j κ1 (2b + 2a (2t+1 − 1)) = 2b + 2a (2t+1 − 1) + j − ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 282 for a > j Hence j+1 κ1 (2b + 2a (2t+1 − 1)) = κ1 (2b + 2a (2t+1 − 1) + j − 1) = 2b + 2a (2t+1 − 1) + j − + 2ν(2 +2 b a (2t+1 −1)+2 j ) (by definition) = + 2a (2t+1 − 1) + j+1 − b So, we obtain the third equality for the case a > j If a = j + 1, then this formula gives κ1a (2b + 2a (2t+1 − 1)) = 2b + 2a+t+1 − This is a particular case of the second equality in the lemma It serves as the starting point for an another inductive procedure The remaining two equalities of the lemma are shown by similar arguments Proof of Lemma 5·9 (i) Let [ j1 , , js−1 ] be the reduced form, which has non-decreasing ··· js−1 , and according to Lemma 5·6, j1 > terms, of Q(I ) From the hypothesis, j1 [log (s − 2)] Using the two formulas just before Lemma 5·12, we can inductively compute δs = deg Q(Is ) We show by induction on s that ν(δs + s) j1 + and ν(σs + 1) j1 Indeed, it is obvious for s = Suppose inductively that it holds for any rank less than s, j1 + and ν(σi + 1) j1 for every i < s By Lemma 5·12, from i.e that ν(δi + i) j js−1 j1 it follows that ν(σs + 1) = ν(κ1 s−1 (σs−1 + 1) + 1) {ν(σs−1 + 1), j1 } = j1 So ν(δs + s) {ν(δs−1 + (s − 1)), s − + j1 } j1 + > [log (s − 2)] + 1, for s Thus (i) is proved (ii) In the general case, we obtain this part by combining Lemmas 5·10 – 5·12 In case s 7, the part follows from the fact that the degree δs in Part (ii) of Lemma 5·10 will explicitly be computed in Lemmas 6·1 – 6·5 We are now ready to give a proof of Theorem 5·1 Proof of Theorem 5·1 According to Lemma 5·9, if δ satisfies the hypothesis of the theorem, then there is no reducible indecomposable of the form Q(s − 2, i , , i s−1 ), whose reduced form has non-decreasing terms, in the degree δ Therefore, by Corollary 5·8, Sq : P(F2 ⊗ H∗ (BVs ))δ → P(F2 ⊗ H∗ (BVs ))s+2δ vanishes G Ls The theorem is proved G Ls Vanishing degrees of the squaring operation in small ranks Using Lemma 5·12 and the inductive formula for σs to compute the degrees of indecomposable monomials Q(Is ), whose reduced forms have non-decreasing terms, we automatically obtain the following five lemmas for s In principle, this method can be applied to any bigger value of s by means of an inductive procedure on s Taking the conclusion j1 > [log (s − 2)] of Lemma 5·6 into account, and using the inductive formulas for δs and σs in the preceding section, we obtain the following lemmas by routine computation L EMMA 6·1 Let Q(I ) = Q(1, i , i ) be an indecomposable monomial in degree δ of D3 with the non-decreasing reduced form [ j1 , j2 ] Then, j1 and δ + = j1 +1 + j2 +3 Squaring operation on the Dickson algebra 283 L EMMA 6·2 Let Q(I ) = Q(2, i , i , i ) be an indecomposable monomial in degree δ of D4 with the non-decreasing reduced form [ j1 , j2 , j3 ] Then, j1 and δ+4= j1 +1 + j2 +3 + j3 +4 , j1 +1 + j2 +5 , j1 j1 j2 < j3 , j2 = j3 L EMMA 6·3 Let Q(I ) = Q(3, i , i , i , i ) be an indecomposable monomial in degree δ of D5 with the non-decreasing reduced form [ j1 , j2 , j3 , j4 ] Then, j1 and ⎧ j +1 j2 < j3 < j4 , ⎨ + j2 +3 + j3 +4 + j4 +5 , j1 j1 +1 j2 +3 j4 +6 +2 +2 , j1 j2 < j3 = j4 , δ+5= ⎩ j1 +1 + j3 +5 + j4 +6 , j1 j2 = j3 j4 L EMMA 6·4 Let Q(I ) = Q(4, i , i , i , i , i ) be an indecomposable monomial in degree δ of D6 with the non-decreasing reduced form [ j1 , j2 , j3 , j4 , j5 ] Then, j1 and ⎧ j1 +1 + j2 +3 + j3 +4 + j4 +5 + j5 +6 , j1 j2 < j3 < j4 < j5 , ⎪ ⎪ ⎪ j +1 j2 +3 j3 +4 j5 +7 ⎪2 + + + , j j2 < j3 < j4 = j5 , ⎪ ⎪ ⎪ j1 +1 j2 +3 j4 +6 j5 +7 ⎪ + + + , j j2 < j3 = j4 j5 , ⎨ j1 +1 j3 +5 j5 +7 +2 +2 , j1 j2 = j3 j4 = j5 − 1, δ+6= ⎪ j1 +1 j3 +5 j4 +6 j5 +6 ⎪ + + + , j j = j j4 < j5 − 1, ⎪ ⎪ ⎪ j1 +1 j3 +5 j4 +5 j4 +6 j4 +7 ⎪ ⎪ + + + + , j j = j < j4 = j5 , 2 ⎪ ⎩ j1 +1 + j2 +8 , j1 j2 = j3 = j4 = j5 L EMMA 6·5 Let Q(I ) = Q(5, i , i , i , i , i , i ) be an indecomposable monomial in degree δ of D7 with the non-decreasing reduced form [ j1 , j2 , j3 , j4 , j5 , j6 ] Then, j1 and ⎧ ⎪ j1 +1 + j2 +3 + j3 +4 + j4 +5 + j5 +6 + j6 +7 , j2 < j3 < j4 < j5 < j6 , ⎪ ⎪ ⎪ j1 +1 + j2 +3 + j3 +4 + j4 +5 + j5 +8 , ⎪ j2 < j3 < j4 < j5 = j6 , ⎪ ⎪ ⎪ ⎪ j1 +1 + j2 +3 + j3 +4 + j5 +7 + j6 +8 , j2 < j3 < j4 = j5 j6 , ⎪ ⎪ ⎪ j1 +1 j2 +3 j3 +9 ⎪ + + , j < j = j = j = j6 , 2 ⎪ ⎪ ⎪ j1 +1 j2 +3 j3 +6 j5 +6 j5 +7 j5 +8 ⎪ + + + + + , j < j = j < j = j6 , ⎪ ⎪ ⎪ j1 +1 j2 +3 j3 +6 j5 +9 ⎪ +2 +2 +2 , j2 < j3 = j4 j5 = j6 − 1, ⎪ ⎪ ⎨ j1 +1 + j2 +3 + j3 +6 + j5 +7 + j6 +7 , j2 < j3 = j4 j5 < j6 − 1, δ+7 = j1 +1 j3 +5 j5 +7 j6 +8 + + + , j = j j = j j6 − 1, ⎪ −1 ⎪ ⎪ j1 +1 j3 +5 j4 +6 j5 +8 ⎪ ⎪ + + + , j = j j < j − = j6 − 1, 2 ⎪ ⎪ j1 +1 j3 +5 j4 +6 j5 +6 j6 +7 ⎪ ⎪ + + + + , j = j j < j − < j6 − 1, 2 ⎪ ⎪ ⎪ j +1 j +5 j +5 j +6 j +9 4 ⎪ +2 +2 +2 +2 , j2 = j3 < j4 = j5 = j6 , ⎪ ⎪ ⎪ j1 +1 j3 +5 j4 +5 j4 +6 j4 +7 j6 +8 ⎪ + + + + + , j2 = j3 < j4 = j5 j6 − 1, ⎪ ⎪ ⎪ j1 +1 j2 +7 j2 +9 ⎪ + + , j = j = j = j = j6 , ⎪ ⎪ ⎩ j1 +1 + j2 +8 + j6 +8 , j2 = j3 = j4 = j5 < j6 Application to the Lannes–Zarati homomorphism In this section, we apply the results of the preceding parts on the squaring operation to investigate the Lannes–Zarati homomorphism, s (F2 , F2 ) → P(F2 ⊗ H∗ (BVs )) ϕs : E xtA G Ls s Let {ai | i 0} be an Sq -family in E xtA (F2 , F2 ), i.e = (Sq )i (a0 ) for any i As the s squaring operations on E xtA (F2 , F2 ) and on P(F2 ⊗ H∗ (BVs )) commute with each other G Ls ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 284 through the Lannes–Zarati homomorphism (see [15]), ϕs (ai ) = (Sq )i (ϕs (a0 )), for any i So, if ϕs (at ) = 0, then ϕs (ai ) = for every i t Therefore, it is important to investigate ϕs (at ) with t small The following, which is also numbered as Corollary 1·6 in the introduction, is a consequence of Corollary 1·3 P ROPOSITION 7·1 If {ai | i for any i > s,∗ 0} is a finite Sq -family in E xtA (F2 , F2 ), then ϕs (ai ) = 0} is a finite Sq -family with the equation Proof Combining the fact that {ai | i i i 0} is a finite Sq ϕs (ai ) = ϕs (Sq ) (a0 ) = (Sq ) (ϕs (a0 )), it implies that {ϕs (ai )| i family in P(F2 ⊗ H∗ (BVs )) By Corollary 1·3, the latter Sq -family is finite with length at G Ls most Thus ϕs (ai ) = 0, for any i > The proposition is proved The following, which is also numbered as Proposition 1·7 in the introduction, is a consequence of Theorem 1·4 and Corollary 1·5 s 0} be an Sq -family in E xtA (F2 , F2 ) Suppose δ = P ROPOSITION 7·2 Let {ai |i Stem(a0 ) satisfies one of the following conditions: (i) ν(δ + s) [log (s − 2)] + for s 3; (ii) δ is not of the form δs given in Theorem 1·4 In particular, δ + s is not of the forms listed respectively in Lemmas 6·3 – 6·5 for s Then ϕs (ai ) = for any i > Proof Applying Theorem 1·4 and Corollary 1·5, from the hypotheses we conclude that Sq vanishes in degree δ of P(F2 ⊗ H∗ (BVs )) So Sq (ϕs (a0 )) = As a consequence, G Ls ϕs (ai ) = ϕs (Sq )i (a0 ) = (Sq )i (ϕs (a0 )) = 0, for any i > The proposition is proved The following is also numbered as Proposition 1·8 in the introduction P ROPOSITION 7·3 If {ai |i then ϕs (ai ) = for any i s 0} is an Sq -family in E xtA (F2 , F2 ) with Stem(a0 ) < 2s−1 , Proof The target of the Lannes–Zarati homomorphism, P(F2 ⊗ H∗ (BVs )), is dual to G Ls F2 ⊗ Ds On the other hand, Ds = F2 [Q s,0 , , Q s,s−1 ] is zero in each degree, which is A less than deg Q s,s−1 = 2s−1 So, if Stem(a0 ) < 2s−1 then ϕs (a0 ) = This implies that ϕs (ai ) = for i The proposition follows The table below reports on the application of Propositions 7·3 and 7·2 to all the known s (F2 , F2 ) listed by Bruner [5] Only the elements of homological degree at most 39 in E xtA initial elements of Sq -families, such as n, x, r but not n , x1 , r1 , appear on the a0 -column The “plus” signal, which is assigned to the Sq -family initiated by a0 , on the fourth or the fifth column means that Proposition 7·3 or Proposition 7·2 can respectively be applied to this Sq -family In particular, applying Proposition 7·2, we get the fifth column of this table by a routine compare of the stems of the elements in question with the degrees listed in Lemmas 6·3–6·5 Squaring operation on the Dickson algebra 285 It should be noted that the Lannes–Zarati homomorphism ϕs has been shown in [12, 15] to vanish in positive stems for the homological degree s = and This is why the table concerns only with s In the table, the elements D11 , H11 are presumed by Bruner [5] to be the images under Sq of D1 and H1 respectively However, this has not been proved so far s a0 Stem(a0 ) ϕs (ai ) = ∀ i as Stem(a0 ) < 2s−1 5 5 5 5 5 Ph Ph n x D1 H1 Q3 D11 x5,77 x5,80 H11 11 31 37 52 62 67 109 125 128 129 + + 6 6 6 6 6 6 6 6 6 6 6 6 r q t y C G D2 A A A” x6,47 x6,53 t1 x6,68 C1 x6,89 G1 x6,94 x6,97 x6,99 x6,101 x6,102 x6,107 x6,110 30 32 36 38 50 54 58 61 61 64 71 76 78 85 106 108 114 124 126 127 128 128 132 134 + ϕs (ai ) = ∀ i > info on Stem(a0 )+ s + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ` N GUY Eˆ˜ N H V HƯNG AND V O˜ T N Q U YNH 286 (Continued) s a0 Stem(a0 ) 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 Pc0 i j k l m B1 B2 Q2 B3 x7,33 x7,34 x7,40 x7,53 m1 x7,57 x7,74 x7,79 x7,81 x7,83 x7,84 x7,88 x7,90 x7,92 x7,93 x7,97 x7,101 x7,103 x7,109 x7,110 x7,118 x7,124 16 23 26 29 32 35 46 48 57 60 63 63 66 75 77 77 87 95 97 99 99 101 103 105 107 112 121 124 127 127 130 133 + + + + + + + + + + + + 8 8 8 8 Pd0 Pe0 N x8,32 x8,33 G 21 P D3 x8,51 22 25 46 62 62 68 69 74 + + + + + + + + ϕs (ai ) = ∀ i as Stem(a0 ) < 2s−1 ϕs (ai ) = ∀ i > info on Stem(a0 )+ s + + + + + + + + + + + + + + + + + + + + Squaring operation on the Dickson algebra (Continued) s a0 Stem(a0 ) 8 8 8 8 8 8 8 8 8 8 x8,57 x8,75 x8,78 x8,80 x8,83 x8,93 x8,105 x8,113 x8,114 x8,115 x8,116 x8,117 x8,118 x8,119 x8,120 x8,124 x8,132 x8,133 x8,136 x8,139 x8,140 77 91 93 94 96 101 112 123 124 125 125 126 126 126 126 127 129 130 131 132 132 s 39 all ϕs (ai ) = ∀ i as Stem(a0 ) < 2s−1 + + + + + + + + + + + + + + + + 287 ϕs (ai ) = ∀ i > info on Stem(a0 ) + s + + + + + + Acknowledgements The authors would like to thank Lˆe M H`a for many valuable discussions [1] [2] [3] [4] [5] [6] [7] [8] [9] REFERENCES J F A DAMS On the non-existence of 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Ls Squaring operation on the Dickson algebra 273 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. .. the dual of the squaring operation on the Dickson invariants It will be applied to investigate the behavior of the squaring operation i s−1 For abbreviation, we denote by Q(I ) the monomial Q