The Intersection of the Admissible Basis and the Milnor Basis of the Steenrod Algebra
نویسندگان
چکیده
We prove a conjecture of K. Monks 4] on the relation between the admissible basis and the Milnor basis of the mod 2 Steenrod algebra A 2 , and generalise the result to the mod p Steenrod algebra A p where p is prime. This establishes a necessary and suucient condition for the Milnor basis element P(r 1 ; r 2 ; : : : ; r k) and the admissible basis element P t 1 P t 2 : : : P t k to coincide. The main technique used is thèstripping' method which utilises the action of the dual algebra A p on A p. 1 The main result We shall prove the following result relating the Milnor basis and the admissible basis of the mod p Steenrod algebra A p. Here !(n) is the smallest integer such that p !(n) > n.
منابع مشابه
On the X basis in the Steenrod algebra
Let $mathcal{A}_p$ be the mod $p$ Steenrod algebra, where $p$ is an odd prime, and let $mathcal{A}$ be the subalgebra $mathcal{A}$ of $mathcal{A}_p$ generated by the Steenrod $p$th powers. We generalize the $X$-basis in $mathcal{A}$ to $mathcal{A}_p$.
متن کاملA note on the new basis in the mod 2 Steenrod algebra
The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations, denoted by $Sq^n$, between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space. Regarding to its vector space structure over $mathbb{Z}_2$, it has many base systems and some of the base systems can also be restricted to its sub algebras. On the contrary, in ...
متن کاملLinking first occurrence polynomials over Fp by Steenrod operations
This paper provides analogues of the results of [16] for odd primes p . It is proved that for certain irreducible representations L(λ) of the full matrix semigroup Mn(Fp), the first occurrence of L(λ) as a composition factor in the polynomial algebra P = Fp[x1, . . . , xn] is linked by a Steenrod operation to the first occurrence of L(λ) as a submodule in P. This operation is given explicitly a...
متن کاملInvariant elements in the dual Steenrod algebra
In this paper, we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${mathcal{A}_p}^*$ under the conjugation map $chi$ and give bounds on the dimensions of $(chi-1)({mathcal{A}_p}^*)_d$, where $({mathcal{A}_p}^*)_d$ is the dimension of ${mathcal{A}_p}^*$ in degree $d$.
متن کاملNilpotence in the Steenrod Algebra
While all of the relations in the Steenrod algebra, A, can be deduced in principle from the Adem relations, in practice, it is extremely difficult to determine whether a given polynomial of elements in A is zero for all but the most elementary cases. In his original paper [Mi] Milnor states “It would be interesting to discover a complete set of relations between the given generators of A”. In p...
متن کامل