ON CONJUGATION INVARIANTS IN

THE DUAL STEENROD ALGEBRA

M. D. Crossley and Sarah Whitehouse

3 April 1998

Abstract. We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra A_{∗}, with a view to determining the subspace, A_{∗}^{χ}, of elements invariant under χ. We give bounds on the dimension of this subspace for each degree and show that, after inverting ξ_{1}, it becomes polynomial on a natural set of generators.

1. Introduction.

The mod 2 dual Steenrod algebra, A_{∗}, being a connected commutative Hopf algebra, has a canonical conjugation or anti-automorphism χ. This map was first studied by Thom [T] but most of what we know today about χ is due to Milnor [M]. Our aim is to study the subspace of A_{∗} consisting of elements invariant under this conjugation map. We denote this subspace by A_{∗}^{χ}. While we are unable to give a complete description of A_{∗}^{χ}, we have established bounds on its dimension in each degree (Theorem 3.1), and we can show that, after inverting the element ξ_{1} ∈ A_{∗}, the invariant subspace becomes polynomial (Theorem 4.1). Finally, our investigations have also lead us to conjecture a set of generators for A_{∗}^{χ} (Conjecture 5.3).

We begin by recalling the structure of A_{∗} and certain well known facts about the map χ, while in section 2 we deduce some elementary properties of χ. In section 3 we derive our bounds on the dimension of the subspace of invariants and in section 4 we study the result of inverting ξ_{1}. The final section contains our conjectures about generators for the invariants.

The structure of the Hopf algebra A_{∗} was determined by Milnor [M]. As an algebra, A_{∗} = 𝔽_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …], where the degree of ξ_{i} is 2^{i} − 1. We use the notation (r_{1}, r_{2}, …, r_{k}) to denote the monomial ξ_{1}^{r_{1}}ξ_{2}^{r_{2}} … ξ_{k}^{r_{k}}. The coproduct ϕ is determined by the formula

ϕ(ξ_{k}) = ∑_{i=0}^{k} ξ_{k−i}^{2^{i}} ⊗ ξ_{i} ,

where ξ_{0} is interpreted as 1.

From [MM] we know that any connected Hopf algebra, H, has a unique bijective linear map, χ : H → H, called the ‘conjugation’, with the following properties:

1) χ(1) = 1,

Typeset by AMS-TEX

2) χ(xy) = χ(y)χ(x) (i.e. χ is an anti-automorphism),

3) If ϕ(a) = ∑ a′_{i} ⊗ a′′_{i} where a ∈ H^{+} (i.e. deg a > 0), then ∑ a′_{i}χ(a′′_{i}) = 0. (Since the coproduct ϕ always satisfies the identity ϕ(a) ≡ a⊗1+1⊗a mod H^{+}⊗

H^{+}, the last property determines χ inductively.)

Furthermore, if the Hopf algebra is either commutative or cocommutative, then χ^{2} is the identity homomorphism.

In the case of the dual Steenrod algebra, property 3 leads inductively to the following formula for χ (Lemma 10 of [M]).

1.1 Lemma. In the dual Steenrod algebra A_{∗},

χ(ξ_{n}) = ∑_{α∈P art(n)} ∏_{i=1}^{l(α)} ξ_{α(i)}^{2^{σ(i)}} ,

where P art(n) denotes the set of all ordered partitions of n; and for a given ordered partition α = (α(1), α(2), …, α(l)) ∈ P art(n), σ(i) denotes the partial sum ∑_{j=1}^{i−1} α(j). □

This lemma enables us to determine χ on an arbitrary element of A_{∗}, by virtue of multiplicativity (which follows from property 2 since A_{∗} is commutative) and linearity.

We end this introduction with a few comments on motivation for the problems discussed in this paper. Expressions like H^{m}(Σ_{n}, π_{∗}(E^{∧n})) arise in spectral sequences for gamma cohomology of an E_{∞}-ring spectrum E [RW]. For E suitably nice, this is H^{m}(Σ_{n}, (E_{∗}E)^{⊗(n−1)}), the Σ_{n} action here being described in [W]; for n = 2, Σ_{2} acts by the usual conjugation on E_{∗}E. This paper is therefore concerned with the very special case of this problem where E = H𝔽_{2} and n = 2.

2. Elementary Properties of conjugation.

Now we make some elementary observations on the properties of the conjugation χ. For a connected Hopf algebra H, we denote by H^{χ} the invariant elements of H under the conjugation map χ : H → H. The identity homomorphism will be denoted by 1, so that H^{χ} = Ker(χ − 1).

2.1 Lemma. If H is commutative, H^{χ} is a subring of H.

Proof. The conjugation χ is always an anti-automorphism. So, when H is commutative, it is a ring homomorphism. □

We denote by H_{d} the degree d part of a graded object (e.g. Hopf algebra) H.

2.2 Lemma. If H is a commutative or cocommutative Hopf algebra over 𝔽_{2}, then dim H_{d}^{χ} ≥ dim H_{d}/2.

Proof. We have χ^{2} = 1 by the (co)commutativity hypothesis. Since we are working over 𝔽_{2}, this gives (χ − 1)^{2} = 0, and so Im(χ − 1) ⊂ Ker(χ − 1). Since dim H_{d} = dim(Im(χ − 1)_{d}) + dim(Ker(χ − 1)_{d}), we have the result. □

2.3 Remark. In the case of the mod 2 (dual) Steenrod algebra, the above dimension constraint is sharp in low degrees, although not in general. It first fails in degree 42. More details are in section 3, particularly Example 3.4.

2.4 Lemma. In a commutative Hopf algebra H over 𝔽_{2}, Im(χ − 1) is an ideal in Ker(χ − 1). In particular, Im(χ − 1) is a subring of H.

Proof. Let x ∈ Ker(χ − 1). Then x(χ(y) − y) = xχ(y) − xy = χ(x)χ(y) − xy = (χ−1)(xy). The second part can be proved directly: (χ(x)−x)(χ(y)−y) = χ(z)−z mod 2, where z = xy + xχ(y). □

In this situation, the higher cohomology H^{n}(Σ_{2}; H) = Ker(χ − 1)/ Im(χ − 1) for n > 0 (where Σ_{2} acts on H by conjugation) has an algebra structure.

We now fix our attention on the dual Steenrod algebra, A_{∗} = 𝔽_{2}[ξ_{1}, ξ_{2}, …]. We shall frequently need to order the monomials in A_{∗} and the right lexicographic ordering turns out to be the most useful.

2.5 Proposition. With respect to right lexicographic ordering of the monomial basis, the matrix of the conjugation map in each degree, χ : (A_{∗})_{d} → (A_{∗})_{d}, is unitriangular.

Proof. It follows from Lemma 1.1 that χ(ξ_{k}) = ξ_{k} + P_{k} where P_{k} is a polynomial in ξ_{1}, …, ξ_{k−1} and hence strictly lower than ξ_{k}. Consequently for any monomial M in ξ_{1}, …, ξ_{k}, we have that χ(M) = M + Q where Q is strictly lower than M. This is because the right lexicographic ordering has the property that if x < x′ and y ≤ y′ then xy < x′y′. It then follows that the matrix is unitriangular. □

2.6 Remark. This unitriangularity property also holds for certain other orderings. A simple example is left lexicographic ordering. More interestingly, if we define the weight w of the monomial ξ_{1}^{r_{1}} … ξ_{k}^{r_{k}} to be r_{1} + r_{2} + ⋯ + r_{k} then we may obtain further orderings by combining weight and lexicographic orderings. For example, the ‘weight/reverse-left lex’ order is defined by a ≺ b if either w(a) < w(b) or w(a) = w(b) and a follows b in left lexicographic ordering. For all such orderings the above proof can be easily modified. In fact, all the proof needs is that for all k, (χ−1)(ξ_{k}) is strictly lower than ξ_{k} (or that (χ−1)(ξ_{k}) is always strictly higher) and that the ordering is ‘multiplicative’, i.e. x < x′ and y ≤ y′ implies that xy < x′y′.

3. Bounds on dimension.

In this section we state and prove the following theorem which gives bounds on the dimension of the invariant subspace A_{∗}^{χ} in a given degree.

3.1 Theorem. Let the dimension of the (dual) Steenrod algebra in degree d be denoted D_{d}. Then D_{d−1}/2 ≤ dim(χ − 1)(A_{∗})_{d} ≤ D_{d}/2 and hence

D_{d}/2 ≤ dim(A_{∗}^{χ})_{d} ≤ D_{d} − (D_{d−1}/2) .

In fact the upper bound on dim(A_{∗}^{χ})_{d} can be improved upon - see Lemma 3.3 and Example 3.4.

The lower bound on dim(A_{∗}^{χ})_{d} was given by Lemma 2.2; the rest of the theorem is a combinatorial corollary of the following result.

3.2 Proposition. Let a monomial (r_{1}, r_{2}, …, r_{k}) (i.e. ξ_{1}^{r_{1}}ξ_{2}^{r_{2}} … ξ_{k}^{r_{k}}) be called ‘uniterminal’ if r_{k} = 1. In (A_{∗})_{d}, the uniterminal monomials have linearly independent images under χ − 1.

Proof. We use right lexicographic ordering and claim that the lowest uniterminal monomial which has (r_{1} +1, r_{2}, …, r_{k−2}, r_{k−1} +2) as a summand in its image under χ−1 is (r_{1}, …, r_{k−1}, 1). Assuming this claim we argue as follows. Let Q be a linear combination of images under χ−1 of uniterminal monomials. So we may write Q as Q = (χ−1)(P) where P is a linear combination of uniterminal monomials. Suppose that (r_{1}, …, r_{k−1}, 1) is the highest monomial which appears in P. Then our claim shows that (χ − 1)(P) will have (r_{1} + 1, r_{2}, …, r_{k−2}, r_{k−1} + 2) as a summand, and so cannot be zero. The proposition then follows.

Now to prove the claim. From Lemma 1.1,

χ(ξ_{k}) = ξ_{k} + ξ_{k−1}^{2}ξ_{1} + ξ_{k−1}ξ_{1}^{2^{k−1}} + P_{k−2} ,

where P_{k−2} is some polynomial in ξ_{1}, …, ξ_{k−2}. Looking at the second term on the right hand side of the above expression, we see that the largest monomial in (χ−1)(r_{1}, …, r_{k−1}, 1) which does not contain a ξ_{k} is (r_{1} +1, r_{2}, …, r_{k−2}, r_{k−1} +2).

Now we need to see that no earlier uniterminal monomial has this term in its image under χ − 1. Suppose it does appear in the image of (j_{1}, …, j_{k′−1}, 1), with (j_{1}, …, j_{k′−1}, 1) < (r_{1}, …, r_{k−1}, 1) so, in particular, k′ ≤ k. If k′ ≤ k − 1 then (χ − 1)(j_{1}, …, j_{k′−1}, 1) will have no summands with ξ_{k′}-exponent greater than 1, so cannot have (r_{1} + 1, r_{2}, …, r_{k−1} + 2) as a summand. If k′ = k then the fact that (j_{1}, …, j_{k′−1}, 1) < (r_{1}, …, r_{k−1}, 1) implies that (j_{1} + 1, j_{2}, …, j_{k−1} + 2) < (r_{1}, r_{2}, …, r_{k−1} + 2). In this case, the image of (j_{1}, …, j_{k′−1}, 1) cannot contain (r_{1}, r_{2}, …, r_{k−1} + 2) as a summand. □

So dim(χ − 1)(A_{∗})_{d} ≥ R_{d} where R_{d} is the number of uniterminal monomials in degree d. In order to complete the proof of Theorem 3.1 we now obtain some information about R_{d}.

3.3 Lemma.

1) R_{d} ≥ D_{d−1}/2 ,

2) R_{d} = D_{d−1} − R_{d−1} .

Proof. We pair up each uniterminal monomial in degree d with another degree d monomial that is not uniterminal by the pairing

(r_{1}, r_{2} …, r_{k−2}, r_{k−1}, 1) ⟷ (r_{1} + 1, r_{2}, …, r_{k−2}, r_{k−1} + 2) .

The monomials left unpaired are characterised by the fact that they begin with zero and are not uniterminal. The number of these is clearly less than or equal to the total number beginning zero, which is D_{d} − D_{d−1}. This gives the first claim. The actual number unpaired is given by (D_{d} − D_{d−1}) − (R_{d} − R_{d−1}), since the number of uniterminal monomials starting with zero in degree d is R_{d} − R_{d−1}. This gives the second part. □

Note that this gives a recursive formula for the R_{d}’s in terms of the D_{d}’s. Alternatively, letting P^{i} be the polynomial algebra 𝔽_{2}[ξ_{1}, …, ξ_{i}] (P^{0} = 𝔽_{2}), it is clear that R_{d} = ∑_{i≥1} dim(P^{i−1})_{d−(2^{i}−1)}.

3.4 Example. In degree 42, A_{∗} has dimension 92 while the dimension of the χinvariants is 47. The number of uniterminal monomials, R_{42}, is 44 while D_{41} is 86. So the bounds in Theorem 3.1 tell us that 46 ≤ dim(A_{∗}^{χ})_{42} ≤ 49, whereas if we are prepared to put in more effort (calculating R_{d} precisely) then we shall obtain a sharper upper bound of 48. Note also that the lower bound here is not sharp; in fact 42 is the first degree in which this happens.

4. Conjugation invariants with ξ_{1} inverted.

In this section we adjoin a formal inverse to ξ_{1}, denoting the resulting object by A_{∗}[ξ_{1}^{−1}]. This is regarded as containing A_{∗} in the usual way. Since ξ_{1} is invariant, A_{∗}[ξ_{1}^{−1}] inherits an action of χ and the subspace of invariants turns out to be much more manageable. In fact we show (Theorem 4.1) that it is a polynomial ring on certain natural invariant elements, ϵ_{2}, ϵ_{3}, · · ·. Clearly A_{∗}^{χ} is the intersection of A_{∗} and A_{∗}[ξ_{1}^{−1}]^{χ}. So one might conclude that, since we have simple descriptions of A_{∗} and A_{∗}[ξ_{1}^{−1}]^{χ}, we can easily obtain a description of A_{∗}^{χ}. However, this turns out to be far from the case - the problem of finding the highest power of ξ_{1} dividing a given polynomial in ξ_{1}, ϵ_{2}, ϵ_{3}, ⋯ seems to be difficult in general. In fact, low degree calculations quickly reveal that the algebra A_{∗}^{χ} is complicated; in particular it is far from being polynomial.

At the end of this section we show how Theorem 4.1 generalises nicely to give a description of the invariants of (A_{∗}/⟨ξ_{1}, …, ξ_{n−1}⟩)[ξ_{n}^{−1}].

Note that A_{∗}[ξ_{1}^{−1}]^{χ} = A_{∗}^{χ}[ξ_{1}^{−1}]. That is to say, if we adjoin ξ_{1}^{−1} to the ring A_{∗}^{χ}, we obtain the same object as if we take the χ-invariants of the ring A_{∗}[ξ_{1}^{−1}].

4.1 Theorem. Let k = 𝔽_{2}[ξ_{1}, ξ_{1}^{−1}]. Then

A_{∗}^{χ}[ξ_{1}^{−1}] = k[ϵ_{2}, ϵ_{3}, …] ,

where ϵ_{2} = ξ_{2}χ(ξ_{2}) and, for n ≥ 3, ϵ_{n} = ξ_{2}ξ_{n} + χ(ξ_{2}ξ_{n}). Furthermore

H^{i}(Σ_{2}; A_{∗}[ξ_{1}^{−1}]) = 0 for i > 0 .

The proof of the second statement is straightforward : in A_{∗} we have (χ−1)ξ_{2} = ξ_{1}^{3}. It follows that in A_{∗}[ξ_{1}^{−1}], 1 is the image under χ − 1 of ξ_{1}^{−3}ξ_{2}. But Im(χ − 1) is an ideal in Ker(χ − 1) and so, since 1 ∈ Im(χ − 1), these two objects must be equal.

Now we consider the first statement. It is evident that the elements ϵ_{n} are invariant and we claim that they are also algebraically independent. This follows from the fact that, for each n ≥ 2, ϵ_{n} has a summand involving ξ_{n}, whereas ϵ_{2}, …, ϵ_{n−1} have no such summands. (The highest right lexicographic monomial of ϵ_{n} is ξ_{1}^{3}ξ_{n} if n ≥ 3 and ξ_{2}^{2} if n = 2.) Thus, since the invariants form a subalgebra, k[ϵ_{2}, ϵ_{3}, …] ⊂ A_{∗}^{χ}[ξ_{1}^{−1}]. The opposite inclusion follows directly from Proposition 4.3.

We use the right lexicographic ordering on monomials in the ξ’s and polynomials in the ξ’s inherit a total order. This is by comparing first their highest (with respect to right lexicographic ordering) monomials; if these are equal their next highest monomials are compared and so on. For example, (1, 2) < (2, 0, 1) and (1, 2) + (1, 1) < (2, 0, 1) + (1).

4.2 Proposition.

1) Consider a monomial (r_{1}, r_{2}, …, r_{k}). If r_{2} is odd then the image of this monomial cannot be expressed as the image of a linear combination of lower monomials.

2) Let x ∈ A_{∗} be invariant and let (r_{1}, r_{2}, …, r_{k}) be the highest monomial appearing in x. Then r_{2} is even.

Proof. We claim that, if r_{2} is odd, then the monomial (r_{1}, r_{2}, r_{3}, …, r_{k}) is the first to have (r_{1} + 3, r_{2} − 1, r_{3}, …, r_{k}) as a summand of its image under χ − 1. Firstly, these two elements are adjacent in the right lexicographic ordering, so by 2.5 this is the first image in which the latter element can occur. Secondly,

χ(r_{1}, r_{2}, r_{3}, …, r_{k}) = χ(r_{1}, 0, r_{3}, …, r_{k})χ(ξ_{2})^{r_{2}}
= χ(r_{1}, 0, r_{3}, …, r_{k})(ξ_{2} + ξ_{1}^{3})^{r_{2}} .

From this, and the fact that \binom{r_{2}}{1} ≡ 1 mod 2, it is clear that (χ−1)(r_{1}, r_{2}, r_{3}, …, r_{k}) does contain a summand (r_{1} + 3, r_{2} − 1, r_{3}, …, r_{k}). (Comparison of exponents of ξ_{k}, ξ_{k−1}, …, ξ_{3} shows that this term cannot arise in any other way.)

The second part follows from the first, for suppose x is invariant, that is (χ − 1)(x) = 0, with highest monomial (r_{1}, r_{2}, …, r_{k}). Then (χ − 1)(r_{1}, r_{2}, …, r_{k}) can be expressed as the image of a linear combination of lower monomials, namely x − (r_{1}, r_{2}, …, r_{k}). If r_{2} is odd then this contradicts the first part. □

4.3 Proposition. If x ∈ A_{∗}^{χ} then there exists some integer s such that ξ_{1}^{s}x ∈ 𝔽_{2}[ξ_{1}, ϵ_{2}, ϵ_{3}, …].

Proof. We assume that x is homogeneous since the general result clearly follows from this case. We proceed by a recursion on a well-founded relation. It is evident that < is well-founded (by which we mean that any non-empty set of polynomials has a minimal element) and we claim that whenever x is such that the proposition is true for all x′ < x then it is true for x. Now consider the set of polynomials for which the proposition is false. If this set is not empty then it must have a minimal element, whose existence contradicts our claim. Hence the proposition is true for all x ∈ A_{∗}^{χ}.

We now prove the claim. Let l = (r_{1}, r_{2}, …, r_{k}) be the leading monomial of x. By 4.2, r_{2} is even. Let z = ξ_{1}^{r_{1}}ϵ_{2}^{r_{2}/2}ϵ_{3}^{r_{3}} … ϵ_{k}^{r_{k}}. Recalling the definition of the ϵ_{n}’s, we see that the leading monomial in z is ξ_{1}^{r_{1}+t}ξ_{2}^{r_{2}}ξ_{3}^{r_{3}} … ξ_{k}^{r_{k}}, where t = 3(r_{3} + r_{4} + ⋯ + r_{k}). This is the leading term of ξ_{1}^{t}x and we set x′ = ξ_{1}^{t}x+z, noting that this is invariant, as it is the sum of two invariants. Since the leading monomial of z is equal to that of ξ_{1}^{t}x, the leading term of x′ will be strictly lower. Denote the leading term of x′ by l′. Homogeneity of x implies that x′ will have the same degree as ξ_{1}^{t}x and it is straightforward to see that since l′ < ξ_{1}^{t}l and deg l′ = deg(ξ_{1}^{t}l), we have l′ < l. Thus x′ < x and, by the hypothesis of the claim, the proposition is true for x′, say ξ_{1}^{s}x′ ∈ 𝔽_{2}[ξ_{1}, ϵ_{2}, ϵ_{3}, …]. Then ξ_{1}^{s+t}x = ξ_{1}^{s}x′ + ξ_{1}^{s}z is also in 𝔽_{2}[ξ_{1}, ϵ_{2}, ϵ_{3}, …] and the claim is proved. □

4.4 Remark. The only properties of ϵ_{n} used in the proof are that ϵ_{n} is invariant and has a certain highest term. In particular we could replace ϵ_{2} by ϵ̃_{2} = (χ − 1)(ξ_{1}^{−1}ξ_{3}). This gives an alternative proof that Im(χ − 1) = Ker(χ − 1) once ξ_{1} is inverted, since ϵ̃_{2} ∈ Im(χ − 1) and ϵ_{n} ∈ Im(χ − 1) for all n ≥ 3 and we can re-run the whole programme of the proof of 4.3 with ‘Im(χ − 1)’ in place of ‘Ker(χ − 1)’.

Now we mention a generalisation of Theorem 4.1. We consider the situation where we kill off the first n − 1 generators, ξ_{1}, …, ξ_{n−1} of A_{∗} and invert the n-th generator : let A_{∗}⟨n⟩ denote (A_{∗}/⟨ξ_{1}, …, ξ_{n−1}⟩)[ξ_{n}^{−1}], where ⟨ξ_{1}, …, ξ_{n−1}⟩ is the ideal generated by ξ_{1}, …, ξ_{n−1}. Note that ξ_{n} is invariant in A_{∗}/⟨ξ_{1}, …, ξ_{n−1}⟩, so A_{∗}⟨n⟩ inherits a well defined map χ. (In fact, ξ_{n}, ξ_{n+1} … ξ_{2n−1} are all invariant.)

4.5 Theorem. Let k = 𝔽_{2}[ξ_{n}, ξ_{n}^{−1}]. Then

A_{∗}⟨n⟩^{χ} = k[ϵ⟨n⟩_{n+1}, ϵ⟨n⟩_{n+2}, …],

where ϵ⟨n⟩_{m} = ξ_{2n}ξ_{m} + χ(ξ_{2n}ξ_{m}) if m ≠ 2n and ϵ⟨n⟩_{2n} = ξ_{2n}χ(ξ_{2n}). Furthermore

H^{i}(Σ_{2}; A_{∗}⟨n⟩) = 0 for i > 0 .

The proof of this result is entirely analogous to that of Theorem 4.1. For the second part, it is sufficient to note that the image under χ − 1 of ξ_{2n} is ξ_{n}^{2^{n}+1} (modulo ξ_{1}, …, ξ_{n−1}). For the first part, one observes that the ϵ⟨n⟩_{m}’s are all invariant and algebraically independent and so k[ϵ⟨n⟩_{n+1}, ϵ⟨n⟩_{n+2}, …] ⊂ A_{∗}⟨n⟩^{χ}. Then one shows that if θ = ξ_{n+1}^{e_{n+1}}ξ_{n+2}^{e_{n+2}} … ξ_{k}^{e_{k}} where e_{2n} is even, then there exists an r such that ξ_{n}^{r}θ can be expressed as the leading term of a monomial in the ϵ⟨n⟩_{m}’s (modulo ξ_{1}, …, ξ_{n−1}). Using this, one runs through the recursion argument of Proposition 4.3 with the following lemma in place of Proposition 4.2.

4.6 Lemma. If θ is the leading term of a polynomial in A_{∗}/⟨ξ_{1}, …, ξ_{n−1}⟩ which is invariant then the exponent of ξ_{2n} in θ is even.

The argument is broadly the same as that of Proposition 4.2 : if the exponent of ξ_{2n} in θ is odd then (χ − 1)θ contains the summand ξ_{n}^{2^{n}+1}ξ_{2n}^{−1}θ and one checks that there cannot exist any monomial θ′ < θ such that (χ − 1)θ′ also contains this summand.

5. Some conjectures.

A natural question to ask is : what is the first (i.e. lowest degree) invariant polynomial which involves ξ_{n} ? If n = 1 it is clearly ξ_{1} and if n = 2 one can see that it is ϵ_{2} = ξ_{2}χ(ξ_{2}). For n ≥ 3, ϵ_{n} = (χ − 1)(ξ_{2}ξ_{n}) is an obvious candidate, being in degree 2^{n} + 2. However, we shall see that, at least for n ≤ 7, we can find an invariant in degree 2^{n} + 1 involving ξ_{n}. The following lemma shows that there are no such invariants in degrees 2^{n} − 1 or 2^{n}.

5.1 Lemma. The monomials ξ_{n} and ξ_{1}ξ_{n} are not summands of any invariant elements.

Proof. The image of ξ_{n} contains the monomial ξ_{1}ξ_{n−1}^{2}, which immediately precedes it in the right lexicographic ordering. Since ξ_{n} is the highest monomial in its degree, nothing else can have this monomial ξ_{1}ξ_{n−1}^{2} in its image. Hence ξ_{n} cannot be a summand of an invariant polynomial (c.f. the proof of Proposition 4.2). Similarly, ξ_{1}ξ_{n} has ξ_{1}^{2}ξ_{n−1}^{2} in its image and is the only monomial which does so. □

However, the above argument cannot be applied to ξ_{1}^{2}ξ_{n}. The highest term in its image is ξ_{1}^{3}ξ_{n−1}^{2}, but this does not immediately precede ξ_{1}^{2}ξ_{n} in the ordering - ξ_{2}ξ_{n−1}^{2} occurs between them and is seen to also have ξ_{1}^{3}ξ_{n−1}^{2} in its image. In fact ξ_{1}^{2}ξ_{n} is a summand (and hence the leading term) of an invariant for small n and we make the following

5.2 Conjecture. For each n ≥ 3, there exists a polynomial d_{n} in degree 2^{n} + 1, invariant under χ and with leading term ξ_{1}^{2}ξ_{n}.

We have been able to construct such elements for n = 3, 4, 5, 6, 7. We describe below one particular choice of such elements but first we need some more notation.

Notation. We define certain elements of A_{∗}, which are evidently invariant.

a_{n} = ξ_{n}χ(ξ_{n}) ,
b_{m_{1},m_{2},...,m_{n}} = (χ − 1)(ξ_{m_{1}}ξ_{m_{2}} … ξ_{m_{n}}) .

The following elements are examples of the d_{n}’s of Conjecture 5.2.

d_{3} = b_{2,3}/ξ_{1} ,
d_{4} = (b_{2,4} + a_{2}^{3})/ξ_{1} ,
d_{5} = (b_{2,5} + a_{2}a_{3}^{2} + a_{2}^{2}b_{3,4})/ξ_{1} ,
d_{6} = (b_{2,6} + a_{2}a_{4}^{2} + a_{3}^{2}b_{3,5} + a_{2}^{6}a_{4} + a_{2}^{4}a_{3}^{3} + a_{2}^{11})/ξ_{1} .

(We have not yet found an expression for d_{7} that is small enough to write on one line.) Note that the above expressions are all quotients by ξ_{1} of some sum starting with b_{2,n}. In fact a conjecture equivalent to 5.2 is

5.2’ Conjecture. For each n ≥ 3, there exists an invariant polynomial x_{n}, not equal to b_{2,n}, but such that

x_{n} ≡ b_{2,n} mod ⟨ξ_{1}⟩ ,

where ⟨ξ_{1}⟩ denotes the ideal generated by ξ_{1}.

We would then obtain d_{n} by setting d_{n} = (b_{2,n} + x_{n})/ξ_{1}. (The invariance of x_{n} implies that of d_{n} since a quotient of one invariant by another is invariant.)

Supposing Conjecture 5.2 to be true, we would hope to describe the generators of the algebra A_{∗}^{χ} as follows.

5.3 Conjecture. The following elements generate the conjugation invariants, A_{∗}^{χ}:

1) ξ_{1},

2) a_{n}, for n ≥ 2,

3) b_{m_{1},m_{2},...,m_{n}} where 2 ≤ m_{1} < m_{2} < ⋯ < m_{n}, n ≥ 2 and either n > 2 or m_{1} > 2,

4) d_{n}, for n ≥ 3.

In support of this Conjecture, we have the following result showing that these elements are all necessary generators, i.e. none of them are decomposable.

5.4 Proposition. The families 1-3 of Conjecture 5.3 are necessary generators. For each n ≥ 3, if d_{n} exists then it is necessary and if it doesn’t exist then b_{2,n} is necessary.

Proof. Evidently ξ_{1} is necessary, so we turn to family 2. Suppose that a_{n} were decomposable, a_{n} = ∑_{i} x_{i}y_{i}. where the x_{i}’s and y_{i}’s are homogeneous invariant polynomials of positive degree. Note that a_{n} has a summand ξ_{n}^{2}. For this to occur in one of the summands x_{i}y_{i} we must have that x_{i} and y_{i} each have a summand ξ_{n}. However, by Lemma 5.1, there is no invariant polynomial which has ξ_{n} as a summand. Hence a_{n} is not decomposable.

Similarly, looking at the summand ξ_{1}^{2}ξ_{n} of d_{n} we see that if d_{n} were decomposable then that would imply the existence of an invariant polynomial with either ξ_{n} or ξ_{1}ξ_{n} as a summand, both of which are precluded by Lemma 5.1. Now suppose n is such that d_{n} does not exist, i.e. there is no invariant polynomial with ξ_{1}^{2}ξ_{n} as a summand. Then we can apply the same argument to the summand ξ_{1}^{3}ξ_{n} of b_{2,n} to conclude that b_{2,n} is not decomposable.

Finally consider b_{m_{1},m_{2},...,m_{n}} where m_{1}, …, m_{n} satisfy the conditions given in 5.3. If m_{1} = 2 then this has a summand ξ_{1}^{3}ξ_{m_{2}} … ξ_{m_{n}}. The decomposability of b_{m_{1},m_{2},...,m_{n}} would then imply the existence of an invariant with a summand ξ_{n_{1}} … ξ_{n_{k}} where k ≥ 2 and 1 ≤ n_{1} < ⋯ < n_{k}. We claim that no such invariant exists and hence that b_{m_{1},m_{2},...,m_{n}} cannot be decomposable if m_{1} = 2. Before proving the claim we will show that the indecomposability of b_{m_{1},m_{2},...,m_{n}} in the case where m_{1} > 2 also follows from it. For if m_{1} > 2 then b_{m_{1},m_{2},...,m_{n}} has a summand ξ_{1}ξ_{m_{1}−1}^{2}ξ_{m_{2}} … ξ_{m_{n}}. Any attempt to write this summand as a product of two monomials will involve at least one factor which, by the claim, cannot occur as a summand in an invariant polynomial.

The proof of the claim is just a generalisation of that of Lemma 5.1. One sees easily that any monomial ξ_{n_{1}} … ξ_{n_{k}}, where k ≥ 2 and 1 ≤ n_{1} < ⋯ < n_{k}, is the highest monomial in its degree with respect to right lexicographic ordering. The image under χ − 1 of this monomial has leading term ξ_{1}ξ_{n_{1}−1}^{2}ξ_{n_{2}} … ξ_{n_{k}}, which immediately precedes it and hence cannot occur in the image of any other monomial. □

We have also verified that the generators of Conjecture 5.3 are sufficient in degrees up to 40.

Conjecture 5.3 would imply that for k > 0, H^{k}(Σ_{2}; A_{∗}) = A_{∗}^{χ}/(χ − 1)A_{∗} is generated by the following elements : ξ_{1}, a_{n} (for n ≥ 2) and d_{n} (for n ≥ 3). These elements determine non-zero classes in A_{∗}^{χ}/(χ−1)A_{∗} since they contain the highest terms (w.r.t. right lexicographic ordering) in their degrees and so they cannot lie in the image (χ − 1)A_{∗}. Conversely, if a product of the elements listed in Conjecture 5.3 has a factor b_{m_{1},m_{2},··· ,m_{n}} then it will lie in (χ − 1)A_{∗} as this is an ideal in A_{∗}^{χ}.

Acknowledgements

We would like to thank Lionel Schwartz and Neil Strickland for helpful conversations and we acknowledge the support of TMR grants from the European Union, held at the Laboratoire d’Analyse, Géometrie et Applications (UMR 7539 au CNRS), Université Paris-Nord.

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Departement de Mathematiques, Institut Galilee, Universite Paris 13, 93430 Villetaneuse, France.

E-mail address: crossley@math.univ-paris13.fr, sarah@math.univ-paris13.fr