Periodic λ-rings

A.A.Davydov

April 21, 1997

Department of mathematics and mechanics, Moscow state University,

Moscow, Russia. e-mail: davydov@mech.math.msu.su

Abstract λ-ring is called n-periodic if its Adams operators satisfies to the following property: ψ^{i+n} = ψ^{i} for any i. The free periodic λ-ring generated by one object is described. From this describtion one can deduce that the order of finite group devides the period in the power of dimension of faithful irreducible representation.

Contents

1 Introduction 1

2 λ-rings 2

3 Free periodic λ-rings 3

4 Representations rings of finite groups 7

1 Introduction

Classical Lagrange theorem states that order of any element of finite group devides the order of the group. In particular the period of the gruop ( greatest common devisor of orders of elements) is a devisor of the order of group. Sylov’s theorems implices that any simple devisor of the order of finite group is the order of some element. Hence the order of finite group devides some power of the period.

It is shown in the present article that this power is equal to the dimension of faithful complex representation (theorem 2) or the dimension of faithful complex representation mines one (theorem 3) if the determinant (maximal external power) of this representation is trivial.

Theorems 2,3 follows from the comporision of coefficients of minor parts in Laurent’s expansions of generating series for the invariants of symmetric powers of faithful representation. The first is Molin’s formula [6], the second is provided by the investigation of properties of λ-structure of the representations ring of finite group.

The notion of λ-ring was introduced by A.Grothendieck as a tool for formulating of general Riemann-Roch theorem [3]. It is convinient to describe the λ-structure of representations ring in terms of some operations, which were introduced by J.Adams [1, 2]. Adams operation on the representations ring satisfies to some periodicity conditions, which period equals the period of group [4]. This condition is used in the article for defining of the notion of periodic λ-ring. The describtion of free semisimple periodic λ-ring (theorem 1) provides the second expression of generating series for the invariants of symmetric powers.

The first section of the article containes general definitions and facts of theory of λ-rins.

The second section is devoted to periodic λ-rins.

The representations rings of finite groups are considered in the third section.

The author was partially supported by RFFI grant 93-01-00149.

2 λ-rings

Let A be a commutative ring without torsion. The set of maps λ^{i} : A → A i ≥ 0, satisfying to the conditions

λ^{0}(x) = 1 λ^{1}(x) = x λ^{i}(x + y) = ∑_{s+t=i} λ^{s}(x)λ^{t}(y) for any x, y ∈ A

(1) is called λ-operations or external powers. It is convinient to write the conditions (1) in terms of generating series λ_{t}(x) = ∑_{i≥0} λ^{i}(x)

λ_{t}(x + y) = λ_{t}(x)λ_{t}(y) for any x, y ∈ A.

Using generating series of external powers we can define another operations:

σ-operations or symmetric powers σ^{i} : A → A i ≥ 0, with generating series

σ_{t}(x) = λ_{−t}(x)^{−1}

and Adams operations ψ^{i} : A → A i ≥ 1, with generating series

ψ_{−t}(x) = −t\frac{λ′_{t}(x)}{λ_{t}(x)}.

The condition (1) implices the following equations:

σ_{t}(x + y) = σ_{t}(x)σ_{t}(y) ψ_{t}(x + y) = ψ_{t}(x) + ψ_{t}(y) for any x, y ∈ A.

The ring A with the set of λ-operations is called λ-ring if corresponding Adams operations are ring homomorphisms.

An element x of λ-ring is finite, if the generating series λ_{t}(x) is a polynomial. The degree of this polynomial is called rank of x. Let us note that λ_{t}(x) = 1 + xt for any element x with rank one. The (semi)group ring Z[G] can be equipped with natural λ-ring structure by setting λ_{t}(x) = 1 + xt for any element x from G. This λ-ring will be called (semi)group.

Free λ-ring Λ(d), generated by finite element of rank d, is isomorphic to the ring of polynomials Z[x, λ^{2}(x), ..., λ^{d}(x)] of external powers of x or to the λ-ring of symmetric polynomials Z[x_{1}, ..., x_{d}]^{S_{d}}, where the variables have rank one [5]. The second isomophism sends λ^{i}(x) to the i-th elementary symmetric function of x_{1}, ..., x_{d}. In particular the maximal external power λ^{d}(x) corresponds to x_{1}...x_{d}.

3 Free periodic λ-rings

Element x of the λ-ring R is n-periodic if

ψ^{i+n}(x) = ψ^{i}(x) for any i ∈ Z.

λ-ring R is n-periodic if all elements of R are n-periodic.

Free n-periodic λ-ring Λ(d, n) generated by finite element x of rank d is a factor of λ-ring Λ(d) = Z[x, λ^{2}(x), ..., λ^{d}(x)] by the ideal, generated by (ψ^{n+i}(x) − ψ^{i}(x)). The next theorem is devoted to the describtion of semisimple free n-periodic λ-ring.

Theorem 1 The natural homomorphism

Λ(d, n)/rad(Λ(d, n)) → Z[x_{1}, ..., x_{d}, x_{i}^{n} = 1]

from factor by (Jackobson) radical of the free n-periodic semisimple λ-ring Λ(d, n) to group λ-ring Z[x_{1}, ..., x_{d}, x_{i}^{n} = 1], which sends x to ∑_{i=0}^{d} x_{i}, is an isomorphism with the subring of S_{d}-invariant elements where symmetric group S_{d} acts by permutation of generators x_{i}.

Proof:

The natural homomorphism

Λ(d, n)/rad(Λ(d, n)) → Z[x_{1}, ..., x_{d}, x_{i}^{n} = 1],  (2)

which sends x to ∑_{i=0}^{d} x_{i}, can be extended to the natural homomorphism

Λ(d) → Z[x_{1}, ..., x_{d}, x_{i}^{n} = 1]

from free λ-ring Λ(d) generated by one object x of dimension d. λ-ring Λ(d) is isomorphic to the subring of S_{d}-invariant elements of semigroup λ-ring Z[x_{1}, ..., x_{d}]. The homomorphism (2) sends the additive generators of Z[x_{1}, ..., x_{d}]^{S_{d}}

t(a_{1}, ..., a_{d}) = \frac{1}{|St_{S_{d}}(a_{1}, ..., a_{d})|} ∑_{σ∈S_{d}} x_{1}^{a_{σ(1)}}...x_{d}^{a_{σ(d)}}

to additive generators of Z[x_{1}, ..., x_{d}, x_{i}^{n} = 1]^{S_{d}}. Hence the homomorphism (2) is surjective.

Now let us prove the injectivity of the homomorphism (2). It will follows from injectivity of homomorphism

Λ(n, d)/rad(Λ(n, d)) → Λ(n, d)/rad(Λ(n, d)) ⊗ C,

semisimplicity of Λ(n, d)/rad(Λ(n, d)) ⊗ C and the fact that any homomorphism of algebras

Λ(d, n)/rad(Λ(n, d)) ⊗ C → C  (3)

can be written as a composition of the homomorphism (2) and some homomorphism

C[x_{1}, ..., x_{d}, x_{i}^{n} = 1]^{S_{d}} → C.

Firstly we will describe the homomorphisms of the form (3). For x from Λ(d, n) the generating series for Adams operations have the following form

ψ_{t}(x) = ∑_{i≥1} ψ^{i}(x)t^{i} = ∑_{i=1}^{n} ψ^{i}(x)t^{i} ∑_{j≥0} t^{j}n = \frac{∑_{i=1}^{n} ψ^{i}(x)t^{i}}{1 − t^{n}} .

Lemma 1 Let A is a commutative algebra over C. Then for any f(t) ∈ A[t] with deg f(t) < d the diffirential equation

\frac{l′(t)}{l(t)} = \frac{f(−t)}{1 − (−t)^{n}} , l(0) = 1

has unique solution

l(t) = ∏_{ε:ε^{n}=1}(1 + εt)^{\frac{1}{n} f(ε^{−1})}.

Proof of the lemma: Using the directly verified equality

\frac{t^{i}}{1 − t^{n}} = \frac{1}{n} ∑_{ε:ε^{n}=1} \frac{ε^{1−i}}{1 − εt}

one can decompose the right side of differential equation into the sum of simple rational fractions

\frac{f(t)}{1 − t^{n}} = \frac{1}{n} ∑_{ε:ε^{n}=1} \frac{εf(ε^{−1})}{1 − εt}.

The lemma follows from the additivity of left side of equation

\frac{(l_{1}(t)l_{2}(t))′}{l_{1}(t)l_{2}(t)} = \frac{l′_{1}(t)}{l_{1}(t)} + \frac{l′_{2}(t)}{l_{2}(t)}

and the formula

\frac{l′(t)}{l(t)} = \frac{a}{1 + bt}, where l(t) = (1 + bt)^{\frac{a}{b}} .

Using the definition of Adams operators

ψ_{−t}(x) = −t\frac{λ′_{t}(x)}{λ_{t}(x)}

and lemma 1 we can write the generating series for external powers as an element of Λ(d, n) ⊗ C[[t]]

λ_{t}(x) = ∏_{ε:ε^{n}=1}(1 + εt)^{α(ε)}, where α(ε) = \frac{1}{n} ∑_{i=1}^{n} ψ^{i}(x)ε^{−i}.

By conditions λ_{t}(x) is a polynomial of degree d. So for any homomorphism of algebras ϕ : Λ(d, n) ⊗ C → C

ϕ(λ_{t}(x)) = ∏_{ε:ε^{n}=1}(1 + εt)^{ϕ(α(ε))}

is also a polynomial.

Let us describe roots of this polynomial. An equality

tl(t)f(t) = l′(t)(1 − t^{n})

in C[t] implies that the roots of l(t) are roots of 1 − t^{n}.

We have that the values of the polynomial ϕ(ψ(t)) = \frac{1}{n} ∑_{i=1}^{n} ϕ(ψ^{i}(x))t^{i} on the n-th roots of unity are non-negative integers for any homomorphism ϕ : Λ(d, n)⊗C → C. We can express the coefficients ϕ(ψ^{i}(x)) of the polynomial ϕ(ψ(t)) throught its values

∑_{ε:ε^{n}=1} ϕ(ψ(ε))ε^{−j} = ∑_{ε:ε^{n}=1} \frac{1}{n} ∑_{i=1}^{n} ϕ(ψ^{i}(x))ε^{i−j} =

\frac{1}{n} ∑_{i=1}^{n} ϕ(ψ^{i}(x)) ∑_{ε:ε^{n}=1} ε^{i−j} = \frac{1}{n} ∑_{i=1}^{n} ϕ(ψ^{i}(x))nδ_{i,j} = ϕ(ψ^{j}(x)).

In particular

d = ϕ(ψ^{n}(x)) = ∑_{ε:ε^{n}=1} ϕ(ψ(ε)).

Thus we have shown that any homomorphism ϕ : Λ(d, n) ⊗ C → C defines the set {ϕ(ψ(ε)), ε^{n} = 1} of non-negative integers which sum equals d. Since ψ^{i}(x) generates the algebra Λ(d, n) the homomorphism ϕ is defined by this set.

To complete the proof of theorem let us define the homomorphism

φ : C[x_{1}, ..., x_{d}, x_{i}^{n} = 1]^{S_{d}} → C

such that ϕ is a composition of (2) and φ. Let φ : {x_{1}, ..., x_{d}} → {ε : ε^{n} = 1} be a map for which ϕ(ψ(ε)) = |φ^{−1}(ε^{−1})|. It is easy to see that this map defines the searching homomorphism.

4 Representations rings of finite groups

We begin with some facts from the theory of representations of finite groups.

The representations ring R_{C}(G) of finite group G (Grothendieck ring of complex group ring) provide an example of λ-ring [2]. External power of the class of representation is a class of its external power. The λ-structure of representations ring can be expressed in more simple form by means of characters.

We can identify the representations ring with the subring (ring of characters) in the ring of complex functions of conjugate classes by corresponding to the representation the traces of elements of the group. In particular, this realization provides the triviality of the radical of representations ring. Adams operations have the following form on the characters [2]:

ψ^{i}(x)(g) = x(g^{i}) for any x ∈ R_{C}(G), g ∈ G.

This describtion implies periodicity of representations λ-ring of finite group [4].

Proposition 1 The period of the representation ring R_{C}(G) of finite group G is equal to the period (exponent) of G, e.g. the smallest positive integer n such that g^{n} = e for any g from G.

Proof: For any character x from R_{C}(G)

ψ^{k+n}(x(g)) = x(g^{k+n}) = x(g^{k}) = ψ^{k}(x(g)),

where n is the period of the group G. Hence the period of λ-ring R_{C}(G) is smaller than n. Conversely consider the regular character ρ (ρ(g) = |G|δ_{e,g})(the character of the regular representation). If N is a period of R_{C}(G) then

ψ^{N}(ρ) = ψ^{0}(ρ) = ρ(e)ϵ = |G|ϵ

and

ρ(g^{N}) = ψ^{N}(ρ)(g) = |G|ϵ(g) = |G|.

That is g^{N} = e and N equals n.

The dimension of the space of invariant elements defines a linear form χ on the representations ring. This form admits the following expression on the character lavel

χ(x) = \frac{1}{|G|} ∑_{g∈G} x(g) for any x ∈ R_{C}(G), g ∈ G.

The generating series of demensions of invariants of symmetric powers of faithful complex representation X : G → GL(V) can be written as follows (Molin’s formula [6]):

χ(s_{t}(x)) = \frac{1}{|G|} ∑_{g∈G} \frac{1}{det(1 − X(g)t)}.  (4)

Now we can prove the general theorem.

Theorem 2 Let n be a period of finite group G and d be a dimension of stirct representation of G. Then n^{d} is devided by the order of the group G.

Proof: Using Molin’s formula (4) we can see that the first coefficient of the Laurent’s expansion of the generating function χ(s_{t}(x)) at t = 1 is \frac{1}{|G|}

χ(s_{t}(x)) = \frac{1}{|G|(1 − t)^{d}} + ...

By another hand, we can write χ(s_{t}(x)) = \frac{f(t)}{(1−t^{n})^{d}} for some f(t) ∈ Z[t]. Consider (S_{d}-invariant) linear form χ on Z[x_{1}, ..., x_{d}] induced by the homomorphism from the free periodic λ-ring Λ(d, n) to R(G), which sends the generator to x, and the inclusion Λ(d, n) ∼ Z[x_{1}, ..., x_{d}]^{S_{d}} ⊂ Z[x_{1}, ..., x_{d}]. The generating function for χ has the following form

χ(t_{1}, ..., t_{d}) = ∑_{i_{s}≥0} χ(i_{1}, ..., i_{d})t_{1}^{i_{1}}...t_{d}^{i_{d}} =

∑_{0≤i_{s}<n} χ(i_{1}, ..., i_{d})t_{1}^{i_{1}}...t_{d}^{i_{d}} ∑_{j_{s}≥0} t_{1}^{nj_{1}}...t_{d}^{nj_{d}} = \frac{f(t_{1}, ..., t_{d})}{∏_{i=1}^{d}(1 − t_{i}^{n})},

where χ(i_{1}, ..., i_{d}) = χ(x_{1}^{i_{1}}...x_{d}^{i_{d}}) and f(t_{1}, ..., t_{d}) = ∑_{0≤i_{s}<n} χ(i_{1}, ..., i_{d})t_{1}^{i_{1}}...t_{d}^{i_{d}}. Using the expansion s^{k}(x) = ∑_{∑_{s} i_{s}=k} x_{1}^{i_{1}}...x_{d}^{i_{d}} in Λ(d, n) we have

χ(s_{t}(x)) = χ(t, ..., t) = \frac{f(t)}{(1 − t^{n})^{d}} ,

where f(t) = f(t, ..., t). So we have

χ(s_{t}(x)) = \frac{f(1)}{n^{d}(1 − t)^{d}} + ....

Now it is enough to remark that f(1) = \frac{n^{d}}{|G|} is integer because the form χ is integer on Z[x_{1}, ..., x_{d}]^{S_{d}}.

In some cases we can estimate the constant f(1) using the additive information about the kernel of the homomorphism

Z[x_{1}, ..., x_{d}, x_{i}^{n} = 1]^{S_{d}} → R_{C}(G),  (5)

which sends the generator to the class of representation x. We will consider the case of trivial determinant of x, when the kernel of (5) contains x_{1}...x_{d}−1.

Theorem 3 Let n be a period of finite group G and d be a dimension of faithful representation of G with trivial determinant (maximal external power). Then n^{d−1} is devided by the order of the group G.

Proof:

From the proof of the theorem (2) we have that the quotient \frac{n^{d}}{|G|} coincides with the value f(1) of the polynomial

f(t) = ∑_{0≤i_{s}<n} χ(x_{1}^{i_{1}}...x_{d}^{i_{d}})t^{i_{1}+...+i_{d}}.

Since the kernel of (5) contains x_{1}...x_{d} − 1 we can write

χ(x_{1}^{i_{1}+i}...x_{d}^{i_{d}+i}) = χ(x_{1}^{i_{1}}...x_{d}^{i_{d}}), for any i.

Hence

f(t) = (∑_{j=1}^{d} ∑_{0≤i_{s}<n,i_{j}=0} χ(x_{1}^{i_{1}}...x_{d}^{i_{d}})t^{i_{1}+...+i_{d}})(∑_{i=0}^{n−1} t^{di}) = g(t) ∑_{i=0}^{n−1} t^{di}.

In particular, \frac{n^{d}}{|G|} = f(1) = g(1)n, for some integer g(1).

References

[1] J.F.Adams, Vector fields on spheres, Ann. Math., 75 (1962), p. 603-632.

[2] M.F.Atiyah,D.O.Tall,  Group  rep-

resentations, λ-rings and J-homomorphism, Topology, 1963, vol. 8, n. 3, p. 253-297.

[3] A.Grothendieck, Special λ-rings, unpublished

[4] M.Kervair, Opération d’Adams en théorie des représentations linéares des groupes finis, Enseign. Math. (1976), ser. 2, vol. 22, n. 1, p. 1-28.

[5] D. Knutson, λ-rings and the representation theory of the symmetric group. Berlin, Springer-Verlag, 1973, Lecture Notes Math., n. 308.

[6] T.Molien, Über die Invarianten der linear Substitutionsgruppen, Sitzungsber. König. Preuss. Akad. Wiss. (1897), 1152-1156.