Formal Group-Theoretic Generalizations of the Necklace Algebra, Including aq-Deformation

199, 703]732 Ž1998. JA977203

JOURNAL OF ALGEBRA ARTICLE NO.

Formal Group-Theoretic Generalizations of the Necklace Algebra, Including a q-Deformation Cristian Lenart* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 Communicated by D. A. Buchsbaum Received May 31, 1997 N. Metropolis and G.-C. Rota Ž Ad¨ . Math. 50, 1983, 95]125. studied the necklace polynomials, and were lead to define the necklace algebra as a combinatorial model for the classical ring of Witt ¨ ectors Žwhich corresponds to the multiplicative formal group law X q Y y XY .. In this paper, we define and study a generalized necklace algebra, which is associated with an arbitrary formal group law F over a torsion free ring A. The map from the ring of Witt vectors associated with F to the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials. We present a combinatorial interpretation for these polynomials in terms of words on a given alphabet. The actions of the Verschiebung and Frobenius operators, as well as of the p-typification idempotent are described and interpreted combinatorially. A q-analogue and other generalizations of the cyclotomic identity are also presented. In general, the necklace algebra can only be defined over the rationalization A m Q. Nevertheless, we show that for the family of formal group laws over the integers Fq Ž X, Y . s X q Y y qXY, q g Z, we can define the corresponding necklace algebras over Z. We classify these algebras, and define isomorphic ring structures on the groups of Witt vectors and the groups of curves associated with the formal group laws Fq . The q-necklace polynomials, which turn out to be numerical polynomials in two variables, can be interpreted combinatorially in terms of so-called q-words, and they satisfy an identity generalizing a classical one. Q 1998 Academic Press

1. THE CLASSICAL NECKLACE ALGEBRA AND RING OF WITT VECTORS In w8x, Metropolis and Rota studied the properties of the so-called necklace polynomials, which are defined for every positive integer n by 1 n d M Ž x, n . [ Ý m x in Q w x x ; n d
ž /

* E-mail address: [email protected]. 703 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

704

CRISTIAN LENART

as usual, m denotes the classical Mobius function. For every positive ¨ integer m, M Ž m, n. represents the number of primiti¨ e necklaces Ži.e., asymmetric under rotation. with n colored beads, where the colors are chosen from a set of size m. Hence, M Ž x, n. are numerical polynomials Ži.e., they take integer values for integer x .. Metropolis and Rota were lead to define for every torsion free commutative ring A with identity the necklace algebra Nr Ž A. Žover A.. This algebra is the set A` of infinite sequences of elements of A with componentwise addition, and multiplication defined by

Ža?b.n[

Ý Ž i , j . a i bj ;

Ž 1.1.

w i , j xsn

here w i, j x and Ž i, j . denote, as usual, the least common multiple and greatest common divisor of i and j, respectively. Note the convention of writing a for an element Ž a 1 , a 2 , . . . . in A` ; similarly, if h is a map from a set X to A` , we write hŽ x . s Ž h1Ž x ., h 2 Ž x ., . . . .. Following w8x, we define a map M: AQ ª AQ` by MnŽ b . [ M Ž b, n., where AQ is the rationalization A m Q. The algebra Nr Ž A. has two remarkable operators for every positive integer r, namely the Verschiebung operator Vr , and the Frobenius operator f r ; the former is defined by Vr , n Ž a . [

½

ai 0

if n s ri otherwise,

Ž 1.2.

and the latter by fr , n a s

d

Ý n ad ,

Ž 1.3.

d

where the summation ranges over the set  d: w r, d x s rn4 . In this setup, the properties of the necklace polynomials proved by Metropolis and Rota can be stated as M Ž xy . s M Ž x . ? M Ž y . M Ž x r . s fr M Ž x .

in Nr Ž Q w x, y x . , in Nr Ž Q w x x . .

Ž 1.4. Ž 1.5.

The algebra Nr Ž A. is closely related to the ring of Witt ¨ ectors W Ž A. Žsee, e.g., w6, pp. 233]234x., and the ring of unital formal power series 1 q tAww t xx under cyclic sum and cyclic product Žsee w8x.. To explain this relation, we introduce the ghost ring GhŽ A., which is just A` with addition

705

THE NECKLACE ALGEBRA

and multiplication defined componentwise. We also define the maps TŽa. [

T : W Ž AQ . ª Nr Ž AQ . ,

Ý Vn M Ž a n . , nG1

wn Ž a . [

w : W Ž AQ . ª Gh Ž AQ . ,

gnŽ a . [

g : Nr Ž AQ . ª Gh Ž AQ . ,

Ý d a dn r d ,

d
Ý dad ,

d
c: Nr Ž AQ . ª 1 q tAQ w t x ,

cŽ a . [

E: Gh Ž AQ . ª 1 q tAQ w t x ,

E Ž a . [ exp

ž

Ł

nG1

ž

an

1 1 y tn

Ý nG1

an n

/

,

tn .

/

THEOREM 1.6 Žcf. w8, 3, 13x.. Ž1. All the abo¨ e maps are ring isomorphisms, and the following diagram is commutati¨ e. c

6

Nr Ž AQ .

1 q tAQ w t x

6

T

6

W Ž AQ .

Ž 1.7.

g w

E

6

6

Gh Ž AQ . Ž2. The image of W Ž A. in 1 q tAQww t xx is precisely 1 q tAww t xx. We also ha¨ e that T ŽW Ž A.. s Nr Ž A. for A s Z, but not in general. Ž3. We ha¨ e that

Ž c(T . Ž a . s

1

Ł

1 y an t n

nG1

.

The following generalization of the cyclotomic identity Ž due to V. Strehl w12x. holds:

Ł

nG1

ž

1 1 y kt n

M Ž m , n.

/

s

Ł

nG1

ž

1 1 y mt n

M Ž k , n.

/

in 1 q tZ w t x , Ž 1.8.

where k, m g Z. There is a group-theoretic generalization of the necklace algebra due to Dress and Siebeneicher. They interpreted the necklace algebra Nr ŽZ. as the Burnside]Grothendieck ring of almost finite cyclic sets in w3x. They also

706

CRISTIAN LENART

interpreted the map T in this context, and were lead to a combinatorial interpretation of the ring structure of W ŽZ.. This enabled them to generalize the ring of Witt vectors W Ž A. in w2x, by defining the Witt]Burnside ring WG Ž A. associated with a profinite group G. There is a corresponding necklace algebra associated with G. In this paper we propose a different generalization of the necklace algebra. Our starting point is the fact that there are groups of Witt vectors associated with every formal group law Žthese concepts will be discussed in the next section., the classical case corresponding to the multiplicative formal group law. Hence, it is natural to expect the existence of a necklace algebra associated with a formal group law. We give a rigorous definition of this algebra in the next section, and we also define the corresponding necklace polynomials, Verschiebung and Frobenius operators. We discuss some formulas and combinatorial interpretations concerning the action of these operators, as well as a generalized cyclotomic identity Žin Sections 2]4.. In Section 5 we discuss the special case corresponding to the formal group laws Fq Ž X, Y . s X q Y y qXY, q g Z, which provides a q-deformation of the classical necklace algebra. I am grateful to Haynes Miller and Mike Hopkins for suggestions concerning formal group laws.

2. FORMAL GROUP LAWS AND GENERALIZATIONS OF WITT VECTORS AND NECKLACE ALGEBRAS In this section, we define the necklace algebra associated with a formal group law. We shall see that the classical case corresponds to the multiplicative formal group law. We start with a brief introduction to formal group laws and Witt vectors associated with them Žcf. w4x.. A Žone-dimensional, commutative. formal group law over a ring A is a formal power series F Ž X, Y . in Aww X, Y xx with the following properties: Ž1. Ž2. Ž3.

F Ž X, 0. s F Ž0, X . s X; F Ž X, Y . s F Ž Y, X .; F Ž X, F Ž Y, Z .. s F Ž F Ž X, Y ., Z ..

A homomorphism between two formal group laws F and G over A is a formal power series f Ž X . in Aww X xx with constant term 0 satisfying f Ž F Ž X , Y . . s GŽ f Ž X . , f Ž Y . . . If f Ž X . is invertible, i.e., f 9Ž0. is a unit in A, then this power series is called an isomorphism; if f 9Ž0. s 1, it is called a strict isomorphism. If

707

THE NECKLACE ALGEBRA

F s G, a homomorphism is called an endomorphism of F, or an automorphism if it is also an isomorphism. The set of endomorphisms of F is closed under addition using F Žsee Ž2.5.. and composition, whence it forms a ring. As in the previous section, we assume that A is a torsion free commutative ring with identity. Therefore, the formal group law F has a logarithm log F Ž X . s

Ý an X n ,

a1 s 1,

a n g AQ,

nG1

that is, a strict isomorphism from F to the additive formal group law X q Y. The compositional inverse of log F Ž X . is denoted by exp F Ž X .. Since A is torsion free, a homomorphism between the formal group laws F and G over A is necessarily of the form exp G Ž b log F Ž X .. for some b g A; if b is a unit, we have an isomorphism, and if b s 1, we have a strict isomorphism. Let us introduce the notation

w b x F Ž X . [ exp F Ž b log F Ž X . . g AQ w X x

for b g AQ. Ž 2.1.

Note that if n is a positive integer, w n x F Ž X . lies in Aww X xx, and can be defined inductively by

w1x F Ž X . s X

and

w n x F Ž X . s F Ž w n y 1x F Ž X . , X . ;

furthermore, wyn x F Ž X . also lies in Aww X xx, and can be defined by F Ž w yn x F Ž X . , w n x F Ž X . . s 0. If w b x F Ž X . has coefficients in A for every b in a certain subalgebra B of A, then F is known as a formal B-module. In this case, the map b ¬ w b x F is a homomorphism from B to the ring of endomorphisms of F extending the homomorphism n ¬ w n x F for n g Z. Note that any formal group law F over a torsion free ring is a formal Z Ž r .-module Žthroughout this work, we use the standard notation Z Ž r . for the ring of integers localized at r, that is,  mrn g Q: Ž n, r . s 14 , and AŽ r . [ A m Z Ž r . .. Indeed, w n x F Ž X . ' nX modŽ X 2 ., whence w n x F Ž X . has a compositional inverse in Z Ž r .ww X xx if n is an integer prime to r; thus F has endomorphisms w1rn x F Ž X . and so, for any integer m, endomorphisms w mrnx F Ž X . s w m x F Žw1rn x F Ž X ... We now present some examples of formal group laws which will be used later. For every integer q we consider the formal group law over Z Fq Ž X , Y . [ X q Y y qXY ,

Ž 2.2.

708

CRISTIAN LENART

with logarithm log q Ž X . s

q ny 1

Ý

n

nG1

X n.

Ž 2.3.

Note that we have written log q Ž X . instead of log FqŽ X ., for simplicity; the same notational convention will be applied throughout. Also note that F0 Ž X, Y . and F1Ž X, Y . are the additive and multiplicative formal group laws, respectively. We will also refer to the formal group laws GŽ X , Y . s

X q Y y 2 XY 1 y XY

and

T Ž X, Y . s

XqY 1 q XY

Ž 2.4.

with logarithms log G Ž X . s

X 1yX

and

log T Ž X . s

X 2 nq1

Ý nG0

2n q 1

.

Note that log G Ž X . is a strict isomorphism over the integers from G to the additive formal group law, and that T gives the addition formula for the hyperbolic tangent. Moreover, it is not difficult to check that XrŽ1 q X . is a strict isomorphism Žover the integers. from T to F2 . There is a universal formal group law FU , which is defined over the so-called Lazard ring L. It is known that LQ is the Q-polynomial algebra Qw m 2 , m 3 , . . . x, where m i are the coefficients of the logarithm of FU ; furthermore, according to Lazard’s theorem, L is a polynomial algebra over the integers in infinitely many variables. We define the map w F : AQ` ª Gh Ž AQ . ,

wnF Ž a . [

Ý an r d a dn r d .

d
The group of Witt vectors W F Ž AQ. associated with F has underlying set AQ`, and is defined by insisting that w F be a group homomorphism. Let C Ž F, A. denote the group of curves in the formal group law F, that is, the group tAww t xx with addition specified by

a Ž t . qF b Ž t . [ F Ž a Ž t . , b Ž t . . .

Ž 2.5.

The third condition in the definition of a formal group law allows us to iterate the notation qF , whence it makes sense to denote by Ý F Ž . the formal sum of the indicated elements. We define the map E F : Gh Ž AQ . ª C Ž F , AQ . ,

E F Ž a . [ exp F Ž a Ž t . . ,

THE NECKLACE ALGEBRA

709

where a Ž t . [ Ý n G 1 a n t n. The map H F : W F Ž AQ. ª C Ž F, AQ. defined by H F [ E F ( w F is known as an Artin]Hasse type exponential map associated with F. It is easy to check that HFŽa. s

F

Ý

an t n .

nG1

For every positive integer r, the Verschiebung operator Vr is defined on W F Ž AQ. and on GhŽ AQ. as in Ž1.2., and on C Ž F, AQ. by Vr a Ž t . s a Ž t r . .

Ž 2.6.

The Frobenius operator f r is defined on GhŽ AQ. and C Ž F, AQ. by fr , n a s ra r n

and

f r a Ž t . s a Ž r t 1r r . qF a Ž r 2 t 1r r . qF ??? qF a Ž r r t 1r r . ,

Ž 2.7.

respectively, where r is a primitive r th root of unity Žsee w4x.. The Frobenius operator is also defined on W F Ž AQ. such that it commutes with H F . Clearly, Vr acts on W F Ž A., GhŽ A., and C Ž F, A., while f r acts on GhŽ A. and C Ž F, A..

C Ž F , AQ . 6

6

wF

HF

6

W F Ž AQ .

Ž 2.8.

EF

Gh Ž AQ .

THEOREM 2.9 Žcf. w4, 7x.. Ž1. Addition in W F Ž AQ. is defined by polynomials with coefficients in A, whence A` is a subgroup of W F Ž AQ. Ž this is the group of Witt ¨ ectors W F Ž A... Ž2. The maps w F , E F , and H F are isomorphisms of abelian groups, and H F restricts to an isomorphism between W F Ž A. and C Ž F, A.. Ž3. The Frobenius operator f r acts on W F Ž A.. The maps w F , E F , and F H commute with the actions of the operators Vr and f r . Furthermore, Vr and f r are endomorphisms of W F Ž A. and C Ž F, A.. Remarks 2.10. Ž1. There is a standard descending filtration on W F Ž A. given by the subgroups V n W F Ž A. for n G 0. Ž2. If F is the multiplicative formal group law F1 over A, then W F Ž A. coincides with the additive group of the ring W Ž A. discussed in Section 1. As pointed out in w4x, it is quite remarkable that in this case we are able to define a multiplicative structure on W F Ž A. such that n ( g F is

710

CRISTIAN LENART

a ring homomorphism, for some map n : GhŽ AQ. ª GhŽ AQ. of the form nnŽ a . s k n a n with k n g Q. In Section 5 we will prove that this actually happens for every formal group law Fq . We now define generalized necklace polynomials associated with a formal group law F; they will enable us to relate the necklace algebra associated with F to the corresponding Witt vectors in a similar way to the classical case. Let us consider the incidence algebra over AQ of the lattice DŽ n. of divisors of n Žsee, e.g., w11x.. Let z F be the element of this algebra defined by

z F Ž d1 , d 2 . [ a d 2 r d 1 , for every d1 , d 2 g DŽ n. with d1 < d 2 . Since a1 s 1, the element z F has a convolution inverse, which will be denoted by m F. It is easy to see that m1 Ž d1 , d 2 . s d1rd 2 m Ž d1 , d 2 . s d1rd 2 m Ž d 2rd1 .. We now define the polynomials M F Ž x, n . [

Ý m F Ž d, n . a d x d

d
in AQ w x x .

Clearly, M 1 Ž x, n. s M Ž x, n., M F Ž x, 1. s x, and M F Ž1, n. s 0 for n ) 1. In order to give a combinatorial interpretation for the polynomials M F Ž x, n., we recall from w8x the polynomials S Ž x, n. [ nM Ž x, n.. Let us also recall that a positive integer n is a period of the word w Žon a given alphabet., if there is a word u such that w s u < w < r n, where < w < denotes the length of w; the smallest period is called the primiti¨ e period. A word with primitive period equal to its length is called aperiodic. It is not difficult to prove, via Mobius inversion, that SŽ m, n. represents the number of aperi¨ odic words of length n on an alphabet with m letters. Necklaces can be defined as equivalence classes of words under the conjugacy relation Žthat is, w ; w9 if and only if there are words u, ¨ such that w s u¨ and w9 s ¨ u.; moreover, primitive necklaces can be defined as equivalence classes of aperiodic words. PROPOSITION 2.11. The polynomials M F Ž x, n. can be expressed in the basis  S Ž x, i .4 of the AQ-module AQw x x by the formula M F Ž x, n . s

Ýt F

d
n

ž / d

, n S Ž x, d . ,

where t F Ž i, n. [ Ý j < i m F Ž1, j . z F Ž j, n.. It turns out that this proposition is a special case of Theorem 3.3, so we postpone the proof until then. Let us note that t F Ž n, n. s 0 for n ) 1, and that t 1 Ž i, n. s 0 unless i s 1; indeed, we can pair the chains in DŽ n.

THE NECKLACE ALGEBRA

711

contributing to t 1 Ž i, n. such that each pair consists of a chain containing i, and the same chain with i removed. We now explain the combinatorial significance of the above formula in terms of a combinatorial object which we call factorized word. This is a word w Žon a given alphabet., together with an expression of the form w s ???

i1 i 2 0

ž ž Ž w . / ??? /

ik

.

Clearly, < w 0 < s < w
Ý Ž y1. k z F Ž d0 , d1 . ??? z F Ž d k , n . s Ý

d k < nrd

stF

n

m F Ž 1, d k . z F Ž d k , n .

ž / d

,n ,

where the first summation ranges over all chains which can be associated with u as above; the first equality follows from the generalization of the formula for the Mobius function of a poset. ¨ We are now able to define the necklace algebra associated with the formal group law F and relate it to the corresponding Witt vectors and curves. In general, this algebra is only defined over AQ, so we denote it by Nr F Ž AQ.. The module structure of Nr F Ž AQ. is the same as that of Nr Ž AQ.. Before discussing the multiplicative structure, let us introduce

712

CRISTIAN LENART

the maps T F : W F Ž AQ . ª Nr F Ž AQ . ,

TFŽa. [

g : Nr Ž AQ . ª Gh Ž AQ . ,

g nF

Ý Vn M F Ž a n . , nG1

F

F

Ž a . [ Ý an r d a d , d
cF Ž a . [

c F : Nr F Ž AQ . ª C Ž F , AQ . ,

F Ý w an x F Ž t n . ;

nG1

here MnF Ž b . [ M F Ž b, n., and the Verschiebung operator Vr on Nr F Ž AQ. is defined as in Ž1.2.. Note that Nr F Ž AQ. and GhŽ AQ. are filtered modules, the filtration being the same as that for W F Ž AQ.. Furthermore, the maps T F and g F are compatible with the corresponding filtrations. For every map n : GhŽ AQ. ª GhŽ AQ. of the form nnŽ a . s k n a n with k n g Q, we define a multiplication in Nr F Ž AQ. by insisting that n ( g F be a ring homomorphism. Considering the formal group law F1 and k n s n, we obtain the necklace algebra defined by Metropolis and Rota. Another important special case corresponds to the formal group law G and k n s 1. It is not difficult to check that g nG Ž a . s

Ý ad ,

d
and that multiplication is defined by

Ža?b.ns

Ý

w i , j xsn

a i bj .

Hence, Z` is a subring of Nr G ŽQ. which coincides with the aperiodic ring ApŽZ. defined in w13x. The ring structure of Nr F Ž AQ. will only be important in Section 5; until then, we regard Nr F Ž AQ. only as an abelian group, unless otherwise specified. PROPOSITION 2.13. Ž1. The maps T F , g F , and c F are isomorphisms of abelian groups, commuting with the action of the Verschiebung operator; furthermore, the following diagram is commutati¨ e. cF

C Ž F , AQ . 6

Nr F Ž AQ .

6

TF

6

gF wF

6

6

W F Ž AQ .

Gh Ž AQ .

EF

Ž 2.14.

713

THE NECKLACE ALGEBRA

Ž2. If F is a formal A-module, then the maps T F and Ž c F .y1 restrict to group isomorphisms from W F Ž A. and C Ž F, A. to A` with pointwise addition. In particular, this happens for e¨ ery formal group law o¨ er Z or Z Ž r . . Proof. Ž1. We note first that g F is invertible, and that its inverse is defined by y1

Ž g F . n Ž a . s Ý m F Ž d, n . a d .

Ž 2.15.

d
We now have

žŽ g

F y1

.

(wF

/

n

Ž a . s Ý mF Ž i , n. i
TnF Ž a . s s

Ý MF i
žÝ ž / ai ,

j
n

a j a ijr j ,

s

i

/

Ý Ý

m F j,

i < n j < nri

n

ž / i

a j a ij

Ý Ý m Ž ij, n . a j a ij s Ý Ý m F Ž k, n . a j a kj r j ; F

i < n ij < n

k
the last equality follows by setting k [ ij. Hence g F (T F s w F. To check the commutativity of the second triangle, we note that

Ž log F ( E F ( g F . Ž a . s Ž g F Ž a . . Ž t . Ž log F ( c F . Ž a . s Ý a i log F Ž t i . .

and

iG1

The coefficients of t n in both power series above are equal to Ý i < n a n r i a i , whence E F ( g F s c F. The map g F is clearly an isomorphism of abelian groups, and commutes with the Verschiebung operator. By using the commutativity of diagram Ž2.14. and Theorem 2.9, we deduce that T F and c F have the same properties as g F. Note that the inverse of T F can be found by using an algorithm similar ot the clearing algorithm in w8x. Ž2. If F is a formal A-module, then c F restricts to a monomorphism from Nr F Ž A. to C Ž F, A.. But every curve in C Ž F, A. can be written uniquely as c F Ž a . for some a in A` , so c F induces an isomorphism between Nr F Ž A. and C Ž F, A.. Indeed, a may be found inductively using the fact that for every k G 1 we have F Ý w an x F Ž t n . ' ak t k

nGk

modŽ t kq1 . .

714

CRISTIAN LENART

In order to complete the proof, it only remains to observe that T F s Ž c F .y1 ( H F . Diagram Ž2.14. for F s F1 is not exactly the same as diagram Ž1.7.. In order to explain how to relate them, we define the homomorphisms

l : Gh Ž AQ . ª Gh Ž AQ . ,

ln Ž a . [ n a n ,

i : C Ž F1 , AQ . ª 1 q tAQ w t x ,

iŽ a Ž t.. [

1 1 y a Ž t.

.

Ž 2.16.

We can easily check that w s l( w1 ,

g s l( g 1 ,

E s i ( E 1 ( ly1 .

c s i ( c1 ,

It is well known that W F Ž A. and C Ž F, A. are values of functors from formal group laws over A to groups, and that the Artin]Hasse exponential map is a natural equivalence of functors. In particular, a homomorphism f Ž X . between formal group laws induces the homomorphism a Ž t . ¬ f Ž a Ž t .. between their groups of curves. Similarly, we can view Nr F Ž AQ. as the value of a functor from formal group laws over AQ to AQ-algebras in a natural way, i.e., such that the maps c F represent a natural equivalence of functors; C Ž F, AQ. is now viewed as the ring defined by insisting that w F (Ž H F .y1 be a ring isomorphism. The homomorphism Nr Ž f .: Nr F Ž AQ. ª Nr G Ž AQ. induced by the homomorphism f Ž X . s exp G Ž b log F Ž X .., b g AQ, between the formal group laws F and G is the unique homomorphism making the following diagram commutative: Nr Ž f .

6

Nr F Ž AQ . gF

gG

Ž 2.17.

6

?b

6

6

Gh Ž AQ .

Nr G Ž AQ .

Gh Ž AQ .

Hence it is specified by

a ¬ b,

where bn s b

Ý z F Ž i , j . mG Ž j, n . a i .

Ž 2.18.

i< j
Clearly, the map Nr Ž f . is a map of filtered modules. A first result which validates our constructions is a formal grouptheoretic generalization of the cyclotomic identity; in some cases, we are able to derive from it nice explicit identities, including a q-analogue of Ž1.8..

715

THE NECKLACE ALGEBRA

PROPOSITION 2.19. The following formal group-theoretic generalization of the cyclotomic identity Ž V. Strehl’s form. holds,

Ý

F

M F Ž u, n .

F

Ž ¨t n . s

nG1

Ý

F

M F Ž ¨ , n.

F

Ž ut n .

in C Ž F , AQ . ,

nG1

Ž 2.20. where u, ¨ g AQ. In particular, for Fq we obtain

Ł

nG1

ž

M qŽ m , n.

1 1 y qkt n

/

s

Ł

nG1

ž

1 1 y qmt n

M qŽ k , n.

/

,

Ž 2.21.

and for T we obtain

g Ž t ; k , m . s g Ž t ; m, k . ,

Ž 2.22.

where k, m g Z, and

g Ž t; i, j. [

Ł nG 1 Ž 1 q it 2 ny1 .

M Ž j, 2 ny1 .

y Ł nG 1 Ž 1 y it 2 ny1 .

M Ž j, 2 ny1 .

Ł nG 1 Ž 1 q it 2 ny1 .

M Ž j, 2 ny1 .

q Ł nG 1 Ž 1 y it 2 ny1 .

M Ž j, 2 ny1 .

.

Proof. For the first part, we use the following identity which holds in Nr F Ž AQ.,

Ý

M F Ž u, n . Vn M F Ž ¨ . s

nG1

M F Ž ¨ , n . Vn M F Ž u . ;

Ý nG1

indeed, the nth term in both sequences is

Ý M F Ž u, d . M F

d
¨,

n

ž / d

.

We apply c F to this identity, using the facts c F Ž Vn M F Ž u . . s Vn c F Ž M F Ž u . . s Vn H F Ž u, 0, 0, . . . . s ut n and c F Ž ua . s

F F Ý w uan x F Ž t n . s Ý w ux F Ž w an x F Ž t n . . s w ux F Ž c F Ž a . . ;

nG1

here u g AQ and a g AQ`.

nG1

716

CRISTIAN LENART

Identity Ž2.21. follows easily from Ž2.20. by noting that 1yq

q

Xm s

žÝ / mG1

Ł Ž 1 y qXm . ,

mG1

and 1 y q Ž w y1 x q Ž X . . s Ž 1 y qX .

y1

.

Combining these identities with the general fact

wkxF Ž X . s

Ý

F

sign Ž k .

1FiF < k <

F

Ž X.

holding for any formal group law F and integer k, we can generalize Ž2.23. to q Ý w k m x q Ž X m . s Ł Ž 1 y qXm . k

m

.

mG1

mG1

This proves Ž2.21.. In order to derive Ž2.22. from Ž2.20., we proceed in a similar way, by noting that

Ý

T

Xm s

mG1

Ł m G 1Ž 1 q Xm . y Ł m G 1Ž 1 y Xm . Ł m G 1Ž 1 q Xm . q Ł m G 1Ž 1 y Xm .

,

Ž 2.24.

and

w y1x T Ž X . s yX s

Ž1 q X .

y1

y Ž1 y X .

y1

Ž1 q X .

y1

q Ž1 y X .

y1

.

Hence Ž2.24. generalizes to Ł m G 1Ž 1 q Xm .

km

y Ł m G 1Ž 1 y Xm .

km

T

mG1

Ł m G 1Ž 1 q Xm .

km

q Ł m G 1Ž 1 y Xm .

km

Ý w k M x T Ž Xm . s

,

k m g Z.

Finally, we note that M T Ž i , m. s

½

M Ž i , m. 0

if m odd otherwise,

since log T Ž X . s X q X 3r3 q X 5r5 q ??? . Let us note that Ž2.21. does not follow from the classical cyclotomic identity Ž1.8., because M q Ž x, n. does not generally coincide with

717

THE NECKLACE ALGEBRA

M Ž qx, n.rq. Indeed, M q Ž x, 1. s x, but for any prime p we have

M q Ž x, p . s M q Ž x, p 2 . s

1 p

2

žq

p 2 y1

q py 1 p

Ž x p y x. ,

2

x p y q 2 py2 x p q Ž q 2 py2 y q p

2

y1

. x/.

3. VERSCHIEBUNG AND FROBENIUS OPERATORS In the previous section, we have defined for all positive integers r the Verschiebung operator Vr , and the Frobenius operator f r on GhŽ R ., W F Ž R ., and C Ž F, R ., where R is one of the rings AQ or A. We have also defined Vr on Nr F Ž AQ. and Nr F Ž A.. We have seen that the isomorphisms in diagram Ž2.14. commute with the actions of these operators. It is natural to define f r on Nr F Ž AQ. in a compatible way with the isomorphisms mentioned above. By the results in the previous section, f r is an endomorphism of Nr F Ž AQ., and even of Nr F Ž A. if F is a formal A-module Žalthough not in general.. Let us recall the well-known identities concerning the interaction of the Verschiebung and Frobenius operators on any of the rings on which they act Žsee w4, 8, 3, 13x., Vr Vs s Vr s ,

fr fs s fr s ,

f r Vr s r Id,

Ž 3.1.

f r Vs s Ž r , s . f r rŽ r , s.VsrŽ r , s. s Ž r , s . VsrŽ r , s. f r rŽ r , s. ; these identities are most easily checked in GhŽ AQ.. In this section, we intend to express and interpret combinatorially the action of the Frobenius operator on Nr F Ž AQ.. THEOREM 3.2. The Frobenius operator f r acts on Nr F Ž AQ. as fr , n a s r Ý t F d < rn

žw

rn r, dx

,

rn d

/

ad .

718

CRISTIAN LENART

Proof. By Ž2.15., we have y1

fr , n a s Ž g F . n sr Ý

ž Ýž

d < rn

sr

d < rn

Ž fr g F Ž a . . s r Ý m F Ž i , n . Ý ar i r d a d

ž

i
Ý

m F 1,

n

ž /z ž

d < ri , i < n

i

F

n rn , i d

//a ž //a

m F Ž 1, j . z F j,

Ý

j < rnr w r , d x

d < ri

d

rn

d

d

/

sr Ý tF d < rn

žw

rn r, dx

,

rn d

/

ad .

The fourth equality follows by setting j [ nri and noting that the conditions d < rnrj and j < n are equivalent to j < rnrw r, d x. Note that if r < d and d / rn, then t F Ž rnrw r, d x, rnrd . s 0. On the other hand, according to the observations about m1 and t 1 in Section 2, we have that t 1 Ž rnrw r, d x, rnrd . s 0 unless w r, d x s rn, in which case it is equal to drrn; hence, we recover the classical formula Ž1.3. for the action of f r on Nr Ž A.. We now interpret combinatorially the action of f r on Nr F Ž AQ. by computing f r, n M F Ž m. for positive integers m, n. THEOREM 3.3. We ha¨ e that f r , n M F Ž x . s r Ý m F Ž d, n . a r d x r d .

Ž 3.4.

d
The abo¨ e polynomial can be expressed in the basis  S Ž x, i .4 of the AQmodule AQw x x by the formula fr , n M F Ž x . s r Ý t F d
ž

n d

, rn S Ž x r , d . .

/

Ž 3.5.

Proof. Formula Ž3.4. follows easily: y1

fr , n M F Ž x . s Ž g F . n

s r Ý m F Ž d, n . a r d x r d . d
y1

Ž f r w F Ž x, 0, 0, . . . . . s r Ž g F . n Ž ar x r , a2 r x 2 r , . . . .

719

THE NECKLACE ALGEBRA

Formula Ž3.5. follows by rewriting its right-hand side:

Ýt F

d
ž

n d

, rn S Ž x r , d . s

/

d
s

ž

Ý Ý

m F Ž 1, i . z F Ž i , rn .

i < nrd

j
Ý Ý

m F Ž 1, i . z F Ž i , rn . x r j

Ý mF

1,

j < n i < nrj

s

j
s

m j, d . x r j

/žÝ Ž žÝ j < d < nri

n

zF

n

m Ž j, d

/ ./

, rn x r j

ž / ž / j

j

Ý m F Ž j, n . ar j x r j s f r , n M F Ž x . . j
Let us note that f r Vs M F Ž x . can be easily computed now, by using Ž3.1.. Proposition 2.11 follows from Ž3.5. by setting r [ 1. Let us also note that t 1 Ž nrd, rn. s 0 unless d s n, in which case it is equal to 1rŽ rn.; hence, Ž3.5. implies Theorem 4 in w8, pp. 100x, namely the fact that f r, n M Ž x . s M Ž x r , n.. We now define the repetition factor of a word w to be the quotient of < w < by the primitive period of w. With this definition, we can interpret Ž3.5. as follows. COROLLARY 3.6. For all positi¨ e integers m, n, 1rr f r, n M F Ž m. in AQ enumerates by type those factorized words of length rn on an alphabet with m r letters, for which r di¨ ides the repetition factor of the root. Proof. Let us recall from the proof of Corollary 2.12 the correspondence between factorized words w s Ž . . . ŽŽ w 0i1 . i 2 . . . . . i k and pairs consisting of an aperiodic word u and a chain  1 s d 0 < d1 < ??? < d k 4 with d k dividing < w
4. THE p-TYPIFICATION IDEMPOTENT Let p be a prime and recall from w4x that a curve a Ž t . in C Ž F, A. is n called p-typical if log F Ž a Ž t .. is of the form Ý n G 0 bn t p . There is a remarkable idempotent « p on C Ž F, AŽ p. ., which is a projection onto the subgroup of p-typical curves; we will call it the p-typification idempotent. It

720

CRISTIAN LENART

is expressed in terms of Vr and f r as 1

«p s

Ý

r

Ž r , p .s1

m Ž r . Vr f r .

The p-typification idempotent has an important role in formal group theory, since the curve « p t in C Ž FU , LŽ p. . is an isomorphism over LŽ p. between the universal formal group law and the universal p-typical formal group law Žsee w4x or w10x.. We can define « p on GhŽ A. Žnot just GhŽ AŽ p. .., W F Ž AŽ p. ., and Nr F Ž AQ.. The action on GhŽ A. is very easy to describe, namely:

an 0

½

«p, n a s

if n s p k otherwise.

In order to describe the action of « p on Nr F Ž AQ., we need some additional notation. First, we denote by ¨ p Ž n. the p-valuation of n Žthat is, the largest integer k such that p k < n.. Now assume that m / p k , k ) 0, and consider the poset Dp Ž m. obtained from the lattice of divisors of m by removing all non-zero powers of p. Let m Fp denote the convolution inverse of z F in the incidence algebra Žover AQ. of this poset. We will write m Fp Ž m. for m pF Ž1, m. if m / p k , k ) 0; otherwise, we set m pF Ž m. s 0. THEOREM 4.1. The idempotent « p acts on Nr F Ž AQ. as ¨ pŽ n .

«p, n a s

m Fp

Ý ks0

n

ž / pk

apk .

Ž 4.2.

In particular, the idempotent « p acts on Nr Ž AŽ p. . by

«p, n a s

p¨ p Ž n. n

m

ž

n p

¨ p Ž n.

/

a p¨ p Ž n. .

Ž 4.3.

Proof. From g F Ž « p a . s « p g F Ž a ., it follows that

Ý an r d « p , d a s

d
½

Ý kis0 a p ky i a p i

if n s p k

0

otherwise.

Ž 4.4.

An easy induction provides « p, p k a s a p k ; hence Ž4.2. holds in this case, according to the convention m pF Ž p k . s 0 for k ) 0. We now use induction

721

THE NECKLACE ALGEBRA

once more and Ž4.4. to prove Ž4.2. for n / p k , k ) 0: ¨ pŽ d .

«p, n a s y

Ý

d < n , d/n

an r d « p , d a s y

¨ pŽ n .

s

ž ž

Ý ls0 ¨ pŽ n .

s

Ý ls0

y

p l < d < n , d/n

y

Ý

m Fp

ls0

a n r d m Fp

n

ž / pl

d < n , d/n

d

ž // pl

an r d

d

ž ž / / Ý

m Fp

ls0

pl

apl

apl n

m Fp Ž 1, i . z F i ,

Ý

ž //

i < nrp l , i/nrp l

¨ pŽ n .

s

Ý

Ý

pl

apl

apl .

Let us now compute m p Ž n. [ n m1p Ž n.. We use a formula in w11x which expresses the Mobius function of a subposet in terms of the Mobius ¨ ¨ function of the whole poset. Assuming that n / p k , k ) 0, we have

m p Ž n . s m D p Ž n. Ž 1, n . ¨ pŽ n .

s

Ý

r

Ž y1. m Ž 1, p l1 . ??? m Ž p l ry 1 , p l . m Ž p l , n .

Ý

ls0 0sl 0-l 1- ??? -l rsl ¨ pŽ n .

s

Ý ls0

m

n

ž / pl

.

Here we have used 0 if k ) 1 y1 if k s 1. Recalling the additional fact that m Ž rs . s m Ž r . m Ž s . if Ž r, s . s 1, we finally have 0 if p < n mp Ž n. s m Ž n. otherwise.

mŽ pk . s

½

½

This is clearly true for n s p k as well, whence Ž4.3. holds. Finally, we interpret combinatorially the action of « p on Nr F Ž AQ. by computing « p, n M F Ž m. for positive integers m, n. THEOREM 4.5. We ha¨ e that ¨ pŽ n .

«p, n M Ž x . s F

Ý ks0

mF Ž p k , n. ap k x p

k

in AQ w x x .

Ž 4.6.

722

CRISTIAN LENART

The abo¨ e polynomial can be expressed in the basis  S Ž x, i .4 of the AQmodule AQw x x by the formula ¨ pŽ n.yk

¨ pŽ n .

«p, n M F Ž x . s

ž

Ý ks0

m F 1,

ž

Ý is0

n p

iqk

/

/

a p iq k S Ž x, p k . .

Ž 4.7.

Proof. Formula Ž4.6. follows easily:

« p , n M F Ž x . s Ž g nF .

y1

s Ž g nF .

y1

Ž « p w F Ž x, 0, 0, . . . . .

ž x, 0, . . . , 0, a

2

p

x p , 0, . . . , 0, a p 2 x p ,0, . . .

/

¨ pŽ n .

s

k

Ý

mF Ž p k , n. ap k x p .

ks0

Formula Ž4.7. follows by noting that its right-hand side can be written as ¨ pŽ n .

Ý is0

m

F

ž / 1,

p

¨ pŽ n.yk

¨ pŽ n .

n

i

api x q

Ý ks1

ž

Ý is0

m F 1,

ž

n p

iqk

/

a p iq k

/

k

Ž xp yxp

ky 1

..

k

Hence the coefficient of x p Žwith 0 F k F ¨ p Ž n.. in the right-hand side is equal to ¨ pŽ n.yk

Ý is0

m

F

ž

1,

n p

iqk

¨ pŽ n.yky1

/

a p iq k y

Ý is0

m F 1,

ž

n p

iqkq1

/

a p iq kq 1

s mF Ž p k , n. ap k ; here the second sum does not appear if k s ¨ p Ž n.. We can interpret Ž4.7. combinatorially as follows. COROLLARY 4.8. For all positi¨ e integers m, n, « p, n M F Ž m. in AQ enumerates by type those factorized words of length n on an alphabet with m letters for which the root length is a power of p. Proof. We recall once again from the proof of Corollary 2.12 the correspondence between factorized words w s Ž . . . ŽŽ w 0i1 . i 2 . ??? . i l and pairs consisting of an aperiodic word u and a chain  1 s d 0 < d1 < ??? < d l 4 with d l dividing < w
THE NECKLACE ALGEBRA

723

5. A q-DEFORMATION OF THE CLASSICAL NECKLACE ALGEBRA In this section we study the necklace algebras Nr q ŽQ. corresponding to the family of formal group laws Fq , q g Z, over Z defined in Ž2.2.. Recall that the multiplicative structure of Nr q ŽQ. depends on the choice of a map n : GhŽQ. ª GhŽQ. of the form nnŽ a . s k n a n with k n g Q; more precisely, this structure is defined by insisting that n ( g q be an algebra map. We choose n to be the map l defined in Ž2.16.. We have already seen that the classical ring of Witt vectors and the necklace algebra of Metropolis and Rota correspond to q s 1 Žin other words, to the multiplicative formal group law.. In order to simplify notation, we set ˜ g q [ n ( g q and ˜ t q Ž d, n. [ qŽ nt d, n.. Theorem 5.8 represents the main result of this section; it states that Nr q ŽZ. is a subalgebra of Nr q ŽQ., and it determines its structure, thus generalizing the classical necklace algebra construction Žwhich can be recovered for q s 1.. The proof of this theorem is based on the following two lemmas. LEMMA 5.1. The polynomials nrd t q Ž d, n. in Qw q x are numerical polynomials for all positi¨ e integers d, n with d < n. Proof. We fix q in Z, and divide the proof into two steps. Step 1. We first show that p k divides ˜ t q Ž p k , np k . if p is a prime and n a positive integer. If q ' 0 Žmod p ., then this is clearly true; indeed, l l l every term of ˜ t q Ž p k , np k . is of the form "q p 1y1 ??? q p sy1 q n p sq 1y1 with m p ly1 l Ž l l 1 q ??? ql sq1 s k, and q is divisible by p since p y l y 1 G 0.. Ž . If q k 0 mod p , we use induction on k, which obviously starts at 0. Partitioning the terms in ˜ t q Ž p k , np k . according to the smallest element different from 1 in the chains from 1 to np k in DŽ np k . corresponding to them, we obtain ky1

˜t q Ž p k , np k . s q n p y1 y q p y1q ny1 y Ý q p y1 ˜t q Ž p kyi , p kyi n . . Ž 5.2. k

k

i

is1

l

ly 1

We know that p l divides q m p y q m p for all positive integers l, m, since l ly 1 M Ž x, p l . is a numerical polynomial; hence p l divides q m p y1 y q m p y1 , by the assumption on q. Applying this fact and induction to Ž5.2., we

724

CRISTIAN LENART

deduce that ˜ t q Ž p k , np k . is congruent modulo p k to qnp

ky 1

y1

y qp

ky1

y

Ý qp

y1

q ny 1 y ˜ t q Ž p ky 1 , np ky 1 .

y1

˜t q Ž p kyi , np kyi . .

ky 1

iy 1

is2

But this is equal to 0, by using Ž5.2. once again. Step 2. We prove that d divides ˜ t q Ž d, n. for every d < n. Given a k prime p dividing d, we have d s mp and n s mrp k for some positive integers k, m, r with Ž p, m. s 1. We now use the identity

t˜ q Ž mp k , mrp k . s

Ý Ž y1. sy1˜t q Ž p i , j1 p i . ??? ˜t q Ž p i , js p i . , Ž 5.3. 1

s

1

s

where the summation ranges over all i t G 0, jt ) 1 with i1 q i 2 q ??? qi s s k, j1 j2 ??? j s s mr, and r < j s . According to the previous step, we have p k
For e¨ ery q / 0, we ha¨ e that

ei ? e j s

j q

Vw i , j x

Ýt q

ž ž d
n d

,

w i, jx i

n S Ž q w i , j xr j , d .

/

/

, nG 1

where e r, s s dr, s . Furthermore, e i ? e j is a sequence of integers di¨ isible by Ž i, j .. Proof. We base our computation of e i ? e j on the formula ei ? e j s Ž ˜ gq.

y1

Ž ˜g q Ž ei . ? ˜g q Ž e j . . .

Ž 5.5.

In order to make the map ˜ g q : Nr q ŽQ. ª GhŽQ. commute with the Verschiebung operator, we have to consider the operator VrX [ rVr . We have

˜g q Ž e i . ? ˜g q Ž e j . s Ž ViX ˜g q Ž e1 . . ? Ž VjX ˜g q Ž e1 . . ,

725

THE NECKLACE ALGEBRA

and ˜ g nq Ž e1 . s q ny 1, whence

˜g q Ž e i . ? ˜g q Ž e j . s ijVw i , j xp s Ž i , j . VwXi , j xp , where pn [ q k ny1q l ny1 , and k [ w i, j xri, l [ w i, j xrj. By Ž5.5., we have e i ? e j s Ž i , j . Vw i , j x Ž ˜ gq.

y1

Žp . .

Ž 5.6.

On the other hand, by Ž2.15. and Ž3.4. we have y1 gq n

Ž˜ .

Ž p . s Ý m Ž d, n . q

d
pd

s k Ý m Ž d, n . q

d

d
s fk , n M q Ž x . ,

q k dy1 kd

q l dy1

x k r ' q l ry1 ;

the ‘‘umbral notation’’ x k r ' q l ry1 means that x k r is replaced by q l ry1 after collecting powers of x in f k, n M q Ž x .. Furthermore, combining this result with Ž3.5., we have y1

Ž ˜g q . n Ž p . s k Ý t q d
s

n

ž

k

Ýt q q d
d

ž

, kn S Ž x k , d .

/

n d

, kn S Ž q l , d . ,

/

x k r ' q l ry1 .

The formula for e i ? e j now follows by combining this result with Ž5.6.. To complete the proof, it is enough to show that NŽ d. [ s

nj

Ž i, j. q 1 q

˜t q

ž

n d

tq ,

ž

n d

,

w i, jx i

w i, jx i

n S Ž q w i , j xr j , d .

/

n S Ž q w i , j xr j , d .

/

is divisible by n for every divisor d of n. We know that n divides qN Ž d . by Lemma 5.1, and the fact that M Ž x, d . s SŽ x, d .rd is a numerical polynomial. Assuming that p r is an arbitrary power of a prime p dividing n, we will now show that p r divides N Ž d .. If q is prime to p, then this is true by the previous remark. Otherwise, we use the fact that q divides SŽ q w i, j xr j, d ., and that p r divides every term of ˜ t q Ž nrd, w i, j x nri . Žessentially because p s m p sy1 divides q for every m, s ) 0.. THEOREM 5.8. Ž1. The map T q induces a group isomorphism between W q ŽZ. and Nr q ŽZ., while the map c q induces a group isomorphism between

726

CRISTIAN LENART

Nr q ŽZ. and C Ž Fq , Z.. In particular, M q Ž x, n. is a numerical polynomial in q and x. Furthermore, we ha¨ e the following expression for the polynomials M q Ž x, n. as a sum of products of numerical polynomials in q and x: M q Ž x, n . s

Ý

d
dt q

n

ž ž // ,n

d

S Ž x, d . d

.

Ž2. We ha¨ e

Ža?b.ns

Ý Ž i , j . Pn , i , j Ž q . a i b j ,

w i , j x< n

where Pn, i, j Ž q ., w i, j x< n, are numerical polynomials in Qw q x gi¨ en by Pn , i , j Ž q . s

j

Ž i, j. q

Ý

tq

d < nr w i , j x

žw

n i, jxd

,

n i

/

S Ž q w i , j xr j , d . .

Hence, there is a Z-algebra structure on Nr q ŽZ.. Ž3. The Verschiebung and Frobenius operators are endomorphisms of the algebra Nr q ŽZ.; the latter acts on Nr q ŽZ. by fr , n a s r Ý t q d < rn

žw

rn r, dx

,

rn d

/

ad .

Proof. Ž1. The fact that T q and c q restrict to group isomorphisms is a special case of Proposition 2.13 Ž2.. The formula for the polynomials M q Ž x, n. follows from Proposition 2.11. The fact that dt q Ž nrd, n. is a numerical polynomial follows from Lemma 5.1. Ž2. This is immediate from Lemma 5.4. Ž3. The Frobenius operator acts on Nr q ŽZ. since it acts on C Ž Fq , Z. and it commutes with the isomorphism c q between these groups. On the other hand, Vr and f r are endomorphisms of GhŽZ. with pointwise addition and multiplication, and they commute with the map ˜ g q ; hence qŽ . they are endomorphisms of Nr Z . The formula for the action follows from Theorem 3.2. Theorem 5.8 shows the existence of necklace algebras Nr q ŽZ. for all q g Z, and completely determines their structure. We have already seen that M q Ž x, n. specialize to the classical necklace polynomials, and that the action of the Frobenius operator specializes to Ž1.3. for q s 1. On the other hand, it is easy to check that the product formula specializes to Ž1.1.. We can now use the maps T q and c q to define multiplicative structures on W q ŽZ. and C Ž Fq , Z.. Furthermore, we can substitute Z with any torsion

THE NECKLACE ALGEBRA

727

free commutative ring with identity. We have already seen in Section 2 that any formal group law over a torsion free ring is a formal Z Ž r .-module; hence, by Theorem 2.13 Ž2., W q ŽZ Ž r . ., Nr q ŽZ Ž r . ., and C Ž Fq , Z Ž r . . are isomorphic rings via the restrictions of the maps T q and c q . On the other hand, we can view q as an indeterminate, and define the corresponding structures appropriately. Hence we have the following immediate corollary of Theorem 5.8, which is essentially a generalization of the classical Theorem 1.6. COROLLARY 5.9. Ž1. There is a ring structure on W q ŽZ. such that the restriction of l ( w q is a ring homomorphism. The map H q pro¨ ides an isomorphic ring structure on C Ž Fq , Z.. Ž2. Let A be a torsion free commutati¨ e ring with identity. There are ring structures on Nr q Ž A., W q Ž A., and C Ž Fq , A. such that the restrictions of l ( g q and l ( w q are ring homomorphisms, and the restriction of H q is a ring isomorphism. The corresponding Frobenius and Verschiebung operators are endomorphisms of these rings. Ž3. If A is a Q-algebra or A s Z or A s Z Ž r . , there are restrictions of the maps T q and c q to the rings Nr q Ž A., W q Ž A., and C Ž Fq , A., and they are isomorphisms. Furthermore, the Verschiebung and Frobenius operators commute with the maps T q and c q. Ž4. The statements in Ž2. also hold if q is ¨ iewed as an indeterminate, and A is an algebra o¨ er the ring of numerical polynomials in q Ž in particular, if it coincides with this ring .. Alternati¨ e Proof of Ž1. Ždue to M. Hopkins w5x.. We can assume that q / 0. Consider the polynomial ring A s Zw a 1 , a 2 , . . . ; b 1 , b 2 , . . . x and the elements a , b in W q Ž AQ.. The nth component gn of the product g [ a ? b is a polynomial in AQ. We wish to investigate when it has integer coefficients, which depends only on the strict isomorphism class of Fq . Recall formula Ž2.3. for the logarithm of Fq . On the other hand, we shall see below that the unique strict isomorphism from the multiplicative formal group law to Fq is given by 1 y Ž1 y x. q

q

.

Ž 5.10.

Over Zw1rq x, formula Ž5.10. tells us that Fq is strictly isomorphic to the multiplicative formal group law. Hence gn has its coefficients in Zw1rq x. Over the localization Z Ž q. of Z at q Žsee the definition at the beginning of Section 2., formula Ž2.3. tells us that Fq is isomorphic to the additive formal group law; indeed, if a prime p divides q, and p k is the highest

728

CRISTIAN LENART

power of p dividing n, then p k also divides q ny 1 since k F n y 1. This implies that gn has its coefficients in Z Ž q. . Combining the information above, we can see that gn has integer coefficients. Remark 5.11. If we are interested just in the existence of ring structures on W q ŽZ. and C Ž Fq , Z., then the argument of M. Hopkins suffices. As in many other situations, combinatorics is useful when one wants some explicit formulas. The necklace algebra Nr q ŽZ. is an isomorphic structure to W q ŽZ. and C Ž Fq , Z. where such formulas can be found. Furthermore, we shall see below that Nr q ŽZ. is not just a purely algebraic construction, but it has a nice combinatorial interpretation as well. Our next task is to classify the algebras Nr q ŽZ. up to isomorphism of filtered algebras. Closely related to this is the classification of the formal group laws Fq , q g Z, up to strict isomorphism over the integers. THEOREM 5.12. The formal group laws Fq are classified up to strict isomorphism o¨ er the integers by the set of prime di¨ isors of q Ž we adopt the con¨ ention that 0 has an empty set of di¨ isors .. Furthermore, Nr q ŽZ. are classified up to isomorphism of filtered algebras in the same way. Proof. Clearly, the additive formal group law F0 is the only one in its strict isomorphism class, so we may assume q / 0. We have log q Ž X . s y

log Ž 1 y qX . q

,

exp q Ž X . s

1 y exp Ž yqX . q

.

Hence the unique strict isomorphism over Q from Fq to Fr is given by

fŽ X. [

1 y Ž 1 y qX .

sXq

rrq

r

Ý Ž y1. ny 1 nG2

Ž r y q . ??? Ž r y Ž n y 1 . q . n!

X n.

Let us assume first that there is a prime p dividing either r or q Žbut not both.. Then the coefficient of X p in f Ž X . is not an integer, because its numerator is not divisible by p. Now assume that the sets of prime divisors of q and r are the same, choose an arbitrary prime p, and consider the coefficient c n of X n in f Ž X .. If p divides both q and r, then the highest power of p dividing the numerator of c n is at least n y 1. On the other hand, the highest power of p dividing n! is N Ž n, p . [ Ý`ks 1 ? nrp k @, where ? x @ is the greatest integer less than or equal to x. But N Ž n, p . -

THE NECKLACE ALGEBRA

729

nrŽ p y 1. F n. Finally, consider a prime p dividing neither q nor r. We note that f Ž X . s qrr w rrq x q Ž X . and rrq, qrr g Z Ž p. , whence f Ž X . has its coefficients in Z Ž p. Žrecall that Fq is a formal Z Ž p.-module.. If Fq and Fr are isomorphic, then the corresponding necklace algebras will be isomorphic as filtered algebras, by the functoriality of the necklace algebra construction Žcf. Section 2.. It remains to show that if q and r have a different set of prime divisors, then Nr q ŽZ. and Nr r ŽZ. are not isomorphic as filtered algebras. Let us assume that there is such an isomorphism h. Recall the notation e k for the vector defined by e k, l s d k, l . Consider the infinite matrix whose rows are the vectors hŽ e k ., for k s 1, 2, . . . . Since h is an isomorphism of filtered algebras, this matrix is upper triangular with "1 on the diagonal. Hence h extends to a filtered isomorphism between Nr q ŽQ. and Nr r ŽQ., and it induces a filtered automorphism ˜ h of GhŽQ.. Now consider the infinite upper triangular matrix with rows ˜ hŽ e k .. Since ˜ hŽ e k . ? ˜ hŽ e k . s ˜ hŽ e k ., this is a 0, 1-matrix, with 1’s on the diagonal. On the other hand, each column contains at most one entry equal to 1; indeed, if ˜ hŽ e k . and ˜ hŽ e l . contain a 1 in the same ˜ ˜ column Ž k / l ., then hŽ e k q e l . ? hŽ e k q e l . / ˜ hŽ e k q e l ., which is a contradiction to ˜ h being a ring homomorphism. So our matrix is the identity matrix, which means that ˜ h is the identity map, and h is the map in Ž2.18. with b s 1, F s Fq , G s Fr . Now consider a prime p dividing q and not dividing r Žor vice versa.. According to Ž2.18., we have h p Ž e1 . s

q py 1 p

y

r py 1 p

,

which is clearly not an integer. This shows that there is no isomorphism of filtered algebras between Nr q ŽZ. and Nr r ŽZ. if q and r have a different set of prime divisors. We conclude with a combinatorial interpretation of the q-necklace polynomials in terms of so-called q-words. Consider an alphabet G, and the free monoid ŽZrŽ q . = G .* generated by ZrŽ q . = G. We define an action of the group ZrŽ q . on ŽZrŽ q . = G .* by letting the generator of ZrŽ q . act as

Ž Ž r1 , a1 . , . . . , Ž rn , an . . ¬ Ž Ž r1 q 1, a1 . , . . . , Ž rn q 1, an . . . We call q-words the orbits of this action. Note that the corresponding equivalence relation is not a congruence, so we cannot define the concatenation of q-words in the usual way. We concentrate instead on defining the periods of a q-word. Let w w x denote the q-word with representative w. We call a positive integer k a period if there is an element w 0 in ŽZrŽ q . = G .* of length k and ri in ZrŽ q . for 1 F i F < w
730

CRISTIAN LENART

It is easy to see that a multiple of a period is not necessarily a period, but the greatest common divisor of two periods is also a period. Hence the periods of w w x still form a sublattice of the lattice of divisors of the length of w, and we can define the primitive period as the zero element of this sublattice. Aperiodic words are then defined as usual. Clearly, for q s 1 one obtains the classical objects. We can visualize the above construction in the following way. Consider a cylinder placed horizontally and n circles on its surface Žthe distances between circles are considered irrelevant .. Then consider q equidistant parallel lines on the surface of the cylinder which intersect each circle in q points. A q-word of length n with letters from G is obtained by attaching a letter to some point Žexactly one. on every circle. Concatenation of q-words is defined in the obvious way, by placing two cylinders Žassumed to be of the same radius. next to each other, such that the parallel lines coincide in pairs, and the right base of the first cylinder coincides with the left base of the second cylinder. Clearly, there are q ways to concatenate two q-words. The periods of a q-word can now be defined in the usual way, namely k is a period of w w x if there is a way to concatenate < w
Ý q n r dy1S q Ž x, d . s q ny 1 x n .

Ž 5.13.

d
Indeed, we have that

Ž l ( g q ( M q . n Ž x . s Ž l ( w q . n Ž x, 0, 0, . . . . s q ny 1 x n . We obtain the following combinatorial interpretation for the q-necklace polynomials. PROPOSITION 5.14. For all positi¨ e integers m, n, S q Ž m, n. represents the number of aperiodic q-words of length n with letters chosen from an alphabet of size m. Proof. We prove the statement concerning S q Ž m, n. by induction on n, which starts at n s 1. Clearly, there are q ny 1 m n q-words of length n with letters chosen from an alphabet of size m. Given an aperiodic q-word of length d dividing n, we can form q n r dy1 q-words of length n by considering all possible ways to concatenate nrd copies of the primitive q-word. Since all q-words of length n arise in this way Ž d being their primitive period., the combinatorial interpretation of S q Ž m, n. follows by induction from Ž5.13..

731

THE NECKLACE ALGEBRA

Unfortunately, for generic q, the obvious actions Žby circular permutations. of ZrŽ n. on q-words of length n fail to partition the primitive q-words into orbits of size n. Thus, although we know that S q Ž m, n. is divisible by n, we are not able to give a combinatorial interpretation of this fact in terms of some objects we would call q-necklaces. It seems quite challenging to find such a combinatorial interpretation which would specialize to the classical one for q s 1. We now derive a q-analogue of the classical identity Ž1.4.. PROPOSITION 5.15. We ha¨ e M q Ž qxy . s qM q Ž x . ? M q Ž y .

in Nr q Ž Q w x, y x . .

Proof. We show that the identity holds after applying the map l ( g q. We use the fact that l ( g q (T q s l ( w q, and the fact that l ( g q is an algebra map from Nr q ŽZ. to GhŽQ.. Hence we have

Ž l ( g q . Ž M q Ž qxy . . s Ž q 2 ny1 x n y n . nG 1 , and

Ž l ( g q . Ž qM q Ž x . ? M q Ž y . . s q Ž l ( g q . Ž M q Ž x . . ? Ž l ( g q . Ž M q Ž y . . s Ž q 2 ny1 x n y n . nG 1 . As far as formula Ž1.5. is concerned, we have already presented a generalization of it in Theorem 3.3, which works for any formal group law. We present here a conjecture, which attempts to provide a different generalization of Ž1.5.. Conjecture 5.16. We ha¨ e that fr , n M q Ž x . s

Ý Qr , n , d Ž q . M q Ž x r , d .

d
in Q w x, q x ,

where Q r, n, d Ž q . in Qw q x are numerical polynomials. REFERENCES 1. J. F. Adams, ‘‘Stable Homotopy and Generalized Homology,’’ Chicago Univ. Press, Chicago, 1972. 2. A. W. M. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vector construction, Ad¨ . Math. 70 Ž1988., 87]132. 3. A. W. M. Dress and C. Siebeneicher, The Burnside ring of the infinite cyclic group and its relation to the necklace algebra, l-rings, and the universal ring of Witt vectors, Ad¨ . Math. 78 Ž1989., 1]41.

732 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

CRISTIAN LENART

M. Hazewinkel, ‘‘Formal Groups and Applications,’’ Academic Press, New York, 1978. M. Hopkins, personal communication, October 1996. S. Lang, ‘‘Algebra,’’ Addison]Wesley, Reading, MA, 1965. C. Lenart, Symmetric functions, formal group laws and Lazard’s theorem, Ad¨ . Math., in press. N. Metropolis and G.-C. Rota, Witt vectors and the algebra of necklaces, Ad¨ . Math. 50 Ž1983., 95]125. J. W. Milnor and J. D. Stasheff, ‘‘Characteristic Classes,’’ Ann. of Math. Stud., Vol. 76, Princeton Univ. Press, Princeton, NJ, 1974. D. C. Ravenel, ‘‘Complex Cobordism and Stable Homotopy Groups of Spheres,’’ Academic Press, New York, 1986. R. P. Stanley, ‘‘Enumerative Combinatorics,’’ Wadsworth & BrooksrCole, Monterey, CA, 1986. V. Strehl, Cycle counting for isomorphism types of endofunctions, Bayreuth. Math. Schr. 40 Ž1992., 153]167. K. Varadarajan and K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, Ad¨ . Math. 81 Ž1990., 1]29.