SOME APPLICATIONS OF INCIDENCE HOPF ALGEBRAS TO

SOME APPLICATIONS OF INCIDENCE HOPF ALGEBRAS TO FORMAL GROUP THEORY AND ALGEBRAIC TOPOLOGY CRISTIAN LENART AND NIGEL RAY

1. Introduction Hopf algebras are achieving prominence in combinatorics through the in uence of G.-C. Rota and his school, who developed the theory of incidence Hopf algebras (see [7], [15], [16]). The aim of this paper is to show that incidence Hopf algebras of partition lattices provide an ecient combinatorial framework for formal group theory and algebraic topology. We start by showing that the universal Hurwitz group law (respectively the universal formal group law) are generating functions for certain leaf-labelled trees (repectively plane trees with colored leaves); the proof uses the combinatorial expression for Lagrange inversion in terms of leaflabelled trees, which is due to M. Haiman and W. Schmitt [5]. A formal group law identity with a combinatorial proof (similar to the one in [8]) is also presented. The relevance of Hopf algebras and formal group theory to algebraic topology (and in particular to K -theory and bordism theory) is well-known. The relevance of the Roman-Rota umbral calculus, as an elegant and illuminating framework for computations, became clear through the work of N. Ray, F. Clarke, A. Baker et al. (see [12], [11], [2], [3]). Applications of incidence Hopf algebra techniques to algebraic topology were given recently by N. Ray and W. Schmitt. In this work, we discuss some of their applications from a dierent point of view, and present other examples of computations (both classical and new) which can be expressed in a very concise form using the incidence Hopf algebra framework. Such applications include: coactions of MU(MU ) and K(K ), expressing the images of the coecients of the universal formal group law under the Hurewicz homomorphism MU ! K(MU ), congruences in MU, and combinatorial models for the dual of the Steenrod algebra. We refer the reader to [18] for all information concerning Hopf algebras. All the prerequisites concerning formal group law theory can be found in [8] x3; we also refer to [6] for an encyclopaedic description. As far as algebraic topology is concerned, the relevant prerequisites are presented in [1] and [19]. However, this paper can be read without prior knowledge of algebraic topology, the only disadvantage being that the motivation for certain computations will be less clear. 1

2

CRISTIAN LENART AND NIGEL RAY

2. Combinatorial Background Let n denote the lattice of partitions of the set [n] := f1; 2; : : : ; ng, and In the Boolean algebra of those partitions of [n] for which all the blocks are intervals (in N). We write n;k and In;k for the subsets of n and In consisting of partitions with k blocks. Consider the family of posets which are isomorphic to a nite product of lattices n . The incidence Hopf algebra of this family modulo isomorphism of posets (see [15], [16]) is the Faa di Bruno Hopf algebra := Z[1; 2; : : : ], with comultiplication and counit speci ed by 1 if n = 0 X (2.1) () jj?1 ; "(n) = 0 otherwise , (n) = 2n+1

for all n 0; here () is the type of the partition , n denotes the isomorphism class of the lattice n+1 , and 0 = 1. Now consider the family of posets which are isomorphic to a nite product of Boolean algebras In. The incidence Hopf algebra of this family modulo isomorphism of posets is the Hopf algebra (H ) := Z[b1; b2; : : : ], with comultiplication speci ed by

(bn) =

X

2In+1

b() bjj?1 =

X

X

k1 n1 +n2 +:::+nk =n+1 ni 1

1 if n = 0 "(bn) =

Yk i=1

!

bni?1 bk?1 ;

(2.2)

0 otherwise , for all n 0; here bn denotes the isomorphism class of the Boolean algebra In+1, and b0 = 1. It is not dicult to check that can be identi ed with a sub-Hopf algebra of (H ) via the monomorphism n 7! (n + 1)!bn. Let R be an evenly graded commutative ring. Consider the set Hom(; R) of graded ring homomorphisms from to R. This set is a group under the operation of convolution, which is dual to the comultiplication in ; more explicitely, we have (f g)(z) = m (f g) (z) and f ?1(z) = f S (z) ; where m denotes the multiplication in R, and S is the antipode of . The map (f g) in Hom(; R R) will be denoted by f ~ g. A sequence of elements = (1 = 0; 1; : : : ) in R with i 2 R2i will be called an umbra. Let in Hom(; R) be the ring homomorphism speci ed by i 7! i, let be its convolution inverse, and let be the umbra speci ed by = ; in this context, if = (1 = 0; 1; : : : ), then is the identity map, and is the antipode of . The correspondence R 7! Hom(; R) is a covariant functor from the category of graded rings to the category of groups. For some group homomorphisms : Hom(; R) ! Hom(; T), as those mentioned in

APPLICATIONS OF INCIDENCE HOPF ALGEBRAS

3

x3, there is a unique ring homomorphism : R ! T such that Hom() = . The above discussion also applies to the group Hom((H ); R) of graded ring homomorphisms from (H ) to R. The following is a well-known result (see [4], Theorem 5.1 and [16], Examples 14.1 and 14.2).

Theorem 2.3.

1. The group Hom(; R) is anti-isomorphic to the group under substitution of Hurwitz series in R ffX gg of the form X (X ) := i?1 X(i) ; i1

where is an umbra in R (if R has no torsion, then we can identify X(i) with X i =i!); the anti-isomorphism is speci ed by 7! (X ). 2. The group Hom((H ); R) is anti-isomorphic to the group under substitution of formal power series in R [[X ]] of the form X r(X ) := ri?1X i ; i1

where r is an umbra in R; the anti-isomorphism is speci ed by r 7! r(X ). We have now established an important notational convention, to which we shall adhere below: a Hurwitz series and a map in Hom(; R) (or a standard formal power series and a map in Hom((H ); R) ) will be associated with an umbra denoted by a Greek letter (or lower case Roman letter). The following notational convention will also be useful: if P lies in [x] and Qb lies in (H )fxg, then P in R[x] and Qr in Rfxg are obtained from P and Qb via the substitutions i 7! i and bi 7! ri, respectively; furthermore, P (1;1;::: ) will be denoted simply by P . Let us now consider the binomial Hopf algebra [x], and the divided power Hopf algebra (H )fxg (see [8], x2 for a detailed discussion). Recall the conjugate Bell polynomials and the Bell polynomials in [x]; they can be expressed combinatorially as follows: Bn(x) = x ( )(n?1) for n 1 ; (2.4) where is the umbra (1; x; x2; : : : ) in [x]. Let us write

Bn(x) =

n X

s(n; k)xk

and

Bn(x) =

n X

S (n; k)xk :

k=1 k=1 The coecients s (n; k) and S (n; k) in 2(n?k) are known as -Stirling numbers of the rst and second kind, respectively; S (n; k) are also known as partial Bell

polynomials. The standard notation s(n; k) and S (n; k) for the classical Stirling numbers of the rst and second kind is consistent with our conventions.

4


Let us consider the umbra x = (1; x=2; x2=3!; : : : ) in ( Q)[x], and the following polynomials in (H )fxg: 0b(x) = 1 ; and nb (x) := x (b x)(bn?1) for n 1 : (2.5) Since is a sub-Hopf algebra of (H ) via the inclusion i, we have the group homomorphism i : Hom((H ); ( Q)[x]) ! Hom(; ( Q)[x]) dual to i, which is speci ed by f 7! f i; more explicitely, we have that i( b) = and i( x) = , whence i(b) = by taking inverses. Hence, identifying [x] and (H )fxg with the corresponding subrings of ( Q)[x], and using (2.4) and (2.5), we deduce that Bn(x) = n! nb (x) : (2.6) A generalization of this result using set systems and their automorphism groups was obtained in [9]. Writing n k X b x ; b n(x) = n;k k! k=1

b = k ! s (n; k ). we clearly have n! n;k

3. Applications to Formal Group Theory Let us denote, as in [8], by ffX; Y gg the -algebra of Hurwitz series in two indeterminates. According to [8] x3, there are formal power series F (X; Y ) 2 1ffX; Y gg and f b (X; Y ) 2 (H )1[[X; Y ]] such that F ((X ); (Y )) = (X + Y ) and f b(b(X ); b(Y )) = b(X + Y ) : Let L be the subring of (H ) generated by the coecients of f b (X; Y ); this is the Lazard ring, which is known to be isomorphic to the complex cobordism ring MU (see [1]). The formal power series F (X; Y ) and f b(X; Y ) are precisely the universal Hurwitz group law (over ) and the universal formal group law (over L). We use the notations X + Y := F (X; Y ) and X +b Y := f b (X; Y ) ; which can be iterated. The coecients of k k Yk Xini X Yk ni X in X and X in i b Xi i n ! i i=1 i=1 i=1 i=1 will be denoted by Fn1 ;:::;nk and fnb1 ;:::;nk , respectively. The rst one lies in , while the second one lies in L.


In [8], we expressed Fn1;:::;nk in terms of as follows: X b b (0; ) (; 1) ; Fn1;:::;nk = 2n

5

(3.1)

here, and throughout this section, n := n1 + : : : + nk , and is the partition of [n] with blocks [n1], n1 +[n2] := fn1 +1; : : : ; n1 + n2g, ..., n ? nk +[nk ]. We are going to present a similar expression (3.6) for fnb1;::: ;nk . Several cancellations occur in these expressions. However, we are able to give a combinatorial interpretation for the coecients of the monomials in Fn1;::: ;nk and fnb1 ;:::;nk in terms of trees, by using the Haiman-Schmitt form of Lagrange inversion [5]. To do this, we need to choose other polynomial generators for and (H ), namely 1; 2; : : : , and b1; b2; : : : . We consider rooted trees with n leaves of two types: rooted trees with leaves labelled 1; 2; : : : ; n, and rooted plane trees with a k-coloring of the leaves of type (n1; : : : ; nk ) (i.e. a coloring with colors 1; : : : ; k such that exactly ni leaves are colored i). We also assume that no non-root vertex has degree 2 (i.e. only one descendent). The number of vertices of a tree T is denoted by jT j. A vertex of a tree of either type is called peripheral if all its descendents are leaves; the set of descendents of a peripheral vertex will be called a peripheral i1 i2 0 0 class. The type (T ) of a tree T of the rst type is de ned as 1 2 : : : , where ij is the number of vertices of T 0 with j + 1 descendents; the type of a tree of the second type is de ned similarly, as a monomial in b1; b2; : : : . We can now state the combinatorial interpretation mentioned earlier. Theorem 3.2. We have that X (3.3) Fn1 ;:::;nk = (?1)jT 0j?n (T 0) ; T X 0

fnb1 ;:::;nk =

T 00

(?1)jT 00j?n (T 00) ;

(3.4)

the rst sum ranges over those trees T 0 of the rst type for which none of the sets of labels corresponding to a peripheral class are contained in a block of the partition ; the second sum ranges over those trees of the second type for which no peripheral class is monochromatic. Furthermore, we have that X (3.5) Fn1 ;:::;nk = (?1)k Ben+k?1;k ; k1

P here Bem;k := (), with summation ranging over those partitions 2 m;k

with no singleton blocks, for which none of the blocks are contained in a block of the partition . Proof. We denote by Tm the set of rooted trees with m leaves labelled 1; 2; : : : ; m; for T 2 Tm, we denote by p(T ) the partition of [m] whose blocks are either peripheral classes or singletons. We write kk for the number of non-singleton blocks of the partition . We also de ne a new partial order on n by insisting

6


that if and only if is obtained from by amalgamating only singleton blocks. With these notations, and using the expression for Lagrange inversion in terms of rooted leaf-labelled trees (see [5] Corollary 1), we have: X b b (0; ) (; 1) Fn1;:::;nk =

0 1 X (b0; ) @ = (?1)jT j?jj (T )A 2n X T 2T jT j?n?kk 0 X 2n

X

j j

=

(?1)

2n T 02Tn : p(T 0 ) (?1)jT 0j?n (T 0) =

X

T 0 2Tn

0

X

; p(T 0 )

(T )

(?1)kk :

To compute the last sum, assume that there are k (possibly k = 0) non-singleton blocks of p(T 0) which are contained in some block of ; then the only partitions satisfying and p(T 0) are those obtained from p(T 0) by splitting all blocks into P singletons, except some of the k blocks mentioned above. Hence the last sum is I [k] (?1)jI j = k;0, which proves (3.3). Formula (3.5) now follows by using the bijection between leaf-labelled rooted trees and partitions established in [5], Theorem 5. To prove (3.4), we must rst nd an analogue of (3.1). We do this by recalling Theorem 5.3 in [9], which provides an expression for the Mobius type function P of a subposet P of n (see [9]) when there is a group G acting on P . We choose P to be [b0; ] [ fb1 g, and G to be the direct product of symmetric groups SB for B 2 (G acts on P in the obvious way). As in [9], we consider the poset A(P ) of all preferential arrangements (; ) of [n] with 2 P (recall that a preferential arrangement of [n] is a pair (; ), where 2 n and is a bijection from [jj] to , inducing a linear order on ). Let A(P ) := A(P )nfb1g and Ab(P ) := A(P )tfb0g. By insisting that G(b0) = fb0g, we obtain a poset action of G on Ab(P ), and hence an induced poset structure on the set of orbits Ab(P )=G. We have (b b Fn1;:::;nk b (3.6) fn1 ;:::;nk = n ! : : :n ! = ? PjG0;j 1) 1 k X bb b b = ?bAb(P )=G(b0; b1) = (0; ) (; 1) : [(;)]2A(P )=G

We can view A(P )=G as the subposet of A(n) consisting of \shues" of preferential arrangements of the sets [n1], n1 +[n2], ..., n ? nk +[nk ], whose blocks are intervals (in N) ordered in the natural way. If we do this, we can then establish a bijection between A(P )=G and the set of pairs (; c), where c is a k-coloring of [n] of type (n1; : : : ; nk ), and is a partition of [n] with monochromatic blocks which are


7

intervals (in N). Indeed, if (; ) is the shue (Bi1j1 ; Bi2j2 ; : : : ) of (B11; B12; : : : ), ..., (Bk1; Bk2; : : : ), then the associated partition is f[jBi1j1 j]; jBi1j1 j+[jBi2j2 j]; : : : g, and all the elements in the r-th block of are colored ir. Finally, we denote by Pm the set of rooted plane trees with m leaves; if we label the leaves of T 2 Pm with 1; : : : ; m, we can de ne p(T ) as before. Using (3.6) and the expression for Lagrange inversion in terms of rooted plane trees (the analogue of Corollary 1 in [5]), we have:

fnb1 ;:::;nk =

X (;)2A(P )=G

b (b0; ) b (; b1)

0 1 X = b (b0; ) @ (?1)jT j?jj (T )A T 2P (;)2A(P )=G X X jT j?n?kk 00 X

=

(?1)

(;c) T 00 2Pn : p(T 00 ) = (?1)jT 00j?n (T 00)

X

(T 00 ;c)

j j

00

X

(T )

(;c): p(T 00 )

(?1)kk :

The last sum is computed as before, and (3.4) follows. Note that for n1 = : : : = nk = 1, the theorem is just the Haiman-Schmitt form of Lagrange inversion.

Example 3.7. In order to express f1b;3, f2b;2, and f1b;1;2, we consider all the plane

trees with 4 leaves, as shown below. The trees on the rst line have type b31 ; those on the second line have type b1b2, except for the last one, whose type is b3 . The three numbers corresponding to a tree represent the number of 2-colorings of type (1; 3), of 2-colorings of type (2; 2), and of 3-colorings of type (1; 1; 2) for the leaves, which satisfy the condition in Theorem 3.2. According to the theorem, we have

f1b;3 = ?8b31+12b1b2?4b3 ; f2b;2 = ?20b31+24b1b2?6b3 ; f1b;1;2 = ?48b31+54b1b2?12b3 :

8


2, 4, 10

2, 4, 10

2, 4, 10

2, 4, 10

0, 4, 8

3, 6, 12

3, 6, 12

2, 4, 10

2, 4, 10

2, 4, 10

4, 6, 12

In [8], we gave a combinatorial proof of a formal group law identity using (3.1). Here we state and prove a similar identity. Proposition 3.8. We have that X nbn?1 = (?1)l()?1 n(l(j) j? 1)! fb ; `n where l() is the length of the partition , and jj = i1! : : : ik ! if = (1i1 : : : k ik ). Proof. Given = (1 : : : m ), we write ! for 1 ! : : :m !. The type of a partition of [n] (as a partition of n) will be denoted by (). The classical Mobius function of n will be denoted by . We have that

! X X (b0; ) (; b1) (; b1) F() (; b1 ) = 2n 2n X b b X b! X

=

(0; ) (; 1)

2n But (; 1) = (?1)jj?1(jj ? 1)!, and

b

n!=(!jj). Hence

(; 1) = (b0; b1 ) = n?1 :

the number of partitions of [n] of type is

1 X(?1)l()?1 n! F (l() ? 1)! ; 1 = nbn?1 = (n?n?1)! (n ? 1)! !jj `n

which implies the identity to be proved. Let us note that the number n(l() ? 1)!=jj is an integer. Indeed, if = i (1 1 : : : kik ), we have that (i1 + 2i2 + : : : + kik ) (i1 + : : : + ik ? 1)! = i1 + : : : + ik ? 1 i1! : : :ik ! i1 ? 1; i2; : : : ik


i1 + : : : + i k ? 1

i1 + : : : + i k ? 1

9

+2 i ; i ? 1; : : : i + : : : + k i ; i ; : : :i ? 1 : 1 2 k 1 2 k Hence, Proposition 3.8 provides another expression of nbn?1 as an integer linear combination of elements in the Lazard ring. 4. Classical Results in Algebraic Topology Restated According to a theorem of Quillen, the ring MU is isomorphic to the Lazard ring, i.e. the ring generated by the coecients of the universal formal group law. It is well-known that the Hurewicz homomorphism MU ! H(MU ) = Z[b1; b2 ; : : : ] is a monomorphism, and that the elements n := (n + 1)!bn all lie in MU. It causes no confusion to identify (H ) with H (MU ), and with the subring of MU generated by 1; 2; : : : , as long as we regard them only as rings, forgetting the coalgebra structure. The element bn in H(MU ) is usually denoted by mn, and n=n! = (n + 1)mn lies in MU, being the cobordism class of C P n , by Mischenko's theorem. It is known that MU(C P+1 ) is isomorphic to the covariant bialgebra of the universal formal group law; more precisely, it is isomorphic to the free MUmodule on generators 1MU ; 2MU ; : : : . It was shown in [12] that the Hurewicz homomorphism MU (C P+1 ) ! (H ^ MU )(C P+1 ) = (H )fxg maps nMU to b n(x). MU Let us consider the Hopf algebroid (see [10]) MU(MU ) = MU[bMU 1 ; b2 ; : : : ] = MU S, where S is the dual of the Landweber-Novikov algebra. We note rst that S is isomorphic to the Hopf algebra (H ) de ned in x2. Let L, R be the left and right units, the comultiplication, and c the conjugation of MU R MU(MU ). We write MU n for (n +1)!bn , n for R(n ), and consider the umbra R R R MU MU MU MU MU b := (1; b1 ; b2 ; : : : ), := (1; MU 1 ; 2 ; : : : ), and := (1; 1 ; 2 ; : : : ). The right unit R is usually expressed via the formula R(X ) = (MU(X )) : By Theorem 2.3 (1) this is equivalent to specifying R as follows: R : 7! R = MU or 7! R = MU ; (4.1) by taking inverses. The comultiplication can be speci ed using the notation in x2 in either of following ways: : bMU 7! bMU ~ bMU or MU 7! MU ~ MU : (4.2) Similarly, the conjugation c is speci ed by c: bMU 7! bMU or MU 7! MU : (4.3) We will present in x5 and x6 more consistent applications, demonstrating the computational advantages of this new point of view. For now, we consider a few simple applications, demonstrating the advantages in simplifying notation and

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proofs. For our applications, we need the umbra := (1; u; u2; : : : ) in K = K (u) (let us recall Z[u; u?1] and R := (1; v; v 2; : : : ) in K (K ), where v = R that the image of K(K ) in K(K ) Q consists precisely of those nite Laurent 1 ] for all m; n 2 Zn f0g). We also series f (u; v) satisfying f (mt; nt) 2 Z[t; t?1; mn need the standard map of ring spectra g : MU ! K representing the universal Thom class in K 0(MU ); the map g : MU ! K is the Todd genus, mapping n to un. Finally, let us recall the fact that K(MU ) = Z[u; u?1; bK1 ; bK2 ; : : : ], MU K and that g : MU(MU ) ! K(MU ) maps bn to bn . We will need the umbra bK := (1; bK1 ; bK2 ; : : : ) and K := (1; K1 ; K2 ; : : : ), where Kn := (n + 1)!bKn . Example 4.4. Let us rst check that the elements Rn are primitive. Indeed, we have : R = MU 7! MU ~ MU = R~ MU = 1 MU = 1 R : Example 4.5. The map f : MU(C P+1 ) ! MU(MU ) induced by the inclusion f : C P 1 ' MU (1) ,! 2MU is speci ed by nMU 7! bMU n?1 . Let us identify MU(C P+1 ) with its image in (H ^ MU ) (C P+1 ), and try to determine the map f. It is a well-known fact in umbral calculus that X b (0; ) Bn (x) : xn = 2n

Applying the map f, we obtain (4.6) f(xn) = ( MU)(n?1 ) = Rn?1 : Hence, the map f can be interpreted as umbral substitution at R. Formula (4.6) and the following commutative diagram show that the elements xn are primitive under the coaction MU(C P+1 ) ! MU(MU ) MU MU(C P+1 ).

MU(C P+1 ) f

?

- MU (MU ) MU MU(C P+1 )

I f

?

MU(MU ) - MU(MU ) MU MU(MU ) It is shown in [3] that K (MU ) is isomorphic to the direct limit

:

MU(C P+1 ) x- MU (C P+1 ) x- MU (C P+1 ) x- : : : ; where the maps are \multiplication by x". This means that there is a Z-linear map MU(C P+1 ) ! K (MU ) sending xn to un?1. We deduce that xn is not divisible (by integers) in MU(C P+1 ), whence the ring of primitive elements in MU(C P+1 ) is precisely Z[x]. This is a simpli ed proof of the result which was rst proved by D. Segal in [17], Theorem 2.1.


11

Example 4.7. The Hurewicz homomorphism h : MU ! K (MU ) can be easily determined using the following commutative diagram: MU

- MU(MU ) HH h HHH g HHj ? R

:

K(MU )

Hence, we can specify h as follows:

7! K

h: which means that

h(n) = and

h(n) =

n X

S (n + 1; k + 1)Kk un?k ;

(4.8)

(?1)k k! sK(n + 1; k + 1) uk :

(4.9)

k=0

n X k=0

7! K ;

or

Example 4.10. We would now like to give a more explicit expression (than the usual one) for the coaction K : K (MU ) ! K(K ) K K(MU ). Let us present rst a simple way of computing the image of bMU n under the map (g ^ g) : MU(MU ) ! K(K ). Since g is a map of ring spectra, we have the

commutative diagram

MU g

?

K

R

- MU(MU ) (g^g)

RK

?

- K(K )

:

Hence (g ^ g)(Rn) = vn. On the other hand, from (4.1) we obtain MU = R. By applying (g ^ g), we deduce (g ^ g) : which means that

MU 7! R ;

Bn(R) = (v ? u) : : : (v ? nu) : (g ^ g)(bMU ) = n (n + 1)! (n + 1)!

12


Let us now recall the commutative diagram

MU(MU ) g

- MU(MU ) MU MU(MU )

:

(g^g) g

?

?

- K(K ) K K(MU )

K

K(MU ) Hence, we have

K :

which means that

K 7! R ~ K ;

1 0 n+1 X X K (bKn ) = (n +k! 1)! @ (b0; ) R(; )A bKk?1 k=1 2n+1;k 11 0 0 n+1 X n+1 X X X k! @ R(; )AA bKk?1 : (b0; ) @ = k=1

(n + 1)!

i=k 2n+1;i

2n+1;k

In consequence, we have the following result: n+1 X k! K K (b ) = n

k=1 (n + 1)!

n+1 X i=k

!

s(n + 1; i) S (i; k) un+1?i vi?k bKk?1 :

(4.11)

5. The Hurewicz Homomorphism MU ! K(MU ) b of the In this section, we intend to compute the images of the coecients fn;m universal formal group law under the Hurewicz homorphism h mentioned in the of the universal Hurwitz title; these coecients are related to the coecients Fn;m = n!m!f b (see [8]). An algorithm for this computation group law by Fn;m n;m appears in [1], but no closed formula is given. All the set partitions considered in this section lie in the lattice n+m , with meet denoted by ^. According to 3.1, we have X b b = Fn;m (0; ) (; 1) ; 0

where 0 is the partition f[n]; n + [m]g of [n + m]. Using (4.8) and (4.9), we have )= h(Fn;m

=

X

; 0

X

0 ^

K(b0; ) K(; ) (; ) K(; b1)

K(b0; ) K(; ) (; 0 ^ ) (0 ^ ; ) K(; b1) :


13

If we sum only over < 0 ^ we get 0, since we have the factor (K )(; 0 ^ ) = 0. Hence X K b )= h(Fn;m (0; 0 ^ ) (0 ^ ; ) K(; b1)

=

X

(b0; ) K

X

(; b1) ujj?j j K

!

1 0min: 0f^n=;m g X K b @ X = (0 ; ) k! nk mk Kjj?k?1 uk A 0 k=0 1 0minfi;jg m n X X K X i j K i+j?k?1 uk A ; = s (n; i) sK(m; j ) @ k! 0

k k where n and m denote the number of blocks of the partition 0 contained in [n] and n +[m], respectively; the third equality follows by counting the partitions with 0 ^ = : concentrating on such? partitions ? with precisely k blocks intersecting both [n] and n + [m], there are nk mk ways of choosing the blocks of to be amalgamated, and k! ways of matching them. Finally, we have the following result: i=1 j =1

k=0

Proposition 5.1. The images of the coecients of the universal formal group b under the Hurewicz homomorphism h are speci ed by law fn;m b )= h(fn;m

n X m X i=1 j =1

1 0minfi;jg X i + j ? k bK kA bK bK @ n;i m;j k; i ? k; j ? k i+j?k?1 u : k=0

6. Congruences in MU In this section, we prove some congruences for the -Stirling numbers S (n; k) in MU2(n?k) modulo a prime p. The main tool will be the Hattori-Stong theorem and the periodicity modulo p of the classical Stirling numbers of the second kind, for which there is a nice proof using group actions (see [14]): S (n; k) S (n ? p + 1; k) (mod p) ; for n > p ? 1 : (6.1) The Hattori-Stong theorem essentially says that the Hurewicz homomorphism h : MU ! K(MU ) is integrality preserving, i.e. for all 2 MU Q we have that 2 MU if and only if (h 1)() 2 K (MU ). This turns out to be a purely algebraic statement, and such a proof can be found in [3]. Proposition 6.2. We have following congruences in MU : 0the(mod p) if n 6 0 (mod p ? 1) S (n; p ? 1) n?p+1 (mod (6.3) p) otherwise,

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in MU2(n?p+1) ;

S (n; p ? 2)

0

(mod p) if n 6 0 (mod p ? 1) n?p+2 (mod p) otherwise,

(6.4)

in MU2(n?p+2) , if p 3; 8 0 (mod p) if n 6 0; ?1; ?2 (mod p ? 1) > < (mod p) if n ?2 (mod p ? 1) S (n; p ? 3) > 3n?n?p+3 (mod p ) if (mod p ? 1) (6.5) : 31pn+3?p+2 ? n?p+3 (mod p) if nn ?0 1(mod p ? 1), in MU2(n?p+3) , if p 5. Proof. Let us compute the image of S (n; k) under the Hurewicz homomorphism h : MU ! K (MU ). By (4.8) we have X b K (0; ) (; ) h(S (n; k)) = 2n;k n

1 0 X X K b @ X K A (; ) = (0 ; ) i=k 2n;i 2n;k n X n?k K =

i=k

S (n; i)S (i; k)u

:

We will rst show that if i p and k p?1, then S K(i;Kk) 0 (mod p). Consider a partition 2 i;k with k p?1 blocks and type () = (K1 )2 : : : (Kj?1)j . If j p, then K() is divisible by p in K(MU ) since Kp?1 is. If j < p, then there are i! j 2 (2!) : : : (j !) (k ? 2 ? : : : ? j )!2! : : : j ! partitions in i;k having the same type as ; but this number is divisible by p under the above hypothesis. Now consider the image of m under h given by (4.8). Using (6.1), we deduce that 8 um (mod p) if m 0 (mod p ? 1) < m K m?1 (mod p) if m 1 (mod p ? 1) h(m) : u + 1 u um + 3K1 um?1 + K2 um?2 (mod p) if m 2 (mod p ? 1),(6.6) in K2m(MU ). According to the above remarks, and using (6.1) again, we have the following congruences, implying (6.3), (6.4), and (6.5), respectively: 0 (mod p) n 6 0 (mod p ? 1) n ? p +1 h(S (n; p?1)) S (n; p?1)u un?p+1 (mod p) ifotherwise;


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h(S (n; p ? 2)) S (n; p ? 2)un?p+2 + S (n; p ? 1)S (p ? 1; p ? 2)K1 un?p+1 8 0 (mod p) if n 6 0; ?1 (mod p ? 1) < (mod p) if n ?1 (mod p ? 1) : unn??pp+2 +2 K n ? p +1 u + 1 u (mod p) if n 0 (mod p ? 1); h(S (n; p ? 3)) S (n; p ? 3)un?p+3 + S (n; p ? 2)S (p ? 2; p ? 3)K1 un?p+2 K(p ? 1; p ? 3)un?p+1 + S ( n; p ? 1) S 8 p) if n 6 0; ?1; ?2 (mod p ? 1) > < u0 n?(mod p+3 (mod p) if n ?2 (mod p ? 1) > 3un?p+3 + 3K un?p+2 (mod p) (mod p ? 1) : 2un?p+3 + 31K1 un?p+2 + (3(K1 )2 ? K2 )un?p+1 (mod p) ifif nn ?0 1(mod p ? 1). We have used the following facts: p ? 2 p ? 1 S (p?1; p?2) = 2 1 (mod p) ; S (p?2; p?3) = 2 3 (mod p); p ? 1 p ? 1p ? 3 1 K K 3(K )2 ? K (mod p) K 2 S (p?1; p?3) = 2 2 ( ) + 2 1 2 1 3 2 in K4(MU ). Congruence (6.3) was proved in [11] using arguments related to the universal formal group law; it had an essential role therein for proving the universal von Staudt theorems. The advantage of this proof is that it also provides (6.4) and (6.5), which seem to be new. 7. Combinatorial Models for the Dual of the Steenrod Algebra In this section, we discuss some connections between the Hopf algebras and (H ), and the dual of the Steenrod algebra. Some of the results are known; nevertheless, we believe that the combinatorial proofs presented here provide new insights. Consider a prime p, and let (n) := pn?1. In [13] it is shown that the subalgebra Z=p[(1) ; (2); : : : ] of the mod p Faa di Bruno Hopf algebra Z=p is actually a sub-Hopf algebra, isomorphic to the polynomial part of the dual of the mod p Steenrod algebra; the proof is based on number-theoretical arguments. We present here an alternative proof, which is purely combinatorial and was inspired from [14]. Consider the cyclic group Cpn acting on [pn ], and hence on the partition lattice n . We want to determine the partitions xed by every element of Cpn . If is such a partition, then Cpn acts on its blocks. Let g be the cycle (1; 2; : : : ; pn), letk B be the block of containing 1, and let hgpk i be the stabilizer of B . From gp B = B , we deduce that f1; 1 + pk ; : : : ; 1 + (pn?k ? 1)pk g B . Furthermore, the sets fi; i + pk ; : : : ; i + (pn?k ? 1)pk g, for i = 1; 2; : : : ; pk , all lie in dierent blocks of , whence they are precisely the blocks of . In consequence, we have

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n + 1 partitions xed by each element of Cpn , one for each k = 0; 1; : : : ; n. The orbit of each partition which is not of the above type has pi elements, where i > 0. Finally, since all the partitions in the same orbit have the same type, (2.1) becomes ((n)) =

n X k=0

p(nk?k) (k)

in Z=p[(1); (2); : : : ] :

(7.1)

Let us now consider the Hopf algebra P := (H ) Z=p, and the ideal J generated by the elements bi, i 6= pn ? 1. Writing b(n) for bpn?1 , we clearly have an isomorphism of algebras P=J = Z=p[b(1); b(2); : : : ]. We intend to show that J is also a coideal (and hence a Hopf ideal), and that in the Hopf algebra Z=p[b(1) ; b(2); : : : ] we have

(b(n)) =

n X k=0

bp(nk?k) b(k) :

(7.2)

We will prove these statements together. Consider arbitrary integers m; k 1 such that m pk . A partition in Im;pk can be represented by a pk -tuple Pk (1; : : : ; pk ) with i 1 and pi=1 i = m. The cyclic group Cpk acts on Im;pk in the obvious way, and there is at most one partition xed by each element of Cpk , namely the one with equal block sizes, if pk divides m; the sizes of all the other orbits are non-zero powers of p. This shows that if k0 is highest power of p dividing m, then

(bm?1) 2 P J +

k0 X k=0

bpm=pk?1 b(k) ; k

whence the desired statements follow. In consequence, we have proved: Proposition 7.3. The polynomial part of the dual of the mod p Steenrod algebra is isomorphic to the sub-Hopf algebra Z=p[(1); (2); : : : ] of the mod p Faa di Bruno Hopf algebra Z=p, as well as to the quotient of (H ) Z=p (i.e. the dual of the Landweber-Novikov algebra tensored with Z=p) by the Hopf ideal J . Let R be an evenly graded commutative ring of characteristic p. Since P=J is a Hopf algebra, then the set Hom(P=J; R) is a group under convolution. We can now derive from Theorem 2.3 (2) an analogue for p-typical power series mod p. Proposition 7.4. The set of formal power series in R[[X ]] of the form X r(X ) = rpi?1X pi i1


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(i.e. p-typical power series) form a group under substitution. There is an antiisomor-phism from Hom(Z=p[b(1); b(2); : : : ]; R) to this group, speci ed by r 7! r(X ).

In consequence, Lagrange inversion for p-typical power series in R[[X ]] is equivalent to computing the antipode of Hom(Z=p[b(1); b(2); : : : ]; R), which is much easier than computing the antipode of Hom((H ); R). References [1] J. F. Adams. Stable Homotopy and Generalised Homology. Chicago Univ. Press, Chicago, 1972. [2] A. Baker. Combinatorial and arithmetic identities based on formal group laws. volume 1298 of Lecture Notes in Math., pages 17{34. Springer-Verlag, Berlin{New York, 1987. [3] F. Clarke, J. Hunton, and N. Ray. Extensions of umbral calculus II: double delta operators and Hattori-Stong theorems. Preprint, University of Manchester, 1995. [4] P. Doubilet, G.-C. Rota, and R. Stanley. On the foundations of combinatorial theory VI: the idea of generating function. In Sixth Berkeley Symposium on Mathematical Statistics and Probability, volume 2, pages 267{318. Univ. of California Press, 1972. [5] M. Haiman and W. Schmitt. Incidence algebra antipodes and Lagrange inversion in one and several variables. J. Combin. Theory Ser. A, 50:172{185, 1989. [6] M. Hazewinkel. Formal Groups and Applications. Academic Press, New York, 1978. [7] S. A. Joni and G.-C. Rota. Coalgebras and bialgebras in combinatorics. Stud. Appl. Math., 61:93{139, 1979. [8] C. Lenart and N. Ray. Hopf algebras of set systems. Preprint, University of Manchester, 1995. [9] C. Lenart and N. Ray. Polynomial invariants of partition systems and posets. Preprint, University of Manchester, 1995. [10] D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres. Academic Press, New York, 1986. [11] N. Ray. Extensions of umbral calculus; penumbral coalgebras and generalised Bernoulli numbers. Adv. Math., 61:49{100, 1986. [12] N. Ray. Symbolic calculus: a 19th century approach to MU and BP . In J. D. S. Jones and E. Rees, editors, Proceedings of 1985 Durham Symposium on Homotopy Theory, volume 117 of London Math. Soc. Lecture Note Ser., pages 195{238, Cambridge, 1987. Cambridge Univ. Press. [13] N. Ray and W. Schmitt. Combinatorics of the Novikov and Steenrod algebras. Preprint, University of Manchester, 1995. [14] B. E. Sagan. Congruences via Abelian groups. J. Number Theory, 20:210{237, 1985. [15] W. Schmitt. Antipodes and incidence coalgebras. J. Combin. Theory Ser. A, 46:264{290, 1987. [16] W. R. Schmitt. Incidence Hopf algebras. J. Pure Appl. Algebra, 96:299, 1994. [17] D. Segal. The cooperation on MU (C P 1 ) and MU (H P 1 ) and the primitive generators. J. Pure Appl. Algebra, 14:315{322, 1979. [18] M. E. Sweedler. Hopf Algebras. W. A. Benjamin, New York, 1969. [19] R. M. Switzer. Algebraic Topology { Homotopy and Homology. Springer-Verlag, Berlin{ New York, 1975.

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Mathematics Department, University of Manchester, Manchester M13 9PL, England

E-mail address :

[email protected]

Mathematics Department, University of Manchester, Manchester M13 9PL, England

E-mail address :

[email protected]