A total Steenrod operation as homomorphism of Steenrod algebra-modules

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

“…By Cartan's formula, it suffices to determine Sq i (t j ) and the instability axioms give Sq 1 (t j ) = t 2 j while Sq k (t j ) = 0 if k > 1. The Steenrod operations and related problems have been investigated by many authors (for instance Brunetti, Ciampella and Lomonaco [6], Brunetti and Lomonaco [7], Singer [41], Silverman [39], Monks [17], Wood [54]). The Steenrod algebra naturally acts on the cohomology ring H * (X) of a CW-complex X.…”

“…We assume that [f ] ω 6 ∈ [(QP ⊗9 n 0 ) >0 (ω 6 )] Σ 9 . Then, we have f ≡ ω6 73 i 80 γ i a i where γ i belong toF 2 .Let us consider the homomorphisms σ d : P ⊗9 −→ P ⊗9 , 1 d 8. An easy calculation shows thatσ 1 (f ) ≡ ω 673 j 79 γ j a j + γ 80 t2 1 t 2 t 3 t 4 t 5 t 2 6 t 7 t 8 t 9 ≡ ω 6 73 j 79 (γ j + γ 80 )a j + γ 80 a 80 , γ By these equalities and the relations σ d (f ) + f ≡ ω 6 0, 1 d 8 we get γ i = 0, 73 i 80.…”

On the modular representations of the general linear group $GL_5$ and the hit problem for the polynomial algebra of five variables

“…Using Theorem 2.2(i), we must have that y ∈ C ⊗5 10 . Following Tin [42], ω(y) belongs to the set (2, 2, 1), (2,4), (4, 1, 1), (4,3) . Therefore, it may be concluded that ω(t) belongs to the set ω (1) , ω (2) , ω (3) , ω (4) .…”

“…According to Proposition 3.1.1, we must have that t = t i t j t k t l y 2 in which y ∈ C ⊗5 10 (4, 3) and 1 ≤ i < j < k < l ≤ 5. By an simple computation, we notice that if x ∈ C ⊗5 10 (4, 3) and t i t j t k t l x 2 v for all v ∈ D 3 and 1 ≤ i < j < k < l ≤ 5, then either t i t j t k t l x 2 = c m , 39 ≤ m ≤ 48 or t i t j t k t l x 2 = c m x 4 1 with suitable monomial x 1 ∈ P ⊗5 3 , and 33 ≤ m ≤ 38, where the monomials c m , 33 ≤ m ≤ 48 are described as in Lemma 3.1.11(ii). We observe that ω 2 (a m ) = 4 0 and ω j (c m ) = 0 for arbitrary j > 2, 33 ≤ m ≤ 38, and the monomials c m are inadmissible.…”

“…space, called Steenrod operations of degrees k. They are stable cohomology operations, that is, they commute with suspension maps. The Steenrod operations and related problems have been investigated by many authors (for instance Brunetti, Ciampella, and Lomonaco [4], Brunetti, and Lomonaco [5], Nicholas [21]). All these Sq k form an algebraic structure which is known as the 2-primary Steenrod algebra A subject to the Adem relations (see also Walker and Wood [44]).…”