Dimension Result for the Polynomial Algebra 𝔽₂[𝑥₁,…,𝑥_𝑛] as a Module over the Steenrod Algebra

Abstract

For 𝑛≥1, let 𝐏(𝑛)=𝔽₂[𝑥₁,…,𝑥_𝑛] be the polynomial algebra in 𝑛 variables 𝑥_𝑖, 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≥0𝐏^𝑑(𝑛) and 𝐐(𝑛) are graded where 𝐏^𝑑(𝑛) denotes the set of homogeneous polynomials of degree 𝑑. A spike of degree 𝑑 is a monomial of the form 𝑥^2𝜆1₁⁻¹⋯𝑥^2𝜆𝑛_𝑛⁻¹ where 𝜆_𝑖≥0 for each 𝑖. In this paper we show that if 𝑛≥2 and 𝑑≥1 can be expressed in the form 𝑑=𝑑(𝜆)=(𝑛−1)(2^𝜆−1) with 𝜆≥2, then dim(𝐐^𝑑(𝜆)(𝑛))≥𝐵(𝑛,𝑑(𝜆))+{∑^𝜆_𝑞=2(𝑛𝑞),if𝜆<𝑛;2^𝑛−(𝑛+1),if𝜆≥𝑛}, where 𝐵(𝑛,𝑑(𝜆)) is the number of spikes of degree 𝑑(𝜆).

1. Introduction

For 𝑛>1 let 𝐏(𝑛) be the mod-2 cohomology group of the 𝑛-fold product of ℝ𝐏^∞ with itself. Then 𝐏(𝑛) is the polynomial algebra 𝐏(𝑛)=𝔽₂𝑥₁,…,𝑥_𝑛 (1) in 𝑛 variables 𝑥_𝑖, 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)=_𝑖>0𝑆𝑞^𝑖𝑢_𝑖∣𝑢_𝑖∈𝐏(𝑛),𝑆𝑞^𝑖∈𝒜. (2) 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 𝒜-generators 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 𝐐^𝑑(𝑛) is zero unless 𝛼(𝑛+𝑑)≤𝑛 or, equivalently, unless 𝑑 can be written in the form 𝑑=∑^𝑛_𝑖=1(2^𝜆𝑖−1), where 𝜆_𝑖≥0. Thus 𝐐^𝑑(𝑛)≠0 only if 𝐏^𝑑(𝑛) contains monomials 𝑣=𝑥^2𝜆1₁⁻¹⋯𝑥^2𝜆𝑛_𝑛⁻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. Preliminaries

In this section we recall some results in Kameko [12] and Singer [2] on admissible monomials and hit monomials in 𝐏(𝑛).

If 𝑏=𝑥^𝑒1₁⋯𝑥^𝑒𝑛_𝑛 is a monomial in 𝐏(𝑛), write 𝑒_𝑖=∑_𝑗≥0𝛼_𝑗(𝑒_𝑖)2^𝑗 for the binary expansion of each exponent 𝑒_𝑖. The expansions are then assembled into a matrix 𝛽(𝑏)=𝛼_𝑗𝑒_𝑖 (5) of digits 0 or 1 with 𝛼_𝑗(𝑒_𝑖) in the (𝑖,𝑗)thposition of the matrix.

We will associate with a monomial 𝑏 two sequences 𝑤(𝑏)=𝑤₀(𝑏),𝑤₁(𝑏),…,𝑤_𝑗(𝑏),𝑒(𝑏)=𝑒,…₁,𝑒₂,…,𝑒_𝑛, (6) where 𝑤_𝑗(𝑏)=∑^𝑛_𝑖=1𝛼_𝑗(𝑒_𝑖) 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 𝑤_𝑗(𝑏)≤𝑛 for all 𝑗. The monomial 𝑏 is said to have length 𝑙 if 𝑤_𝑙(𝑏)≠0 and 𝑤_𝑗(𝑏)=0 for all 𝑗>𝑙.

Given two sequences 𝑝=𝑢₀,𝑢₁,…,𝑢_𝑙,0,0,⋯,𝑞=𝑣₀,𝑣₁,…,𝑣_𝑙,0,0,…, (7) we say that 𝑝<𝑞 if there is a positive integer 𝑘 such that 𝑢_𝑖=𝑣_𝑖 for all 𝑖<𝑘 and 𝑢_𝑘<𝑣_𝑘. 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,…,𝑏_𝑟∈𝐏(𝑛) with 𝑏_𝑗<𝑏 for each 𝑗, 1≤𝑗≤𝑟, such that 𝑏≡^𝑟_𝑗=1𝑏_𝑗mod𝒜⁺𝐏(𝑛). (8)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.

Definition 4. Let 𝑑 be a positive integer. Define a linear mapping ℎ∶𝐏^2𝑑+𝑛(𝑛)→𝐏^𝑑(𝑛) by ℎ(𝑏)=𝑐if𝑏=𝑥₁𝑥₂⋯𝑥_𝑛𝑐²0otherwise (9) for any monomial 𝑏∈𝐏^2𝑑+𝑛(𝑛).
Then ℎ induces a homomorphism ℎ_∗∶𝐐^2𝑑+𝑛(𝑛)→𝐐^𝑑(𝑛).

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_∗∶𝐐^2𝑑+𝑛(𝑛)→𝐐^𝑑(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).

Definition 6. A spike 𝑣=𝑥^2𝜆1₁⁻¹⋯𝑥^2𝜆𝑛_𝑛⁻¹ is called a minimal spike if its weight order is minimal with respect to other spikes of degree 𝑑 or, equivalently, if 𝜆₁≥𝜆₂≥⋯≥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.

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)(2^𝜆−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 dim𝐐^𝑑(𝜆)(𝑛)((𝜆+≥𝐵𝑛,𝑑))_𝜆𝑞=2𝑛𝑞2𝑖𝑓𝜆<𝑛_𝑛−(𝑛+1)𝑖𝑓𝜆≥𝑛. (10)

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 𝑐_𝑞 be the monomial 𝑐_𝑞=𝑥^2𝑞−1₁⁻¹⋯𝑥^{(2𝜆−2𝑞−𝑖}_𝑖⁾⁻¹⋯𝑥^2𝜆_𝑞⁻²𝑥^2𝜆−1_𝑞+1⋯𝑥^2𝜆_𝑛⁻¹ (11) in 𝐏^𝑑(𝜆)(𝑛) and let 𝑊𝑛,𝑞=𝜎∈𝑆_𝑛∣𝜎𝑗<𝜎(𝑘)if𝑗<𝑘≤𝑞and𝜎(𝑠)<𝜎(𝑟)if𝑞<𝑠<𝑟. (12)

Lemma 9. Let 𝑞,𝜆 be integers such that 𝜆≥𝑞 and let 𝑐_𝑞∈𝐏^𝑑(𝜆)(𝑛). Then for any given 𝜎∈𝑊(𝑛,𝑞) the monomial 𝑐_𝑞(𝜎)=𝑥^2𝑞−1_𝜎(1)⁻¹⋯𝑥^{(2𝜆−2𝑞−𝑖}_𝜎(𝑖)⁾⁻¹⋯𝑥^2𝜆−2_𝜎(𝑞)𝑥^2𝜆−1_𝜎(𝑞+1)⋯𝑥^2𝜆−1_𝜎(𝑛) (13) is admissible.

The following example illustrates the definitions previously mentioned and typical monomials defined in the lemma.

Example 10. If 𝑛=3, 𝜆=3, and 𝑞=2, then 𝑊(3,2)={(1),(23),(123)} and in matrix notation 𝑐₂(1)=𝑐100011111²()=𝑐23100111011₂(123)=.111100011 (14)

Note that 𝑊=𝑛𝑞𝑛,𝑞 (15) and that if 𝑛=𝑞 then the only element 𝜎 in 𝑊(𝑛,𝑞) is the identity (13).

4. Proof of Lemma 9

We first show that if 𝑞=𝑛, then 𝑐_𝑛=𝑐_𝑛(1) is admissible, that is, that 𝑏=𝑏(𝑛)=𝑐_𝑛(1)=𝑥^2𝑛−1₁⁻¹𝑥^{2𝜆−2𝑛−2}₂⁻¹⋯𝑥^{2𝜆−2𝑛−𝑖}_𝑖⁻¹⋯𝑥^2𝜆_𝑛⁻² (16) is admissible. We will suppress mentioning the fact that 𝑏(𝑛) is also a function of 𝜆.

Suppose that 𝜆≥𝑛 is given and let 𝑚∈𝐏^𝑑(𝜆)(𝑛) be the spike 𝑚=𝑚(𝑛)=𝑥⁰₁𝑥^2𝜆₂⁻¹⋯𝑥^2𝜆_𝑗⁻¹⋯𝑥^2𝜆_𝑛⁻¹ (17) so that 𝑤(𝑚)=𝑛−1,𝑛−1^𝜆,…,𝑛−1. (18) Let 𝐃^𝑑(𝜆)(𝑛) denote the subspace of 𝐏^𝑑(𝜆)(𝑛) spanned by monomials 𝑎=𝑥^𝑡1₁⋯𝑥^𝑡𝑛_𝑛 such that 𝑤(𝑎)=𝑤(𝑚). Further let 𝐃^𝑑(𝜆)(𝑛,𝑏) denote the subspace of 𝐃^𝑑(𝜆)(𝑛) for which 𝑡_𝑖≠2^𝜆−1 for any 𝑖. Let 𝜋∶𝐏^𝑑(𝜆)(𝑛)⟶𝐃^𝑑(𝜆)(𝑛) (19) denote the projection of 𝐏^𝑑(𝜆)(𝑛) onto its summand 𝐃^𝑑(𝜆)(𝑛) and let 𝜋_𝑏∶𝐃^𝑑(𝜆)(𝑛)⟶𝐃^𝑑(𝜆)(𝑛,𝑏) (20) denote the projection of 𝐃^𝑑(𝜆)(𝑛) onto 𝐃^𝑑(𝜆)(𝑛,𝑏). Let 𝐇^𝑑(𝜆)(𝑛,𝑏) be the subspace of 𝐃^𝑑(𝜆)(𝑛,𝑏) spanned by 𝑏₁+𝑏₂∣𝑏₁,𝑏₂monomialsin𝐃^𝑑(𝜆)(𝑛,𝑏). (21) We claim that 𝜋_𝑏𝜋𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)=𝐇^𝑑(𝜆)(𝑛,𝑏). (22) Clearly this shall suffice to show that 𝑏 is admissible (it being the monomial of least order in 𝐃^𝑑(𝜆)(𝑛,𝑏)).

Our proof of (22) is by induction on 𝑛 and 𝜆 and splits into two cases depending on whether 𝜆=𝑛 or 𝜆>𝑛.

Case 1 (i) (𝜆=𝑛). In this case our argument is by induction on 𝑛, starting with the case 𝑛=2. If 𝑛=2 then (22) is easily seen to be true since 𝜋_𝑏(2)𝜋𝒜⁺𝐏(2)∩𝐏³(2) (23) is spanned by {𝑥²₁𝑥₂+𝑥₁𝑥²₂}.

Assume that (22) is true when the number of variables is 𝑛−1, where 𝑛 is fixed and 𝑛−1≥2.

Let 𝜏=(12⋯𝑛)∈𝑆_𝑛 and for each 𝑗, 1≤𝑗≤𝑛, let 𝐃_𝑗^𝑑(𝜆)(𝑛,𝑏) be the vector subspace of 𝐃^𝑑(𝜆)(𝑛,𝑏) generated by monomials 𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛𝑥^𝑒1_𝜏𝑗(1)⋯𝑥^𝑒𝑛−1_{𝜏𝑗(𝑛−1)}𝑥^2𝜆−1_𝜏⁻¹_𝑗(𝑛)², (24) where 𝑥^𝑒1₁⋯𝑥^𝑒𝑛−1_𝑛−1∈𝐃^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1)) and 𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛 is obtained from the monomial 𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛 by deleting the factor 𝑥_𝑗. It is easy to see that 𝐃^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1)) is isomorphic to 𝐷_𝑗^𝑑(𝜆)(𝑛,𝑏) for each 𝑗. In fact the linear mapping ℎ_𝑗∶𝐃^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1))⟶𝐃_𝑗^𝑑(𝜆)(𝑛,𝑏) (25) given on monomials by ℎ_𝑗𝑥^𝑒1₁⋯𝑥^𝑒𝑛−1_𝑛−1=𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛𝑥^𝑒1_𝜏𝑗(1)⋯𝑥^𝑒𝑛−1_{𝜏𝑗(𝑛−1)}𝑥^2𝜆−1_𝜏⁻¹_𝑗(𝑛)² (26) is an isomorphism for each 𝑗. It is easy to check that 𝐃^𝑑(𝜆)(𝑛,𝑏)=^𝑛_𝑗=1ℎ_𝑗𝐃^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1)) (27) and further that 𝜋_𝑏(𝑛)𝜋(𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)) is equal to 𝜋_𝑏(𝑛)𝜋𝑆𝑞¹𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)_𝑛𝑗=1ℎ_𝑗𝐇^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1)). (28) It is therefore sufficient to show that we can find monomials 𝑢_𝑗∈𝐷_𝑗^𝑑(𝜆)(𝑛) such that 𝑢₁+𝑢₂,…𝑢_𝑗+𝑢_𝑗+1,…𝑢_𝑛−1+𝑢_𝑛 (29) all belong to 𝜋_𝑏(𝑛)𝜋(𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)).

We choose 𝑢_𝑗=ℎ_𝑗(𝑏(𝑛−1)) for each 𝑗; that is, 𝑢_𝑗=𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛𝑥^2𝑛−2_𝜏⁻¹_𝑗(1)⋯𝑥^2𝜆−1_𝜏⁻²_{𝑗(𝑛−1)}𝑥^2𝜆−1_𝜏⁻¹_𝑗(𝑛)². (30)

We first illustrate the argument in the special case 𝑛=𝜆=4. In this case 𝑏(3)=𝑥³₁𝑥⁵₂𝑥⁶₃,𝑢₁=𝑥₁¹⁴𝑥⁷₂𝑥₃¹¹𝑥₄¹³,𝑢₂=𝑥₁¹³𝑥₂¹⁴𝑥⁷₃𝑥₄¹¹,𝑢₃=𝑥₁¹¹𝑥₂¹³𝑥₃¹⁴𝑥⁷₄, and 𝑢₄=𝑥⁷₁𝑥₂¹¹𝑥₃¹³𝑥₄¹⁴=𝑏(4).

Consider 𝑢₁∈𝐃₁⁴⁵(4,𝑏(4)). Put 𝑟₁=𝑥₁¹⁴𝑥₂¹³𝑥⁷₃𝑥₄¹¹∈𝐃₁⁴⁵(4,𝑏(4)). (31) By the induction hypothesis, 𝑢₁+𝑟₁∈𝜋_𝑏(4)𝜋𝒜⁺𝐏(4)∩𝐏⁴⁵(4). (32) But 𝑟₁+𝑢₂∈𝜋_𝑏(4)𝜋(𝑆𝑞¹(𝐏(4))∩𝐏⁴⁵(4)). Put 𝑟₂=𝑥₁¹¹𝑥₂¹⁴𝑥₃¹³𝑥⁷₄∈𝐃₂⁴⁵(4,𝑏(4)). (33) By the induction hypothesis, 𝑢₂+𝑟₂∈𝜋_𝑏(4)𝜋𝒜⁺𝐏(4)∩𝐏⁴⁵(4). (34) But 𝑟₂+𝑢₃∈𝜋_𝑏(4)𝜋(𝑆𝑞¹(𝐏(4))∩𝐏⁴⁵(4)). Put 𝑟₃=𝑥⁷₁𝑥₂¹¹𝑥₃¹⁴𝑥₄¹³∈𝐃₃⁴⁵(4,𝑏(4)). (35) By the induction hypothesis 𝑢₃+𝑟₃∈𝜋_𝑏(4)𝜋𝒜⁺𝐏(4)∩𝐏⁴⁵(4). (36) But 𝑟₃+𝑢₄∈𝜋_𝑏(4)𝜋(𝑆𝑞¹(𝐏(4))∩𝐏⁴⁵(4)), which concludes the argument in this special case.

The argument to show that 𝑢_𝑗+𝑢_𝑗+1∈𝜋_𝑏(𝑛)𝜋(𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)) in general proceeds as follows.

Consider the element 𝑢_𝑗=𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛𝑥^2𝑛−2_𝜏⁻¹_𝑗(1)⋯𝑥^2𝜆−1_𝜏⁻²_{𝑗(𝑛−1)}𝑥^2𝜆−1_𝜏⁻¹_𝑗(𝑛)² (37) in 𝐃_𝑗^𝑑(𝜆)(𝑛,𝑏). Let 𝑟_𝑗∈𝐃_𝑗^𝑑(𝜆)(𝑛,𝑏) be the monomial 𝑟_𝑗=𝑥₁⋯𝑥_𝑗⋯𝑥_𝑛×𝑥^2𝑛−2_𝜏⁻¹_𝑗+1(1)⋯𝑥^2𝜆−1_𝜏⁻⁴_{𝑗+1()𝑛−2}𝑥^2𝜆−1_𝜏⁻²_{𝑗+2()𝑛−1}𝑥^2𝜆−1_𝜏⁻¹_𝑗(𝑛)². (38) By the induction hypothesis, 𝑢_𝑗+𝑟_𝑗∈𝜋_𝑏(𝑛)𝜋𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛). (39) But 𝑟_𝑗+𝑢_𝑗+1∈𝜋_𝑏(𝑛)𝜋(𝑆𝑞¹(𝐏(𝑛))∩𝐏^𝑑(𝜆)(𝑛)) which shows that 𝑢_𝑗+𝑢_𝑗+1∈𝜋_𝑏(𝑛)𝜋(𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)) as required.

Case 2 (ii) (𝜆≥𝑛). Our argument is by induction on both 𝑛 and 𝜆. In the case 𝑛=2<𝜆, 𝜋_𝑏(2)𝜋𝒜⁺𝐏(2)∩𝐏^𝑑(𝜆)(2) (40) is known to be spanned by 𝐇^𝑑(𝜆)(2,𝑏(2)).

We have seen that (22) is true if 𝜆=𝑛. Assume therefore that (22) is true when the number of variables is 𝑛−1 where 𝑛 is fixed and 𝑛−1≥2 and for 𝜆−1, where 𝜆 is fixed and 𝜆−1≥𝑛.

Let 𝜏∈𝑆_𝑛 be defined as above, let 𝐃_𝑗^𝑑(𝜆)(𝑛,𝑏) be the vector subspace of 𝐃^𝑑(𝜆)(𝑛,𝑏) defined in the same manner as above, and let ℎ_𝑗∶𝐃^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1))⟶𝐃_𝑗^𝑑(𝜆)(𝑛,𝑏) (41) be the mapping given by (26).

For each 𝑖, 1≤𝑖≤𝑛, let 𝑓_𝑖∶𝐃^{𝑑(𝜆−1)}(𝑛,𝑏(𝑛))⟶𝐃^𝑑(𝜆)(𝑛,𝑏(𝑛)) (42) be the linear mapping given on monomials by 𝑓_𝑖𝑥^𝑒1₁⋯𝑥^𝑒𝑛_𝑛=𝑥^2𝑒1₁⁺¹⋯𝑥^{2𝑒𝑖−1+1}_𝑖−1𝑥^2𝑒𝑖_𝑖𝑥^{2𝑒𝑖+1+1}_𝑖+1⋯𝑥^2𝑒𝑛_𝑛⁺¹. (43) It is easy to check that 𝐃^𝑑(𝜆)(𝑛,𝑏(𝑛))=^𝑛_𝑗=1ℎ_𝑗𝐃^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1))_𝑛𝑖=1𝑓_𝑖𝐃^{𝑑(𝜆−1)}(𝑛,𝑏(𝑛)) (44) and further that 𝜋_𝑏(𝑛)𝜋(𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)) is equal to 𝜋_𝑏(𝑛)𝜋𝑆𝑞¹𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)_𝑛𝑗=1ℎ_𝑗𝐇^{𝑑(𝜆−1)}(𝑛−1,𝑏(𝑛−1))_𝑛𝑖=1𝑓_𝑖𝐇^{𝑑(𝜆−1)}(𝑛,𝑏(𝑛)). (45) It is therefore sufficient to show that we can find monomials 𝑢_𝑗∈𝐃_𝑗^𝑑(𝜆)(𝑛) and monomials 𝑣_𝑖∈𝑓_𝑖(𝐃^{𝑑(𝜆−1)}(𝑛,𝑏(𝑛))) such that 𝑢₁+𝑢₂,…,𝑢_𝑗+𝑢_𝑗+1,…,𝑢_𝑛−1+𝑢_𝑛,𝑣₁+𝑣₂,…,𝑣_𝑖+𝑣_𝑖+1,…,𝑣_𝑛−1+𝑣_𝑛, (46) and 𝑣₁+𝑢₁ all belong to 𝜋_𝑏(𝑛)𝜋(𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)). But this is a direct calculation, similar to the one for the case 𝜆=𝑛, which makes use of induction on 𝑛 and 𝜆 and is omitted.

Finally we show that if 𝑞<𝑛 and 𝜆≥𝑞, then 𝑐_𝑞(𝜎) is admissible. Let 𝑑(𝜆) denote 𝑑(𝜆,𝑞)=(𝑞−1)(2^𝜆−1). We first note that 𝑢_𝑞=𝑐_𝑞(1)=𝑥^2𝑞−1₁⁻¹𝑥^{2𝜆−2𝑞−2}₂⁻¹⋯𝑥^{2𝜆−2𝑞−𝑖}_𝑖⁻¹⋯𝑥^2𝜆_𝑞⁻² (47) is an admissible monomial in 𝐏^𝑑(𝜆)(𝑞) (by Cases 1 and 2).

Now given that 𝜎∈𝑊(𝑛,𝑞), let 𝑔^𝜎_𝑞∶𝐏𝑞⟶𝐏(𝑛) (48) be the linear mapping given on monomials by 𝑔^𝜎_𝑞𝑥^𝑒1₁⋯𝑥^𝑒𝑞_𝑞=𝑥^𝑒1_𝜎(1)⋯𝑥^𝑒𝑞_𝜎(𝑞)𝑥^2𝜆−1_𝜎(𝑞+1)⋯𝑥^2𝜆−1_𝜎(𝑗)⋯𝑥^2𝜆−1_𝜎(𝑛). (49)

Then for all 𝑢∈𝐏(𝑞) and 𝜃∈𝒜⁺ for which 𝜃(𝑢)∈𝐏^𝑑(𝜆)(𝑞) we have 𝑔^𝜎_𝑞𝜋_𝑢𝑞𝜋(𝜃(𝑢))=𝜋_𝑏𝜋𝜃𝑔^𝜎_𝑞(𝑢) (50) so that 𝜋𝒜⁺𝐏(𝑛)∩𝐏^𝑑(𝜆)(𝑛)=_𝜎,𝑞𝑔^𝜎_𝑞𝐇^𝑑(𝜆)𝑞,𝑢_𝑞. (51) Since 𝐃^𝑑(𝜆)(𝑛)=_𝜎,𝑞𝑔^𝜎_𝑞𝐃^𝑑(𝜆)𝑞,𝑢_𝑞 (52) and each 𝑔^𝜎_𝑞 preserves the order of monomials, we see that 𝑐_𝑞(𝜎)=𝑔^𝜎_𝑞(𝑢_𝑞) is admissible for each pair 𝜎,𝑞. This completes the proof of the lemma.

5. Proof of Theorem 8

Let 𝐂^𝑑(𝜆)(𝑛) denote the subspace of 𝐐^𝑑(𝜆)(𝑛) with basis the union of 𝐵(𝑛,𝑑(𝜆)) (the spikes of degree 𝑑(𝜆)) and the collection ℬ_𝑛(𝑑(𝜆))=𝑐_𝑞(𝜎)∣2≤𝑞≤𝑛,𝜎∈𝑊𝑛,𝑞 (53) of monomials defined in Lemma 9. Then dim𝐐^𝑑(𝜆)(𝑛)≥dim𝐂^𝑑(𝜆)(𝑛)=𝐵(𝑛,𝑑(𝜆))+ℬ_𝑛(𝑑(𝜆((𝜆+))=𝐵𝑛,𝑑))^{min{𝜆,𝑛}}_𝑞=2𝑊((𝜆+𝑛,𝑞=𝐵𝑛,𝑑))^𝜆_𝑞=2𝑛𝑞if2𝜆<𝑛_𝑛−(𝑛+1)if𝜆≥𝑛. (54)

We note that if 𝜆≥𝑛 and 𝐂_∗(𝑛) is the subspace of 𝐂(𝑛) generated by ℬ_𝑛, then the mapping 𝑓∶𝐂_∗^𝑑(𝜆)(𝑛)⟶𝐂_∗^𝑑(𝜆+1)(𝑛) (55) defined on monomials by 𝑓(𝑐_𝑞(𝜎))=𝑐_𝑞(𝜎)(𝑥_𝜎(2)⋯𝑥_𝜎(𝑛))^2𝜆 is an isomorphism between the vector subspaces 𝐂_∗(𝑛).

This completes the proof of Theorem 8.

Remark 11. The result of this paper in part fulfills the requirements of the following theorem, obtained in [14], whose proof needed a more detailed but separate treatment of Theorem 8.

Theorem 12. Suppose that 𝑛≥2 and 𝑑>0 may be expressed in the form 𝑑=∑^𝑛−1_𝑖=1(2^𝜆𝑖−1) with 𝜆₁>𝜆₂>⋯>𝜆_𝑛−2≥𝜆_𝑛−1≥1. Then dim𝐐^𝑑(𝑛)≥^{min{𝜆𝑛−1,𝑛}}_𝑞=1𝑛𝑞dim𝐐^𝑑′(𝑛−1), (56) where 𝑑^′=∑^𝑛−1_𝑖=1(2^{𝜆𝑖−𝜆𝑛−1}−1).

Disclosure

The result in Section 3 is based on work in the author’s Ph.D. thesis [13] authored under the supervision of Dr. R. M. W. Wood.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.