Nguyen Sum
We compute the action of the primitive Steenrod-Milnor operations on generators of algebras of invariants of subgroups of general linear group GL_n=GL(n,F_p) in the polynomial algebra with p an odd prime number.
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Let $p$ be an odd prime number. Denote by $GL_n = GL(n,\mathbb F_p)$ the general linear group over the prime field $\mathbb F_p$. Each subgroup of $GL_n$ acts on the algebra $P_n=E(x_1,\ldots,x_n)\otimes \mathbb F_p(y_1,\ldots,y_n)$ in the usual manner. We grade $P_n$ by assigning $\dim x_i=1$ and $\dim y_i=2.$ This algebra is a module over the mod $p$ Steenrod algebra $\mathcal A_p$. The purpose of the paper is to compute the action of the Steenrod-Milnor operations on the generators of $P_2^{GL_2}$. More precisely, we explicitly determine the action of $St^{(i,j)}$ on the Dickson invariants …
Product formulas for conjugated motivic milnor basis
研究给出共轭动机米尔诺基中模2动机斯廷罗德代数的乘积公式，为计算动机亚当斯谱序列的E2页提供有效工具。
The algebra ${\mathsf A}_q$ of Steenrod $q$th powers, where $q = p^e$ is a power of a prime $p$, is isomorphic to a subalgebra ${\mathsf A}'_q$ of the algebra of Steenrod $p$th powers ${\mathsf A}_p$. The filtration of ${\mathsf A}_p$ by powers of its augmentation ideal was studied by J. P. May in his Princeton thesis of 1964. We extend some of May's results to ${\mathsf A}_q$ and obtain a convenient set of defining relations for the graded algebra $E^0({\mathsf A}_q)$. In the case $q=p$, we recover the observation of S. B. Priddy that the subalgebra $E^0({\mathsf A}_p(n-2))$ of $E^0({\mathsf …
Normalized Derivations for Steenrod--Milnor Operations on the Dickson Algebra and Applications
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Let $p$ be an odd prime number. We study the problem of determining the module structure over the mod $p$ Steenrod algebra $\mathcal A(p)$ of the Dickson algebra $D_n$ consisting of all modular invariants of general linear group $GL(n,\mathbb F_p)$. Here $\mathbb F_p$ denotes the prime field of $p$ elements. In this paper, we give an explicit answer for $n=2$. More precisely, we explicitly compute the action of the Steenrod-Milnor operations $St^{S,R}$ on the generators of $D_n$ for $n=2$ and for either $S=\emptyset, R=(i)$ or $S=(s), R=(i)$ with $s,i$ arbitrary nonnegative integers.
Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology.
This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over F_2 by Steenrod operations, J. Algebra 246 (2001), 739--760] for odd primes p. It is proved that for certain irreducible representations L(lambda) of the full matrix semigroup M_n(F_p), the first occurrence of L(lambda) as a composition factor in the polynomial algebra P=F_p[x_1,...,x_n] is linked by a Steenrod operation to the first occurrence of L(lambda) as a submodule in P. This operation is given explicitly as the image of an admissible monomial in the Steenrod algebra A_p …
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本文给出奇素数情形下本原 Milnor 运算作用于 $GL_n$ 在多项式代数 $P_n$ 中 Dickson 不变量代数生成元的公式。
We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $ξ_k$ in the category of associative algebras freely kills the higher generators $ξ_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us…
In the paper "The Steenrod algebra and its dual", J.Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme $G_p$ represented by the dual Hopf algebra of the mod $p$ Steenrod algebra. Then, $G_p$ assigns a graded commutative algebra $A_*$ over a prime field of finite characteristic $p$ to a set of isomorphisms of the additive formal group law over $A_*$, whose group structure is given by the composition of formal power series. The aim of this paper is to show some group theoretic properties of $…
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We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.
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本文研究了$H^{*}(V^{\oplus 5})$的hit问题，验证了Singer猜想在特定度数下的成立，并提供了新算法的实现。
Let ΩX be the space of Moore loops on a finite, q-connected, n-dimensional CW complex X, and let R be a subring of Q containing 1/2. Let p(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M / Torsion M. We show that the inclusion of the sub-Lie algebra P of primitive elements of FH_*(ΩX;R) induces an isomorphism of Hopf algebras UP = FH_*(ΩX;R), provided p(R) > n/q - 1. Furthermore, the Hurewicz homomorphism induces an embedding of F(π_*(ΩX)\otimes R) in P, with torsion cokernel. As a corollary, if X is elliptic, then FH_*(ΩX;R) is a finitely-generate…
Let $F_2$ be the prime field of two elements and let $GL_s:= GL(s, F_2)$ be the general linear group of rank $s.$ Denote by $\mathscr A$ the Steenrod algebra over $F_2.$ The (mod-2) Lambda algebra, $Λ,$ is one of the tools to describe those mysterious "Ext-groups". In addition, the $s$-th algebraic transfer of William Singer \cite{Singer} is also expected to be a useful tool in the study of them. This transfer is a homomorphism $Tr_s: F_2 \otimes_{GL_s}P_{\mathscr A}(H_{*}(B\mathbb V_s, F_2))\to {\rm Ext}_{\mathscr {A}}^{s,s+*}(F_2, F_2),$ where $\mathbb V_s$ denotes the elementary abelian $2$…
A derived Milnor-Moore theorem
该文通过 Koszul 对偶构造包络 Hopf 代数函子，证明其在有理范畴全忠实，并在色同伦论中将其单位与 Goodwillie 完备化等同，推广了 Milnor-Moore 定理。
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研究五阶彼得森击中问题，通过精确稀疏消元等方法确定余击中模相关值，证明第五个辛格上同调转移是同构，找到无定向配边中49次幂的几何生成元，明确其与代数不变量的边界关系。
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本文通过开发计算Kameko同态核的GL(q)-不变量的新算法，证明了在双次数(6,6+36)上Singer猜想不成立。
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