Dimension Result for the Polynomial Algebra <svg style="vertical-align:-4.47127pt;width:124.3px;

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2013, Article ID 150704, 6 pages http://dx.doi.org/10.1155/2013/150704 Research Article Dimension Result for the Polynomial Algebra F2[𝑥1, . . . , 𝑥𝑛] as a Module over the Steenrod Algebra Mbakiso Fix Mothebe Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana Correspondence should be addressed to Mbakiso Fix Mothebe; Received 12 August 2013; Accepted 19 November 2013 Academic Editor: Alexander Rosa Copyright © 2013 Mbakiso Fix Mothebe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For 𝑛 ≥ 1, let P(𝑛) = F2 [𝑥1 , . . . , 𝑥𝑛 ] be the polynomial algebra in 𝑛 variables 𝑥𝑖 , of degree one, over the field F2 of two elements. The mod-2 Steenrod algebra A acts on P(𝑛) according to well-known rules. Let A+ P(𝑛) denote the image of the action of the positively graded part of A. A major problem is that of determining a basis for the quotient vector space Q(𝑛) = P(𝑛)/A+ P(𝑛). Both P(𝑛) = ⊕𝑑≥0 P𝑑 (𝑛) and Q(𝑛) are graded where P𝑑 (𝑛) denotes the set of homogeneous polynomials of degree 𝑑. A spike of degree 𝜆1 𝑑 is a monomial of the form 𝑥12 −1 𝜆𝑛 ⋅ ⋅ ⋅ 𝑥𝑛2 −1 where 𝜆 𝑖 ≥ 0 for each 𝑖. In this paper we show that if 𝑛 ≥ 2 and 𝑑 ≥ 1 can be expressed 𝑛 𝜆 in the form 𝑑 = 𝑑(𝜆) = (𝑛 − 1)(2 − 1) with 𝜆 ≥ 2, then dim(Q𝑑(𝜆) (𝑛)) ≥ 𝐵(𝑛, 𝑑(𝜆)) + {∑𝑞=2 ( ), if 𝜆 < 𝑛; 2𝑛 − (𝑛 + 1), if 𝜆 ≥ 𝑛}, 𝑞 where 𝐵(𝑛, 𝑑(𝜆)) is the number of spikes of degree 𝑑(𝜆). 𝜆 1. Introduction For 𝑛 > 1 let P(𝑛) be the mod-2 cohomology group of the 𝑛fold product of RP∞ with itself. Then P(𝑛) is the polynomial algebra P (𝑛) = F2 [𝑥1 , . . . , 𝑥𝑛 ] (1) in 𝑛 variables 𝑥𝑖 , each of degree 1. P(𝑛) is a module over the mod-2 Steenrod algebra A according to well-known rules. A polynomial 𝑢 is said to be hit if it belongs to the set A+ P (𝑛) = {∑𝑆𝑞𝑖 𝑢𝑖 | 𝑢𝑖 ∈ P (𝑛) , 𝑆𝑞𝑖 ∈ A} . (2) 𝑖>0 The problem of determining A+ P(𝑛) is called the hit problem and has been studied by several authors [1–3]. A closely related problem is that of determining a basis for the quotient vector space Q (𝑛) = P (𝑛) A+ P (𝑛) (3) which has also been studied by several authors [4–9]. Some of the motivation for studying these problems is mentioned in [6]. It stems from the Peterson conjecture proved in [3] and various other sources [10, 11]. The following result is useful for determining Agenerators for P(𝑛). Let 𝛼(𝑚) denote the number of digits 1 in the binary expansion of 𝑚. Theorem 1 (Wood [3]). Let 𝑢 ∈ P(𝑛) be a monomial of degree 𝑑. If 𝛼(𝑛 + 𝑑) > 𝑛 then 𝑢 is hit. Thus Q𝑑 (𝑛) is zero unless 𝛼(𝑛 + 𝑑) ≤ 𝑛 or, equivalently, unless 𝑑 can be written in the form 𝑑 = ∑𝑛𝑖=1 (2𝜆 𝑖 − 1), where 𝜆 𝑖 ≥ 0. Thus Q𝑑 (𝑛) ≠ 0 only if P𝑑 (𝑛) contains monomials 𝜆1 𝜆𝑛 V = 𝑥12 −1 ⋅ ⋅ ⋅ 𝑥𝑛2 −1 (4) called spikes. We note that a spike can never appear as a term in a hit polynomial. Q(𝑛) has been explicitly calculated by Peterson [7] for 𝑛 = 1, 2, by Kameko [12] for 𝑛 = 3, and independently by Kameko [5] and Sum [8] for 𝑛 = 4. 2 International Journal of Mathematics and Mathematical Sciences 2. Preliminaries Then ℎ induces a homomorphism ℎ∗ : Q2𝑑+𝑛 (𝑛) → Q (𝑛). 𝑑 In this section we recall some results in Kameko [12] and Singer [2] on admissible monomials and hit monomials in P(𝑛). 𝑒 If 𝑏 = 𝑥11 ⋅ ⋅ ⋅ 𝑥𝑛𝑒𝑛 is a monomial in P(𝑛), write 𝑒𝑖 = 𝑗 ∑𝑗≥0 𝛼𝑗 (𝑒𝑖 )2 for the binary expansion of each exponent 𝑒𝑖 . The expansions are then assembled into a matrix 𝛽 (𝑏) = (𝛼𝑗 (𝑒𝑖 )) (5) of digits 0 or 1 with 𝛼𝑗 (𝑒𝑖 ) in the (𝑖, 𝑗)th position of the matrix. We will associate with a monomial 𝑏 two sequences 𝑤 (𝑏) = (𝑤0 (𝑏) , 𝑤1 (𝑏) , . . . , 𝑤𝑗 (𝑏) , . . .) , 𝑒 (𝑏) = (𝑒1 , 𝑒2 , . . . , 𝑒𝑛 ) , (6) where 𝑤𝑗 (𝑏) = ∑𝑛𝑖=1 𝛼𝑗 (𝑒𝑖 ) for each 𝑗 ≥ 0. 𝑤(𝑏) is called the weight vector of the monomial 𝑏 and 𝑒(𝑏) is called the exponent vector of the monomial 𝑏. Note that 𝑤𝑗 (𝑏) ≤ 𝑛 for all 𝑗. The monomial 𝑏 is said to have length 𝑙 if 𝑤𝑙 (𝑏) ≠ 0 and 𝑤𝑗 (𝑏) = 0 for all 𝑗 > 𝑙. Given two sequences 𝑝 = (𝑢0 , 𝑢1 , . . . , 𝑢𝑙 , 0, 0, ⋅ ⋅ ⋅ ) , 𝑞 = (V0 , V1 , . . . , V𝑙 , 0, 0, . . .) , (7) we say that 𝑝 < 𝑞 if there is a positive integer 𝑘 such that 𝑢𝑖 = V𝑖 for all 𝑖 < 𝑘 and 𝑢𝑘 < V𝑘 . We are now in a position to define an order relation on monomials. Definition 2. Let 𝑎, 𝑏 be monomials in P(𝑛). We say that 𝑎 < 𝑏 if one of the following holds: (i) 𝑤(𝑎) < 𝑤(𝑏), (ii) 𝑤(𝑎) = 𝑤(𝑏) and 𝑒(𝑎) < 𝑒(𝑏). Note that the order relation on the set of sequences is the lexicographical one. Following Kameko [4] we define the following. Definition 3. A monomial 𝑏 ∈ P(𝑛) is said to be inadmissible if there exist monomials 𝑏1 , 𝑏2 , . . . , 𝑏𝑟 ∈ P(𝑛) with 𝑏𝑗 < 𝑏 for each 𝑗, 1 ≤ 𝑗 ≤ 𝑟, such that Let 𝛽(𝑑) = min{𝑚 ∈ Z | 𝛼(𝑚 + 𝑑) ≤ 𝑚}. In [5, Theorem 4.2] Kameko proved the following. Theorem 5 (Kameko). Let 𝑑 be a positive integer. If 𝛽(2𝑑+𝑛) = 𝑛, then ℎ∗ : Q2𝑑+𝑛 (𝑛) → Q𝑑 (𝑛) is an isomorphism. From Wood’s theorem and the above result of Kameko the problem of determining A-generators for P(𝑛) is reduced to the cases 𝛽(𝑑) < 𝑛. We recall the following result of Singer on hit polynomials in P(𝑛). 𝜆1 Theorem 7 (Singer). Let 𝑏 ∈ P(𝑛) be a monomial of degree 𝑑, where 𝛼(𝑛 + 𝑑) ≤ 𝑛. Let V be the minimal spike of degree 𝑑. If 𝑤(𝑏) < 𝑤(V), then 𝑏 is hit. 3. Main Result In this section we state our main result, Theorem 8, which was obtained in [13]. The proof is deferred until Section 5. Theorem 8 is the basis of a more general result, obtained in [14], which we state at the conclusion of this paper. Let 𝜆 ≥ 2 be an integer and let 𝑑(𝜆) = (𝑛 − 1)(2𝜆 − 1). Let 𝐵(𝑛, 𝑑(𝜆)) denote the number of spikes of degree 𝑑(𝜆). In [13] it is shown that one has the following. Theorem 8. If 𝑛 ≥ 2, then dim (Q𝑑(𝜆) (𝑛)) ≥ 𝐵 (𝑛, 𝑑 (𝜆)) 𝜆 𝑛 { {∑ ( ) + {𝑞=2 𝑞 { 𝑛 {2 − (𝑛 + 1) 𝑟 𝑏 ≡ ( ∑𝑏𝑗 ) mod A+ P (𝑛) . (8) 𝑗=1 𝑏 is said to be admissible if it is not inadmissible. Clearly the set of all admissible monomials in P(𝑛) is a minimal set of A-generators of P(𝑛). Clearly a spike is admissible. We shall require the following result due to Kameko. 𝑐 ℎ (𝑏) = { 0 2 if 𝑏 = 𝑥1 𝑥2 ⋅ ⋅ ⋅ 𝑥𝑛 𝑐 otherwise for any monomial 𝑏 ∈ P2𝑑+𝑛 (𝑛). (9) 𝑖𝑓 𝜆 < 𝑛 (10) 𝑖𝑓 𝜆 ≥ 𝑛. A general formula for computing 𝐵(𝑛, 𝑑) for an arbitrary value of 𝑑 can be found in [15]. The result is a consequence of the following lemma. For each 𝑞, 2 ≤ 𝑞 ≤ 𝑛, and any given 𝜆 ≥ 𝑞, let 𝑐𝑞 be the monomial 𝑞−1 Definition 4. Let 𝑑 be a positive integer. Define a linear mapping ℎ : P2𝑑+𝑛 (𝑛) → P𝑑 (𝑛) by 𝜆𝑛 Definition 6. A spike V = 𝑥12 −1 ⋅ ⋅ ⋅ 𝑥𝑛2 −1 is called a minimal spike if its weight order is minimal with respect to other spikes of degree 𝑑 or, equivalently, if 𝜆 1 ≥ 𝜆 2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆 𝑠 ≥ 0 and 𝜆 𝑗−1 = 𝜆 𝑗 only if 𝑗 = 𝑠 or 𝜆 𝑗+1 = 0. In [11, Theorem 1.2] Singer proved the following. 𝑐𝑞 = 𝑥12 −1 𝜆 𝑞−𝑖 ⋅ ⋅ ⋅ 𝑥𝑖(2 −2 )−1 𝜆 𝜆 𝜆 2 −1 ⋅ ⋅ ⋅ 𝑥𝑞2 −2 𝑥𝑞+1 ⋅ ⋅ ⋅ 𝑥𝑛2 −1 (...truncated)