A NEW ACTION OF THE KUDO-ARAKI-MAY ALGEBRA ON THE DUAL OF THE SYMMETRIC ALGEBRAS, WITH
APPLICATIONS TO THE HIT PROBLEM
DAVID PENGELLEY AND FRANK WILLIAMS Abstract. The hit problem for a cohomology module over the Steenrod algebra A asks for a minimal set of A-generators for the module. In this paper we consider the symmetric algebras over the field Fp, for p an arbitrary prime, and treat the equivalent problem of determining the set of A*-primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo-Araki-May algebra K, and consider the K-module structure of the primitives, which form a sub K-algebra of the dual of the infinite symmetric algebra. Our examples show that the K-action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem. We also note connections to the Singer transfer.
1. Introduction and Theorem The hit problem for a cohomology module over the Steenrod algebra A, foran arbitrary prime
p, asks for a minimal set of A-generators for the module,i.e., a basis for its quotient by those elements hit under the left action of the
positive dimensional elements of A. For p = 2 the hit problem for symmetricalgebras has been studied by Janfada and Wood [9, 10] (see below); and quotients had earlier been studied by Peterson [19]. The infinite symmetricalgebra over a prime
p can be regarded as the cohomology H*(BU ; Fp)[ H*(BO; F2), if p = 2]. We shall adopt the point of view of Alghamdi, Crabb,and Hubbuck [1] and consider the equivalent problem of determining the
set P of A-annihilated elements for the corresponding A-action on the dual M = H*(BU ; Fp) [H*(BO; F2)], i.e., the kernel of the downward right actionon
M of the positive dimensional elements of A. These are also the primitiveelements for the A*-coaction on
M , and so we will henceforth call themprimitives, as in literature such as [2, 3, 8, 17]. Since the downward A-action
satisfies the Cartan formula, the primitive elements P form a subalgebra of M . And the infinite loop space multiplication on BU [BO] makes M intoa Hopf algebra over the Dyer-Lashof algebra R, or equivalently over the
Kudo-Araki-May bialgebra K [14, 15].Recall that
M is a polynomial algebra with generators an (n >= 1) dualto the powers
cn1 [wn1 , if p = 2] of the elementary symmetric function of
1991 Mathematics Subject Classification. 16W22,16W30,16W50,55R40,55R45,55S05,55S10,55S12,57T05,57T25.
1
2 DAVID PENGELLEY AND FRANK WILLIAMS degree one. (The reader should assume henceforth that the occurrence in anyexpressions of
an with n <= 0 is interpreted as zero or omitted. In particular,note there is no nonzero element
a0.) We shall now state our main theorem,proved in a later section, which provides a method of producing primitive
elements for all primes, often indecomposable in the algebra P, by definingan alternative action of R on
M , or equivalently and more simply, an actionof the Kudo-Araki-May bialgebra K.
Theorem. The formulas
d0 (1) = 1, di (1) = 0 for i > 0, and for n >= 1 and i >= 0
di(an) = 8!:
apn if i = 0,- apn+(p-1) if i = 1,0 if
i > 1,
iterated and extended on products in M by the Cartan formula using
\Delta dm = X
i+j=m
di \Omega dj,
respect the K Adem relations, and thus produce a graded action by the bial-gebra K (equivalently R) on the algebra
M , where the di are the standardgenerators of K. The entire structure thus becomes a K-algebra, i.e., an
algebra over the bialgebra K.The action of K commutes up to Verschiebung with the right action of the Steenrod algebra. Consequently, the subalgebra P of M consisting ofprimitive elements is closed under the K-action, hence becomes a sub-K- algebra.
Our primary motivation for and application of this theorem is to generatenew elements of P from known ones by applying elements of K (equivalently
R), and this we shall explore below in examples that use this new action toobtain and analyze families of primitive elements, especially at odd primes. It is nontrivial that the formulas above respect the Adem relations thatdetermine K, thus producing an action of K on
M , rather than just of thefree bialgebra on the elements di, and this will be exploited in our examples.Singer [20] developed a method for finding primitives at the prime 2, and
the primitives we will produce for p = 2 coincide with his. We shall commentbelow on the connection between our methods and his.
We shall also discuss how our action incorporates on the duals of the finitesymmetric algebras the operators introduced by Kameko [7, 11], Crossley [6], and Janfada and Wood [9, 10].
2. Background It is easy to solve the hit problem for the cohomology of a single projectivespace. Most work on the hit problem has been done on the cohomology of
products of projective spaces (the canonical polynomial algebras over A on
THE HIT PROBLEM FOR SYMMETRIC ALGEBRAS 3 one-dimensional generators), where the problem is quite difficult. Initialresults for such products were obtained by Peterson [18], and he made a conjecture [19] for p = 2 about exactly which degrees contain A-generators(albeit not their rank), which was proven by Wood [22]. A good reference list of the literature on attempts to solve the problem for projective spacesis given in [13]. As we noted above, for
p = 2 the hit problem for symmetricalgebras has been studied by Janfada and Wood [9, 10].
Since the polynomial generators an (n >= 1) are the images of the positivedegree elements of the homology of the projective space C
P (1) = BU (1) ae BU [RP (1) = BO (1) ae BO], by the Cartan formula the downward A-action preserves the length of monomials in the generators
an. So if webigrade M by length l and topological degree n, we have
P i : Ml,n ! Ml,n-(p-1)i for the right action of the Steenrod powers from A (we use complex topolog-ical grading for
p odd, i.e., an 2 M1,n = H2n(BU (1) ; Fp)). Thus M splitsby length as an A-module. Clearly too, P is a bigraded subalgebra of
M .We now review a few basic properties [14, 15] of the Kudo-Araki-May
algebra K, which will be one of our main tools in this paper. The bigradedbialgebra K over a prime
p is generated by elements di 2 K1,(p-1)i, i = 0, 1, . . . (if p is odd we here too use the complex grading in the secondcoordinate) subject to their own set of (Adem) relations
didj = X
k `
(p - 1) (k - j) - 1
pk - i - (p - 1) j 'di+pj-pkdk, (i > j).
The degree of elements in K satisfies the condition that multiplication isa map K
l1,r \Omega Kl2,n ! Kl1+l2,pl1n+r. So the first index is the length of amonomial, and multiplication is skew-additive on the second index. N.B:
The identity element is 1 2 K0,0, quite distinct from d0 2 K1,0. A monomial dI = di1 * * * dil is admissible provided the multi-index I = (i1, . . . , il) isnondecreasing, and we recall that the admissibles form a basis for K.
A graded homology K-module N is one that satisfies the requirementK l,r \Omega Nn ! Npln+r. A graded homology K-module structure on N (complexgraded if
p is odd) corresponds to an action of the Dyer-Lashof algebra R by
iterating the conversion formula Qi(xn) = (-1)i-n di-n(xn) on an element xn 2 Nn. Note that this is not merely a reindexing between K and R, sincethe degree of the underlying homology element changes repeatedly during a
composition. The algebra structures of K and R are very different.We also recall that the algebra K has a compatible coproduct determined by
\Delta dm = X
i+j=m
di \Omega dj,
and that, just as the Dyer-Lashof action on the homology of an infinite loopspace satisfies a Cartan formula on homology products using a coproduct in R, so does the corresponding action of K via its own coproduct. We call such
4 DAVID PENGELLEY AND FRANK WILLIAMS a combined structure of a graded homology K-module with a product anda compatible Cartan formula a homology K-algebra. Clearly the K-action defined in our main theorem makes M with its topological degree into ahomology K-algebra.
3. Finite symmetric algebras From the definition, we see that di : Ml,n ! Ml+(p-1)(l-i),pn+(p-1)i. Ex-amining the relations between
i and l in this formula yields the followingproposition.
Proposition 3.1. Applying dl to a monomial in M of length l in the polyno-mial generators
ak preserves the length of the monomial, making the length l subspaces of both M and P into left Fp [dl] modules as well as right A-modules. In fact they are clearly each free modules over F
p[dl], since dl is amonic endomorphism on the subspace of length l, and raises degree.More generally,
di on a monomial of length l is zero if i > l, of length lif i = l, and of greater length if i < l.
We now see how this K-module structure applies to finite symmetric alge-bras. The subspace of
M with basis the monomials in the an of total length<= l corresponds to H* (BU (l) ; Fp) [H* (BO (l) ; F2)], so the results in thispaper also imply results on the hit problem for their duals
H *(BU (l); Fp)[ H* (BO (l) ; F2)], the algebras of symmetric polynomials in finitely manyvariables; and also for the corresponding Thom spaces
M U (l) [M O (l)],whose homology basis is the monomials of length exactly l. When i < l, theK-action allows us to transfer information about primitives from each length
degree to higher lengths. We will illustrate this in our examples of new fam-ilies of primitive elements. The sparseness of the new K-action formulas on generators (greatly simplified from the Dyer-Lashof action), combined withthe Cartan formula on products, makes calculation relatively easy, as we shall see in the examples.The fact that P is endowed with the structure of a module over K does more than simply produce new primitives from known ones. Specifically, theAdem relations in K will help reveal K-connections between our examples of primitives in the module P, and in the last section of the paper wewill illustrate aspects of the K-module structure, by showing that P has nontrivial K-module relations.At this point we note that our K-action contains within it the duals on symmetric algebras of the operators k introduced by Kameko [11] at theprime 2 for studying the hit problem for the cohomology of products of projective spaces. The operators k are left inverses of his operators OEkgiven by
OEk(f ) = oekf 2, which have been utilized by many researchers (oekis the k-th elementary symmetric function, i.e., ck [wk] in our notation),and have been adapted for mod 2 symmetric algebras by Janfada and Wood
[9, 10] (see also Crossley's map [6] analogous to *2 for H* (CP1 * CP1; Fp)at odd primes).
THE HIT PROBLEM FOR SYMMETRIC ALGEBRAS 5 We note that *k : Ml,n ! Mk,2n+k for l <= k, and one can compute that
*k (ai1 * * * ail) = ak-l1 a2i1+1 * * * a2il+1
= ak-l1 dl (ai1 * * * ail) .
Thus on the length k subspace Mk,*, we see that *k coincides exactly with dk.For algebras of coinvariants in mod 2 homology, the dual Kameko operators *k on Mk,* have together been called Sq0, and used to study theSinger transfer from the primitives to the cohomology of the Steenrod algebra [2, 3, 4, 5, 7, 8, 20, 21]. We wonder if the entire K-action on thesymmetric coinvariants, not just the single operator
dk on each correspond-ing length k subspace, may be useful in relation to the transfer. Singer hasalready extended the action of the dual
Sq0 of the Kameko operators onthe symmetric coinvariants to an action of the "bigraded Steenrod algebra"
[20], and we explain at the end how this is equivalent to the K-action aboveat the prime 2.
Finally, let us consider the state of knowledge on the hit problem for sym-metric algebras in
l variables. For l = 1, the hit problem is easily solvedfor all primes (see below). For the prime 2, the hit problem for symmetric
algebras was solved for l = 2, 3 by Janfada and Wood [10]. We, on the otherhand, shall use our K-action to give examples for all primes
p of numerousfamilies of primitives ranging over many lengths. For p odd, these are en-tirely new. While some of these families specialize to zero at the prime 2,
those that remain nontrivial coincide with the elements given by Singer [20].In a paper in preparation [16] we shall treat the hit problem of determining all the primitives in H* (BU (2) ; Fp) at any prime, making use of Proposi-tion 3.1. Additionally, Janfada and Wood [9] formulated and proved a Peterson conjecture for the finite symmetric algebras H* (BO (l) ; F2), whichis formally identical, surprisingly, to the conjecture proven for products of projective spaces. At the end of this paper we will comment on an analogat odd primes.
4. Examples of primitives and of K-module relations in P In this section we shall illustrate some examples of primitives, showinghow the K-action can be used to generate more primitives from existing ones,
and also begin to see some nontrivial relations in the K-module structure ofP, the algebra of primitives. Recall that
d0 always acts as the p-th power, so p-th powers of primitives always produce more primitives, and we will leavethis largely unspoken in what follows. In fact we are particularly interested
in finding new indecomposable primitives in P. Example 4.1 (Length one primitives). The primitives of length one are easily determined from the formula am * P r = \Gamma m-r(p-1)r \Delta am-r(p-1). A basis
6 DAVID PENGELLEY AND FRANK WILLIAMS for P in length one is then{
ajpn-1 | 1 <= j <= p - 1, n >= 0, (j, n) 6= (1, 0)} . And from this, since our K-action formula says d1 (ajpn-1) = -ajpn+1-1, weobtain the set
{ak | 1 <= k <= p - 1} as a basis for the free Fp [d1]-module structure that P has in length one.From Proposition 3.1 or the definition,
di for i > 1 will be zero on thelength one primitives, so this is the end of their story from the point of view
of the K-action.
Of course products of these length one primitives provide some higherlength primitives in P, but we shall now seek indecomposable primitives of higher lengths. Example 4.2. Define
fi (k, l) = kakal+p-1 - lalak+p-1 for 1 <= k < l <= p - 1. (Note that this set is empty when p = 2.)Each of these length two elements is verified by inspection to be an indecomposable primitive in P. Proposition 3.1 ensures that the length twoprimitives form a free F
p[d2]-module. We compute dn2 fi (k, l) = ka(k+1)pn-1a(l+p)pn-1 - la(l+1)pn-1a(k+p)pn-1.
This example gives us many new primitive elements of monomial lengthtwo, which are dual to A-indecomposables in
H*(BU (2); Fp). (We shall seein [16] that there are many more length two primitives.)
Of course the full K-action on the fi (k, l) might yet produce more prim-itives, of higher length. Since K has as basis the admissible monomials, we need only consider action by admissibles. Combined with Proposition 3.1and the fact that
d0 always acts as the p-th power, we see that to understandwhat the fi (k, l) can produce, it remains only to consider how d1 acts onthem, which will yield primitives of higher length. This we will do below.
Example 4.3. Define
ff(k, l) = kapkap2+lp-1 - lapk+p-1alp+p-1, for 1 <= k, l <= p - 1. N.B: While for fi (k, l) we required k < l, here there isno such restriction. Note that
ff(k, l) has length p + 1. A short calculationchecks that all such ff(k, l) are primitive. Clearly they are also indecompos-able in P. (Note that when
p = 2, ff(1, 1) = a21a5 + a22a3 is a length threeindecomposable primitive in degree 7, cf. [20, p. 559].)
Now let us explore further using the K-action, to see what additionalprimitives may arise, and also to begin to see relations in the K-module structure of P. Since K has as basis the admissibles, we need only consideraction by these and compare results to find K-module relations.
THE HIT PROBLEM FOR SYMMETRIC ALGEBRAS 7 Consider first the action of d1 on the set {fi (k, l) | 1 <= k < l <= p - 1} .The operation
d1 will raise their length from two to p + 1, the same lengthas the ff(k, l).We observe that
d1fi (k, l) = -ff(k, l) + ff(l, k), a linear combination of primitives already discovered above.We examine the action of K on
ff(k, l). We see from the Cartan formula(refining Proposition 3.1) that di (ff(k, l)) = 0 unless i = 0, 1, p, or p + 1.Further, for s >= 0, one calculates easily that for k < l,
dpdsp+1ff(k, l) = d0ds+12 fi (k, l) , in length 2p, the p-th power of an element of length two from Example 4.2.Thus
dpdsp+1 (ff(k, l)) does not produce new indecomposable elements of P,but since it is admissibles in K that appear on both sides, this does display
nontrivial K-module relations in the structure of P.However, applying
d1 and dp+1 to ff(k, l) does yield new primitive ele-ments. Iterating these operations of K on
ff(k, l), we find the following setof algebraically indecomposable primitives in P:\Phi
dr1dsp+1ff(k, l) | r, s >= 0 and 1 <= k, l <= p - 1\Psi =
{kap
r+1
kps+(ps-1)alpr+s+1+(pr+s+2-1) - la
pr+1 kps+(ps+1-1)alpr+s+1+(pr+s+1-1)|
r, s >= 0 and 1 <= k, l <= p - 1},
of length pr+1 + 1. Remark. For p = 2, the primitives in Example 4.3 were first obtained bySinger [20].
Example 4.4 (More relationships between ff's and fi's in S). Finally we areready to consider the action of
d1 on all the primitives dn2 fi (k, l). For n = 0this yielded - ff(k, l) + ff(l, k) from Example 4.3. We intend to generalizethis phenomenon. To calculate
d1dn2 (fi (k, l)) for n >= 1 efficaciously, we firstnote that in K, dp+1d1 = d1d2 is easily computed from the K Adem relationsgiven earlier. Using this on
d1dn2 fi (k, l), we can repeatedly pass d1 across d2, replacing d2 each time with dp+1, producing
d1dn2 fi (k, l) = dnp+1d1fi (k, l) = -dnp+1ff(k, l) + dnp+1ff(l, k). Thus we obtain for the d1dn2 fi (k, l) sums of primitives derived at the end ofExample 4.3 with
r = 0.
5. Proof of the main theorem We begin with motivation for the action defined above in the theorem.
8 DAVID PENGELLEY AND FRANK WILLIAMS
To get new primitives in M from known ones, we might na"ively hope touse the Nishida relations [12] connecting the standard R and A actions:
(Qr(x)) * P s = X
i (-1)
i+s `(p - 1) (r - s)
s - pi 'Q
r-s+i \Gamma x * P i\Delta ,
noting that they imply that an element x in the kernel of all *P i also haseach
Qr(x) in the kernel of all *P s. But, of course, *P 0 is the identity, sothere are no such
x. We need to look more carefully at the structure of Mitself over R (equivalently over K).
We begin by translating the Dyer-Lashof operations in the Nishida for-mula into operations in K. We obtain
(dj(xn)) * P s = X
i `
(p - 1) (j + n - s)
s - pi 'dj+pi-s \Gamma xn * P
i\Delta .
Our goal is to modify this K-action so that it will produce new primitivesfrom old. Recall from above that the
i = 0 term in the Nishida formulaobstructs our goal, but observe that eliminating all but the the last term on
the right, for which i = s/p, would yield
(dj (xn)) * P s = dj ixn * P s/pj , which would admirably achieve our goal. Thus we aim greatly to simplifythe K-action formula to this end.
According to Kochman [12], the values of the Dyer-Lashof operations onthe polynomial generators of
M = H*(BU ; Fp) [H*(BO; F2) if p = 2] aregiven, for n >= 1, by
Qr (an) = (-1)r+n+1 `r - 1n 'an+(p-1)r + decomposables. In terms of K, this is
dj (an) = -`n + j - 1n 'apn+(p-1)j + decomposables. We note in particular that d1(an) = -apn+(p-1)+ decomposables, and since d0 is always the p-th power we have d0(an) = apn. Calculation of the A-actionshows that the leading terms here are compatible with our aim above, so we
make the vastly simplified definition of a new action in the main theoremabove based just on these leading terms. In our definition of the new action, it also seems necessary for the higher dj to be zero on the an in orderto have length degree respected via
di : Ml,n ! Ml+(p-1)(l-i),pn+(p-1)i, asneeded to prove Proposition 3.1. Henceforth we use only this action of the
K-generators on M . Our goal is to prove that this new definition respects theAdem relations, creating an actual action by K that preserves the subalgebra P of primitives.
THE HIT PROBLEM FOR SYMMETRIC ALGEBRAS 9 Remark. The alternative action could equivalently (but less conveniently)be defined using the Dyer-Lashof algebra by the formula
Qj(an) = 8!:
apn if j = n, apn+(p-1) if j = n + 1, and0 otherwise.
The proof of our main theorem will be accomplished in two steps. Thefirst is the verification that we have an actual action by K. The second is the simplified Nishida formula we need to ensure that the primitives P forma sub K-algebra under this action.
Proof of Theorem. We shall use formal power series over Fp. Let d(u) =P1
i=0 diui, and note for applying the Cartan formula that it is grouplike,i.e., \Delta d (u) = d (u) \Omega d (u). Let a(y) = P1
j=1 ajyj, recalling that there isno a0. Also let a(p-1)(y) denote the (p - 1)-st derivative of a(y), and recallWilson's Theorem to observe that the alternative K-action on the generators
of M can be expressed in terms of formal power series by
d(u)a(yp) = [a(y)]p + ua(p-1)(y). Also recall from [14] and [15] that the Adem relations in K can be ex-pressed by the formal power series identity
d(u)d((u - v)p-1v) = d(v)d((u - v)p-1u). The proof will be in two steps.
Step one. The K-action formulas in the theorem satisfy the Adem rela-tions in K and hence define a homology K-algebra structure on
M .We first compute
d(u)d((u - v)p-1v)a(yp
2 ) = d(u) n[a(yp)]p + (u - v)p-1va(p-1)(yp)o
= [d(u)a(yp)]p + (u - v)p-1vd(u)a(p-1)(yp) = [(a(y))p + ua(p-1)(y)]p + (u - v)p-1v ia(p-1)(y)j
p + (u - v)p-1vud
1a(p-1) (yp)
= (a(y))p
2 + \Gamma up + (u - v)p-1v\Delta ia(p-1)(y)jp + (u - v)p-1vud
1a(p-1) (yp) .
Now up+(u-v)p-1v = (up - vp)+(u-v)p-1v+vp = (u-v)p+(u-v)p-1v+vp = (u-v)p-1u+vp, i.e., it is symmetric in u and v. So d(u)d((u - v)p-1v)a(yp
2 ) is symmetric
in u and v, and hence is equal to d(v)d((u - v)p-1u)a(yp
2 ). Thus the Adem
relations are satisfied on the polynomial generators, and so, by the definitionof the K-action on all of
M through the Cartan formula, they are satisfiedon all of M .
Step two. The K-action formulas in the theorem satisfy the identity
(dix) * P k = di ix * P k/pj
10 DAVID PENGELLEY AND FRANK WILLIAMS for any element x 2 M , and hence the primitives become a sub-K-algebraunder the K-action.
We could prove this formula by brute force calculation with binomialcoefficients. However, we prefer to continue with a formal power series approach. Let P (z) = Pr>=0 P rzr, for which \Delta P (z) = P (z) \Omega P (z). Notethat the downward A-action on
M
am * P r = `m - r (p - 1)r 'am-r(p-1) is expressed in formal power series by the identity
a(x) * P (x-py) = a(x + y). We remark that taking the kth partial derivative with respect to x of bothsides of this equation yields
a(k)(x) * P (x-py) = a(k)(x + y). We may now check the interaction of the A-action with the K-action essen-tially by verifying an identity of formal power series:
First we check the identity
(diaj) * P k = di iaj * P k/pj via the formal power series computation
[d(u)a(xp)] * P (x-py) = h[a(x)]p + ua(p-1)(x)i * P (x-py)
= [a(x + y)]p + ua(p-1)(x) * P (x-py) = [a(x + y)]p + ua(p-1)(x + y) = d(u)a(xp + yp)
= d(u) ha(xp) * P (x-p
2 yp)i .
Now the identity for general x 2 M follows by calculation from the iden-tity above for the
aj, since both the A-action and the K-action satisfy Cartanformulas.
This completes the proof of the theorem. \Lambda
6. Closing comments Remark (Singer's action of the bigraded Steenrod algebra). A graded co-homology K-module
M is one in which the action satisfies Km,r \Omega Ms ! Mpms-r. Such an action by K determines an action by Steenrod operations [14, 15]. When p = 2, this takes the form Sqix = dr-ix, where x is a classof grade
r, for an appropriate grading. If we take for our grading the lengthgrading in
M, then the formulas for our K-action do clearly create a graded
THE HIT PROBLEM FOR SYMMETRIC ALGEBRAS 11 cohomology K-module structure on M , and translate to
Sqi(an) = 8!:
a2n if i = 1, a2n+1 if i = 0, and0 if
i > 1,
since Sqi(an) = d1-i(an) for all generators. This is the action created bySinger [20] of his "bigraded Steenrod algebra" (note that
Sq0 acts nontriv-ially, so this is not an action of the Steenrod algebra A).
We have here a highly curious and unusual situation with the bigraded M . The single K-module structure we have defined makes M simultaneouslyinto a graded cohomology K-module with respect to its length grading, and
a graded homology K-module with respect to its topological grading, asnoted earlier. Taken together this is what enables
M to have essentiallyequivalent actions of both the Dyer-Lashof algebra and Singer's bigraded
Steenrod algebra, so that the two actions produce the same primitives. Remark (The Peterson conjecture and odd primes). A natural odd primary"Peterson conjecture" for
M is that S, in length degree n, is concentratedin topological degrees d such that d + n has no more than n non-zero digitsin its p-ary expansion, since this has been verified for p = 2 by Janfada andWood [9]. Moreover, our results here are consonant at all primes with this
conjecture, in that action by any di multiplies the total degree d + n by p.Although this conjecture is false in general for odd primes, in a future paper [16] we shall show that it holds for H*(BU (2); F3), and in somewhat alteredform for
H*(BU (2); Fp), p >= 5.
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