Hit polynomials and the canonical antiautomorphism of the Steenrod algebra
Hit polynomials and the canonical antiautomorphism of the Steenrod algebra
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命中多项式和 Steenrod 代数的规范反自同构
DOI：
10.1090/s0002-9939-1995-1254854-8
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发表时间：
1995
影响因子：
0.8
通讯作者：
J. H. Silverman
中科院分区：
文献类型：
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作者：
J. H. Silverman
本文推广了Davis（Proc.Amer.Math.Soc.44（1974），235-236）关于mod-2 Steenrod代数si（2）的反自同构的一个公式，并在此基础上建立了i-ii-i积Sq（0，...，0，a）·Sq（0，.，0，B）。本文还对作者和Singer（On the action of Steenrod squares on polynomial algebras II，J. Pure Appl. Algebra（to appear））关于j/（2）-作用在¥2[xx > ···，*s]上的一个猜想作了推广，并利用反自同构公式证明了推广猜想的几种情况。我们讨论了这两个猜想之间的关系，并给出了蒙克斯证明原猜想的一个特例的充分条件。最后，我们在一个表格中说明了已知为真的假设的特殊情况的相对强度。1.上同调运算的mod-2 Steenrod代数sA（2）是一个连通的Hopf代数，因此允许一个唯一的反自同构x [M][158]。本文推广了Davis [Dav 74]的一个反自同构公式，并利用这个结果研究了s>A（2）在W ~ 2 [xx，…，xs]，EP°°与自身的5重拷贝的mod-2上同调代数。sA（2）的Milnor基由几乎所有为0的非负整数序列的集合索引。让A？是这样的序列的集合。如果S eA？其中对于i > N，sx = 0，对应的基元素表示为Sq（S）= Sq（sx，.，sN）;iXs维度为\Sq（S）\ = £~，（2' l）sj。对于f> 1，定义它：A7 -A7 by（sx，.，sN）·，rtN）与r. = {sj> i**jt，i，0，t不整除i。设Sq，（S）= fSq（i，（S））.则| Sqt（S）|= ε（2* l）sj ;具体地，| Sqt（s）|=（2 * l）s。元素{Sq，（S）：S e S？} 1993年5月19日编辑收到。1991年数学学科分类。小学55 S 05、55 S10;中学20 J 05。
In this paper, we generalize a formula of Davis (Proc. Amer. Math. Soc. 44 (1974), 235-236) for the antiautomorphism of the mod-2 Steenrod algebra si (2), in the process formulating the analogue of the Adem relations for i-i i-i products Sq(0, ... , 0, a) ■ Sq(0, ... , 0, b). We also state a generalization of a conjecture by the author and Singer (On the action of Steenrod squares on polynomial algebras II, J. Pure Appl. Algebra (to appear)) concerning the j/(2)-action on ¥2[xx > ••• , *s] and use the antiautomorphism formula to prove several cases of the generalized conjecture. We discuss the relationship between the two conjectures and make explicit a sufficient condition for Monks's work to prove a special case of the original conjecture. Finally, we illustrate in a table the relative strengths of the special cases of the conjectures known to be true. 1. Statement of results The mod-2 Steenrod algebra sA(2) of cohomology operations is a connected Hopf algebra, and as such admits a unique antiautomorphism x [MÍ158]. In this paper we generalize an antiautomorphism formula of Davis [Dav74] and use the result to study the action of s>A(2) on W2[xx, ... , xs], the mod-2 cohomology algebra of the 5-fold copy of EP°° with itself. The Milnor basis of sA (2) is indexed by the set of sequences of non-negative integers almost all of which are 0. Let A? be the set of such sequences. If S eA? with s¡ = 0 for i > N, the corresponding basis element is denoted Sq(S) = Sq(sx,... ,sN);iXs dimension is \Sq(S)\ = £~ ,(2' l)sj . For / > 1, define it : A7 —► A7 by (sx, ... , sN) ■-» (rx, ... , rtN) with r. = {sj> i**jt, ' \0, t does not divide i. Let Sq, (S) =f Sq(i,(S)). Then \Sqt (S)\ = £(2* l)sj ; in particular, \Sq, (s)\ = (2' l)s. The elements {Sq,(S) : S e S?} form a basis for a Hopf Received by the editors May 19, 1993. 1991 Mathematics Subject Classification. Primary 55S05, 55S10; Secondary 20J05.