DSpace at VNU: The hit problem for the Dickson algebra

The Hit Problem for the Dickson Algebra Author(s): Nguyễn H V Hung and Tran Ngọc Nam Source: Transactions of the American Mathematical Society, Vol 353, No 12 (Dec., 2001), pp 5029-5040 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2693915 Accessed: 13-01-2016 00:09 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/2693915?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive We use information technology and tools to increase productivity and facilitate new forms of scholarship For more information about JSTOR, please contact support@jstor.org American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society http://www.jstor.org This content downloaded from 150.135.239.97 on Wed, 13 Jan 2016 00:09:25 UTC All use subject to JSTOR Terms and Conditions TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 12, Pages 5029-5040 S 0002-9947(01)02705-2 Article electronically published on May 22, 2001 THE HIT PROBLEM FOR THE DICKSON ALGEBRA X NGUYEN H V HUNG AND TRAN NGOC NAM Dedicated to Professor Franklin P Peterson on the occasion of his 70th birthday Let the mod Steenrod algebra, A, and the general linear group, with Ixil = in the usual manner GL(k,F2), act on Pk := F2[xl, ,xk] We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans Amer Math Soc 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra Dk (Pk)GL(kF2) is A-decomposable in Pk for arbitrary k > This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in QoS? are the elements of Hopf invariant one and those of Kervaire invariant one ABSTRACT INTRODUCTION Let Pk F2 [xl, ,Xk] be the polynomial algebra over (the field of two elements) F2 in k variables, each of degree The general linear group GLk GL(k,F2) acts on Pk in the usual manner Dickson proves in [1] that the ring of invariants, Dk (Pk)GLk, is also a polynomial algebra Dk "- 2[Qk,k-1** Qk, where Qk,s denotes the Dickson invariant of degree 2k - 25 It can be defined by the inductive formula + Vk * Qk-l,si Qk,s = Qk-l,s-1 where, by convention, Qk,k = 1, Qk,s = Vk= for s < and fJ (Alxl?+?*+Ak-lXk-1 + Xk) Aj EF2 Let A be the mod Steenrod algebra The usual action of A on Pk commutes with that of GLk So Dk is an A-module One of the authors has been interested in the homomorphism jk: F2 ?(Pk)GLk A >' (F2 A Pk)GLk which is induced by the identity map on Pk (see [3]) Observing that ji is an isomorphism and i2 is a monomorphism, he sets up the following 1.1 (Nguyen H V Hu'ng [3]) ik = in positive degrees for k > Let Dk+ and A+ denote respectively the submodules of Dk and A consisting of all elements of positive degree Then Conjecture 1.1 is equivalent to Dk+ C A+ Pk Conjecture Received by the editors September 29, 1999 and, in revised form, February 22, 2000 2000 Mathematics Subject Classification Primary 55S10; Secondary 55P47, 55Q45, 55T15 Key words and phrases Steenrod algebra, invariant theory, Dickson algebra This work was supported in part by the National Research Project, No 1.4.2 (@2001 American Mathematical 5029 This content downloaded from 150.135.239.97 on Wed, 13 Jan 2016 00:09:25 UTC All use subject to JSTOR Terms and Conditions Society NGUYEN 5030 H V HUNG AND TRAN NGOC NAM for k > (see [3]) In other words, it predicts that every GLk-invariant element of positive degree is hit by the Steenrod algebra acting on Pk for k > Conjecture 1.1 is related to the hit problem of determination of F2 0Pk This A problem has first been studied by F Peterson [9], R Wood [14], W Singer [12], and S Priddy [10], who show its relationships to several classical problems in cobordism theory, modular representation theory, Adams spectral sequence for the stable homotopy of spheres, and stable homotopy type of classifying spaces of finite groups The tensor product F2 (Pk has explicitly been computed for k < The cases A k = and are not difficult, while the case k = is complicated and was solved by M Kameko [8] It seems unlikely that a very explicit description of F2 Pk A for general k will appear in the near future There is also another approach, the qualitative one, to the problem By this we mean giving conditions on elements of Pk to show that they go to zero in F2 0Pk, i.e belong to A+ Pk Peterson's A conjecture, which was established by Wood [14], claims that F2 0Pk = in degree A d such that a(d + k) > k Here a(n) denotes the number of ones in the dyadic expansion of n Recently, W Singer, K Monks, and J Silverman have refined the method of R Wood to show that many more monomials in Pk are in A+ Pk (See Silverman [11] and its references.) Conjecture 1.1 presents a large family, whose elements are predicted to be in A+ Pk In [3], one of the authors proves the equivalence of Conjecture 1.1 and a weak algebraic version of the conjecture on spherical classes stating that: There are no spherical classes in QoS? except the elements of Hopf invariant one and those of Kervaire invariant one He also gives two proofs of Conjecture 1.1 for the case k = In this paper, we establish this conjecture for every k > That Conjecture 1.1 is no longer valid for k = and is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes We have D+ C A+ *Pk for k > Main Theorem Recently, F Peterson and R Wood privately informed us that they had proved the theorem for k = and probably for k The readers are referred to [4] and [5] for some problems, which are closely related to the main theorem Additionally, the problem of determination of F2 Dk and its applications have been studied by A Hu'ng and Peterson [6], [7] The paper contains five sections Section is a preparation on the action of the Steenrod squares on the Dickson algebra We prove the main theorem in Section by means of two lemmata, which are later shown in Section and Section respectively PRELIMINARIES The action of the Steenrod squares on Dk is explicitly described as follows Theorem 2.1 ([2]) Sqi(Q-, ) Qk,r for Qk,rQk,t for i s)for k, s i=2s2r2 i = 2k 2- r