ASYMPTOTIC BEHAVIOUR OF LIE POWERS AND LIE MODULES

ROGER M. BRYANT, KAY JIN LIM, AND KAI MENG TAN

Abstract. Let V be a finite-dimensional F G-module, where F is a field of prime characteristic p and G is a group. We show that, when r is not a power of p, the Lie power L^{r}(V) has a direct summand B^{r}(V) which is a direct summand of the tensor power V^{⊗r} and which satisfies dim B^{r}(V)/ dim L^{r}(V) → 1 as r → ∞. Similarly, for the same values of r, we obtain a projective submodule C(r) of the Lie module Lie(r) over F such that dim C(r)/ dim Lie(r) → 1 as r → ∞.

1. Introduction

Let F be a field of prime characteristic p, G a group, and V a finite-dimensional (right) F G-module. Write T (V) for the tensor algebra of V: thus T (V) = ⨁_{r=0}^{∞} V^{⊗r}. Let L(V) denote the Lie subalgebra of T (V) generated by V: thus L(V) = ⨁_{r=1}^{∞} L^{r}(V), where L^{r}(V) = L(V) ∩ V^{⊗r}. We call L^{r}(V) the rth Lie power of V. The tensor power V^{⊗r} is an F G-module (under the ‘diagonal’ action of G) and L^{r}(V) is a submodule of V^{⊗r}. The dimension of L^{r}(V) is given by a formula of Witt (see [MKS, Theorem 5.11]):

dim L^{r}(V) = \frac{1}{r} ∑_{d|r} μ(d)n^{r/d},  (1.1)

where n = dim V and μ denotes the Möbius function.

The modules L^{r}(V) have been extensively studied in recent years. We refer to the paper of Bryant and Schocker [BS1] and the works cited there for details of progress on the problem of describing L^{r}(V) up to isomorphism. When r is not divisible by p, L^{r}(V) is a direct summand of V^{⊗r} (see, for example, [DE, §3.1]). Although this does not hold in general, our first main result shows that, in an asymptotic sense, ‘most’ of L^{r}(V) is a direct summand of V^{⊗r} provided that r is not a power of p.

Theorem 1.1. Suppose that dim V > 1. Let A be the set of all positive integers r such that r is not a power of p. Then, for each r ∈ A, there is a direct summand B^{r}(V) of L^{r}(V) such that B^{r}(V) is a direct summand of V^{⊗r} and

lim_{\begin{matrix}r→∞ \\ r∈A \end{matrix}} \frac{dim B^{r}(V)}{dim L^{r}(V)} = 1.

The modules B^{r}(V) are the modules denoted by B_{r} in the decomposition theorem of [BS1] and their dimensions are given by a recurrence formula. When r = p^{m} with m > 0, we have B_{r} = 0. Thus, although we conjecture that a result like Theorem 1.1 holds without any restrictions on r, we cannot use [BS1] when r is a power of p.

Date: November 2010.

2000 Mathematics Subject Classification. 17B01, 20C30 (primary), 20C20 (secondary).

Supported by EPSRC Standard Research Grant EP/G024898/1 and EP/G025487/1, and by MOE Academic Research Fund R-146-000-135-112.

The other main result of this paper concerns the Lie module Lie(r). This is a module for the symmetric group 𝔖_{r} and can be defined as follows. Let E_{n} be a vector space over F of finite dimension n, where n ⩾ r, and let {e_{1}, …, e_{n}} be a basis for E_{n}. Then Lie(r) is the subspace of L^{r}(E_{n}) spanned by all elements of the form [e_{1π}, e_{2π}, …, e_{rπ}] where π ∈ 𝔖_{r} and [e_{1π}, e_{2π}, …, e_{rπ}] denotes the ‘leftnormed’ Lie product in which [e_{1π}, e_{2π}] = e_{1π}⊗e_{2π}−e_{2π}⊗e_{1π} and [e_{1π}, e_{2π}, …, e_{rπ}] = [[e_{1π}, e_{2π}, …, e_{(r−1)π}], e_{rπ}]. Clearly Lie(r) is a (right) F 𝔖_{r}-module, where the action of 𝔖_{r} comes from its right action on {1, …, r}. Furthermore, the definition of Lie(r) is independent of the choice of n with n ⩾ r. The dimension of Lie(r) is (r − 1)! (as follows from [MKS, Theorem 5.11] by taking n_{1} = n_{2} = ⋯ = 1).

The main motivation for our result on Lie(r) comes from the work of Selick and Wu [SW1]. These authors considered the problem of finding natural homotopy decompositions of the loop suspension of a p-torsion suspension and proved that this problem is equivalent to the algebraic problem of finding natural coalgebra decompositions of the primitively generated tensor algebras over the field with p elements. They determined (see [SW1, Theorem 6.5]) the finest coalgebra decomposition of a tensor algebra (over an arbitrary field). Their result can be described as a functorial Poincaré-Birkhoff-Witt theorem. However, in order to compute the factors in this decomposition, it is necessary to know a maximal projective submodule, Lie^{max}(r), of the Lie module Lie(r).

The projective modules for the symmetric groups over fields of positive characteristic p are not known in general. Their structure depends on the decomposition matrices for symmetric groups and the determination of these is a famous open problem. However, according to [SW2], even if the modules Lie^{max}(r) cannot be computed precisely, it is of interest to know how quickly their dimensions grow and whether or not the growth rate is exponential. Determination of Lie^{max}(6) and Lie^{max}(8) in characteristic 2 in [SW2] suggests that Lie^{max}(r) is relatively large compared with Lie(r). If this is true in general, it has the desirable consequence that the factors in the functorial PBW theorem are relatively small.

Some progress has been made in the understanding of Lie^{max}(r) in the case where r = pk with p ∤ k. Erdmann and Schocker [ES] established, amongst other things, that there is a one-to-one correspondence between the non-projective indecomposable summands of Lie(pk) and the indecomposable summands of Lie(k). Erdmann and Tan [ET] provided an upper bound for dim Lie^{max}(pk) and showed that the ratio of this upper bound to dim Lie(pk) approaches 1 as k tends to infinity. They conjectured that the same fact holds for dim Lie^{max}(pk)/ dim Lie(pk).

Our second main result establishes this conjecture and considerably more.

Theorem 1.2. Let A be the set of all positive integers r such that r is not a power of p. Then

lim_{\begin{matrix}r→∞ \\ r∈A \end{matrix}} \frac{dim Lie^{max}(r)}{dim Lie(r)} = 1.

The theorem shows, in particular, that dim Lie^{max}(r) grows exponentially with r when r is not a power of p. As in the case of Theorem 1.1, we conjecture that the restriction on r can be removed. However, our methods work only for r ∈ A. In the proof, we quickly reduce to the case where F is an infinite field and then consider the n-dimensional natural module E_{n} for the general linear group GL_{n}(F), where n ⩾ r. The image of L^{r}(E_{n}) under the Schur functor is the Lie module Lie(r) (see Section 3). With B^{r}(E_{n}) as in Theorem 1.1, the image C(r) of B^{r}(E_{n}) under the Schur functor is a projective submodule of Lie(r). We describe the dimension of C(r) by means of a recurrence formula and show that dim C(r)/ dim Lie(r) → 1 as r → ∞ with r ∈ A.

Theorem 1.1 will be proved in Section 2 and Theorem 1.2 in Section 3. Throughout this paper F denotes a field of prime characteristic p. All of our algebras and vector spaces are taken over F, unless otherwise stated, and all tensor products are also taken over F. All modules are right modules.

2. Modular Lie powers

In this section we shall prove Theorem 1.1.

Let G be a group and let V be a finite-dimensional F G-module. The decomposition theorem of [BS1] yields, for each positive integer r, a certain direct summand B_{r} of L^{r}(V) such that B_{r} is a direct summand of V^{⊗r}. Here we write B^{r}(V) instead of B_{r}. We shall use the following result, where we write p^{i}U for the direct sum of p^{i} copies of a module U.

Theorem 2.1 ([BS2, Theorem 4.2]). Let k be a positive integer not divisible by p. Then, for each non-negative integer m, we have

p^{m}B^{p^{m}k}(V) ⊕ p^{m−1}B^{p^{m−1}k}(V)^{⊗p} ⊕ ⋯ ⊕ B^{k}(V)^{⊗p^{m}} ≅ L^{k}(V^{⊗p^{m}}).  (2.1)

For positive integers n and r, let

w(n, r) = \frac{1}{r} ∑_{d|r} μ(d)n^{r/d}.

Lemma 2.2. For all n and r, we have

n^{r}/r − n^{r/2}/2 ⩽ w(n, r) ⩽ n^{r}/r.

Proof. The result is clear if r = 1, so we assume r ⩾ 2. By [R, Theorem 7.1], rw(n, r) is the number of words of length r on an alphabet of cardinality n that cannot be written as a power of a shorter word. Clearly rw(n, r) ⩽ n^{r}. If u is a word of length r of the form v^{d}, where d > 1, then v has length at most r/2. Hence, for a given d, the number of possibilities for v is at most n^{r/2}. The number of possibilities for d is at most r/2. Therefore n^{r} − rw(n, r) ⩽ (r/2)n^{r/2}, which gives the required result. □

Let n = dim(V) and suppose that n ⩾ 2. By (1.1), we have dim L^{r}(V) = w(n, r). For each r ⩾ 1, we define

b_{r} = \frac{dim B^{r}(V)}{dim L^{r}(V)} = \frac{dim B^{r}(V)}{w(n, r)}.

Since B^{r}(V) is a submodule of L^{r}(V) we have 0 ⩽ b_{r} ⩽ 1. In order to prove Theorem 1.1, we need to prove that b_{r} → 1 as r → ∞ with r ∈ A. Note that for r ∈ A we can write r = p^{m}k where m ⩾ 0, p ∤ k and k ⩾ 2. Suppose, for the rest of this section,

4 ROGER M. BRYANT, KAY JIN LIM, AND KAI MENG TAN

that m and k satisfy these conditions. By direct computations of the dimensions in (2.1), we have

∑_{i=0}^{m} p^{m−i}(b_{p^{m−i}k})^{p^{i}}w(n, p^{m−i}k)^{p^{i}} = w(n^{p^{m}}, k).  (2.2)

For i = 0, 1, …, m, define

a_{i} = \frac{w(n, p^{m−i}k)^{p^{i}}}{p^{i}w(n, p^{m}k)} .

Then, on dividing (2.2) by p^{m}w(n, p^{m}k) and re-arranging, we obtain

b_{p^{m}k} = \frac{w(n^{p^{m}}, k)}{p^{m}w(n, p^{m}k)} − ∑_{i=1}^{m} a_{i}(b_{p^{m−i}k})^{p^{i}}.  (2.3)

Lemma 2.3. Suppose that 0 < s ⩽ i ⩽ m. Then

\frac{a_{i}}{a_{i−s}} ⩽ p^{−s} (\frac{2p^{s}}{(p^{m−i}k)^{p^{s}−1}} )^{p^{i−s}} .

Proof. We have

\frac{a_{i}}{a_{i−s}} = \frac{w(n, p^{m−i}k)^{p^{i}}}{p^{s}w(n, p^{m−i+s}k)^{p^{i−s}}} = p^{−s} (\frac{w(n, p^{m−i}k)^{p^{s}}}{w(n, p^{m−i+s}k)})^{p^{i−s}} .

Write q = p^{m−i+s}k. Then, by Lemma 2.2,

\frac{w(n, p^{m−i}k)^{p^{s}}}{w(n, p^{m−i+s}k)} ⩽ \frac{(p^{m−i}k)^{−p^{s}}n^{q}}{n^{q}/q − n^{q/2}/2} = \frac{(p^{m−i}k)^{−p^{s}}qn^{q/2}}{n^{q/2} − q/2} .

Since n ⩾ 2 and q ⩾ 6, we have n^{q/2}/(n^{q/2} − q/2) ⩽ 2. Thus

\frac{w(n, p^{m−i}k)^{p^{s}}}{w(n, p^{m−i+s}k)} ⩽ 2(p^{m−i}k)^{−p^{s}}p^{m−i}kp^{s} = \frac{2p^{s}}{(p^{m−i}k)^{p^{s}−1}} .

This gives the required result. □ Corollary 2.4. Suppose that m ⩾ 1.

(i) For i = 2, 3, …, m − 1, we have a_{i}/a_{i−1} ⩽ 1.

(ii) We have a_{1} ⩽ 2/(p^{m−1}k)^{p−1} and a_{m} ⩽ 2/k^{p^{m}−1}. Proof.

(i) By Lemma 2.3 with s = 1, we have

\frac{a_{i}}{a_{i−1}} ⩽ p^{−1} (\frac{2p}{(p^{m−i}k)^{p−1}} )^{p^{i−1}} ⩽ (\frac{2}{p^{(m−i)(p−1)}} )^{p^{i}−1} ⩽ 1.

(ii) By Lemma 2.3 with i = s = 1, we have

a_{1} = \frac{a_{1}}{a_{0}} ⩽ p^{−1} \frac{2p}{(p^{m−1}k)^{p−1}} = \frac{2}{(p^{m−1}k)^{p−1}} .

Similarly, with i = s = m,

a_{m} = \frac{a_{m}}{a_{0}} ⩽ p^{−m} \frac{2p^{m}}{k^{p^{m}−1}} = \frac{2}{k^{p^{m}−1}} .

□

Lemma 2.5. We have

\frac{w(n^{p^{m}}, k)}{p^{m}w(n, p^{m}k)} ⩾ 1 − \frac{k}{2n^{p^{m}k/2}} .

Proof. By Lemma 2.2, we have

\frac{w(n^{p^{m}}, k)}{p^{m}w(n, p^{m}k)} ⩾ \frac{n^{p^{m}k}/k − n^{p^{m}k/2}/2}{p^{m}n^{p^{m}k}/(p^{m}k)} = 1 − \frac{k}{2n^{p^{m}k/2}} .

□

Proof of Theorem 1.1. Suppose that m ⩾ 1. Recall that b_{r} ⩽ 1 for all r. By (2.3) and Corollary 2.4(i), we have

1 ⩾ b_{p^{m}k} ⩾ w(n^{p^{m}}, k)/(p^{m}w(n, p^{m}k)) − (m − 1)a_{1} − a_{m}.

Hence, by Corollary 2.4(ii) and Lemma 2.5,

1 ⩾ b_{p^{m}k} ⩾ 1 − \frac{k}{2n^{p^{m}k/2}} − \frac{2(m − 1)}{(p^{m−1}k)^{p−1}} − \frac{2}{k^{p^{m}−1}} .

Also, for m = 0, we have b_{k} = w(n, k)/w(n, k) = 1. Thus b_{r} → 1 as r → ∞ with r ∈ A. □

3. Lie modules

In this section we shall prove Theorem 1.2.

In order to prove the theorem we shall apply the Schur functor to the modules appearing in a special case of the isomorphism (2.1). Since the Schur functor, in its usual form, requires the field F to be infinite, we begin by showing that it is enough to prove Theorem 1.2 in the case where F is infinite.

Let G be a finite group and let M be a finite-dimensional F G-module. We write M^{max} for a maximal projective submodule of M. Thus M^{max} is a maximal projective direct summand of M and is uniquely determined up to isomorphism. Let K be an extension field of F. Then K ⊗ M and K ⊗ M^{max} are KG-modules.

Lemma 3.1. Suppose that F is a splitting field for G. Then K ⊗ M^{max} ≅ (K ⊗ M)^{max}.

Proof. For modules U and V, we write U | V to denote that U is isomorphic to a direct summand of V. Clearly K ⊗ M^{max} is projective and K ⊗ M^{max} | K ⊗ M. Thus K ⊗ M^{max} | (K ⊗ M)^{max}. Since F and K are splitting fields for G, it follows from [HB, Theorem 10.18] that if P is a projective indecomposable F G-module then K ⊗ P is a projective indecomposable KG-module and, furthermore, every projective indecomposable KG-module is isomorphic to some such K ⊗ P. Hence there is a projective F G-module Q such that K ⊗ Q ≅ (K ⊗ M)^{max}. In particular, K ⊗ Q | K ⊗ M and it follows, by [HB, Theorem 1.21], that Q | M. Thus Q | M^{max} and so (K ⊗ M)^{max} | K ⊗ M^{max}. The result follows. □

In Section 1 we defined the F 𝔖_{r}-module Lie(r) for each positive integer r. Let Lie_{K}(r) denote the K𝔖_{r}-module defined in the same way over K. Thus K⊗Lie(r) ≅ Lie_{K}(r). As in Section 1, let Lie^{max}(r) be a maximal projective submodule of Lie(r). It is well-known that every field is a splitting field for 𝔖_{r} (see [J, Theorem 11.5]). Thus, by Lemma 3.1, K⊗Lie^{max}(r) is isomorphic to a maximal projective submodule of Lie_{K}(r). Consequently, if Theorem 1.2 holds over K, it holds over F. Therefore it suffices to prove Theorem 1.2 in the case where F is infinite. Hence, from now on, we take F to be infinite.

Suppose that n ⩾ r and let E_{n} be the n-dimensional natural module for the general linear group GL_{n}(F) with standard basis {e_{1}, …, e_{n}}. We identify 𝔖_{r} with the subgroup of GL_{n}(F) consisting of all those elements which permute {e_{1}, …, e_{r}} and fix e_{r+1}, …, e_{n} pointwise.

Let Λ(n, r) be the set of all n-tuples (α_{1}, …, α_{n}) of non-negative integers such that α_{1} + ⋯ + α_{n} = r. The elements of Λ(n, r) are called weights. For t_{1}, …, t_{n} ∈ F^{×} = F \ {0} let d(t_{1}, …, t_{n}) denote the diagonal matrix in GL_{n}(F) with entries t_{1}, …, t_{n} down the diagonal.

Let M be a finite-dimensional F GL_{n}(F)-module such that M is a homogeneous polynomial module of degree r (see [G]). For α = (α_{1}, …, α_{n}) ∈ Λ(n, r) the αweight space of M is denoted by M^{α} and is the subspace of M defined by

M^{α} = {u ∈ M | u · d(t_{1}, …, t_{n}) = t_{1}^{α_{1}} ⋯ t_{n}^{α_{n}}u, ∀t_{1}, …, t_{n} ∈ F^{×}}.

By [G, (3.2c)], we have

M = ⨁_{α∈Λ(n,r)} M^{α}.  (3.1)

Let ω_{r} ∈ Λ(n, r) be defined by ω_{r} = (1, …, 1, 0, …, 0) where there are r entries equal to 1 and n − r entries equal to 0. For σ ∈ 𝔖_{r} ⩽ GL_{n}(F) we have

σd(t_{1}, …, t_{n})σ^{−1} = d(t_{1σ}, …, t_{rσ}, t_{r+1}, …, t_{n}).

It follows that M^{ω_{r}} is invariant under the restriction to 𝔖_{r} of the action of GL_{n}(F) on M. Hence M^{ω_{r}} is an F 𝔖_{r}-module.

Let mod_{F} (n, r) be the class of all finite-dimensional homogeneous polynomial F GL_{n}(F)-modules of degree r and let mod(F 𝔖_{r}) be the class of all finite-dimensional F 𝔖_{r}-modules. Let

f_{r} : mod_{F} (n, r) → mod(F 𝔖_{r})

be the map defined by f_{r}(M) = M^{ω_{r}} for all M ∈ mod_{F} (n, r). This map f_{r} is called the ‘Schur functor’. (It is an exact functor between the two module categories: see [G, Chapter 6].) It is easy to verify that

f_{r}(M ⊕ N) = f_{r}(M) ⊕ f_{r}(N)  (3.2)

for all M, N ∈ mod_{F} (n, r). Furthermore, it is clear from the definition of the Lie module in Section 1 that

f_{r}(L^{r}(E_{n})) = Lie(r),

for all n such that n ⩾ r.

Let r_{1}, …, r_{l} be positive integers such that r_{1} + ⋯ + r_{l} = r. We may regard 𝔖_{r} as the group of all permutations of a set of cardinality r written as the disjoint union of sets of cardinalities r_{1}, …, r_{l}. Thus we may regard 𝔖_{r_{1}} × ⋯ × 𝔖_{r_{l}} as a subgroup of 𝔖_{r}. The following lemma is an immediate consequence of a slightly stronger version given in [DE].

Lemma 3.2 ([DE, §2.5, Lemma]). Let n ⩾ r and let r_{1}, …, r_{l} be positive integers such that r_{1} + ⋯ + r_{l} = r. For i = 1, …, l, let M_{i} ∈ mod_{F} (n, r_{i}). Then

f_{r}(M_{1} ⊗ ⋯ ⊗ M_{l}) ≅ Ind_{𝔖_{r_{1}}×···×𝔖_{r_{l}}}^{𝔖_{r}} (f_{r_{1}}(M_{1}) ⊠ ⋯ ⊠ f_{r_{l}}(M_{l})).

Lemma 3.3. Let n ⩾ qk where q and k are positive integers. Then

dim f_{qk}(L^{k}(E_{n}^{⊗q})) = (qk)!/k.

Proof. Let I(n, q) be the set of all ordered q-tuples (i_{1}, …, i_{q}) where i_{1}, …, i_{q} ∈ {1, …, n}. For θ ∈ I(n, q), where θ = (i_{1}, …, i_{q}), write

e_{θ} = e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{q}} ∈ E_{n}^{⊗q}.

Thus {e_{θ} | θ ∈ I(n, q)} is a basis for E_{n}^{⊗q}.

The F GL_{n}(F)-module L^{k}(E_{n}^{⊗q}) is spanned as a vector space by the elements [e_{θ_{1}}, …, e_{θ_{k}}] with θ_{1}, …, θ_{k} ∈ I(n, q). It is easily verified that each of these elements belongs to some weight space of L^{k}(E_{n}^{⊗q}). Thus, by (3.1), (L^{k}(E_{n}^{⊗q}))^{ω_{qk}} is spanned by those elements [e_{θ_{1}}, …, e_{θ_{k}}] which belong to it. These are the elements [e_{θ_{1}}, …, e_{θ_{k}}] such that each of e_{1}, …, e_{qk} occurs once and only once among the tensor factors of e_{θ_{1}}, …, e_{θ_{k}}.

Let Φ be the set of all k-element subsets {θ_{1}, …, θ_{k}} of I(n, q) such that each q-tuple θ_{j} (1 ⩽ j ⩽ k) has q distinct entries and {1, …, qk} is the disjoint union of these k sets of entries. For each ϕ ∈ Φ let W_{ϕ} be the subspace of L^{k}(E_{n}^{⊗q}) spanned by the elements [e_{θ_{1}}, …, e_{θ_{k}}] where {θ_{1}, …, θ_{k}} = ϕ. It follows that

f_{qk}(L^{k}(E_{n}^{⊗q})) = (L^{k}(E_{n}^{⊗q}))^{ω_{qk}} = ⨁_{ϕ∈Φ} W_{ϕ}.

It is easily verified that |Φ| = (qk)!/k!. Also, by [MKS, Theorem 5.11], dim W_{ϕ} = (k − 1)! for all ϕ ∈ Φ. Thus

dim f_{qk}(L^{k}(E_{n}^{⊗q})) = (qk)!(k − 1)!/k! = (qk)!/k.
□

We consider B^{r}(V), as in Section 2, in the special case where G = GL_{r}(F) and V = E_{r} and we define

C(r) = f_{r}(B^{r}(E_{r})).

The following result will enable us to use F GL_{n}(F)-modules for arbitrary values of n with n ⩾ r.

Lemma 3.4. For all n ⩾ r, we have

C(r) ≅ f_{r}(B^{r}(E_{n})).

Proof. We regard E_{r} as a subspace of E_{n} with GL_{r}(F) ⩽ GL_{n}(F) in the obvious way. Let d_{n,r} : mod_{F} (n, r) → mod_{F} (r, r) be the truncation map (see [G, §6.5]). For all M ∈ mod_{F} (n, r), we have

d_{n,r}(M) = ⨁_{β∈Λ(r,r)^{∗}} M^{β}

where Λ(r, r)^{∗} is the subset of Λ(n, r) consisting of those weights (β_{1}, …, β_{n}) such that β_{r+1} = ⋯ = β_{n} = 0 (see [G, (6.5c)]). In particular, f_{r}(d_{n,r}(M)) = f_{r}(M). It is a consequence of [BS1, Theorem 4.2] that d_{n,r}(B^{r}(E_{n})) = B^{r}(E_{r}). The result follows. □

By definition, C(r) is a submodule of Lie(r). However, B^{r}(E_{r}) is a direct summand of E_{r}^{⊗r} and, by [G, (6.3d)], f_{r}(E_{r}^{⊗r}) is a regular F 𝔖_{r}-module. Thus, by (3.2), C(r) is a projective F 𝔖_{r}-module, and so dim C(r) ⩽ dim Lie^{max}(r). Hence, in order to prove Theorem 1.2, it suffices to prove that

lim_{\begin{matrix}r→∞ \\ r∈A \end{matrix}} \frac{dim C(r)}{dim Lie(r)} = 1.

For each r ⩾ 1, we define

c_{r} = \frac{dim C(r)}{dim Lie(r)} = \frac{dim C(r)}{(r − 1)!}.  (3.3)

Thus 0 ⩽ c_{r} ⩽ 1. It suffices to prove that c_{r} → 1 as r → ∞ with r ∈ A.

Suppose that m is a non-negative integer and k is a positive integer such that p ∤ k and k ⩾ 2. Take n ⩾ r = p^{m}k and apply f_{r} to the isomorphism (2.1) with V = E_{n}. By (3.2) we obtain

⨁_{i=0}^{m} p^{m−i}f_{r}(B^{p^{m−i}k}(E_{n})^{⊗p^{i}}) ≅ f_{r}(L^{k}(E_{n}^{⊗p^{m}})).  (3.4)

By Lemma 3.2, Lemma 3.4 and (3.3), we have

dim f_{r}(B^{p^{m−i}k}(E_{n})^{⊗p^{i}}) = \frac{(p^{m}k)!}{((p^{m−i}k)!)^{p^{i}}} (dim C(p^{m−i}k))^{p^{i}} = \frac{(p^{m}k)!}{(p^{m−i}k)^{p^{i}}} (c_{p^{m−i}k})^{p^{i}}.

Hence, by (3.4) and Lemma 3.3, we obtain

∑_{i=0}^{m} \frac{p^{m−i}(p^{m}k)!}{(p^{m−i}k)^{p^{i}}}(c_{p^{m−i}k})^{p^{i}} = \frac{(p^{m}k)!}{k}.  (3.5)

For i = 0, 1, …, m, define

a′_{i} = (p^{m−i}k)^{−(p^{i}−1)}.

Then (3.5) may be written as

c_{p^{m}k} = 1 − ∑_{i=1}^{m} a′_{i}(c_{p^{m−i}k})^{p^{i}}.  (3.6)

Lemma 3.5. Suppose that 0 ⩽ s ⩽ i ⩽ m. Then

\frac{a′_{i}}{a′_{i−s}} = p^{−s} (\frac{p^{s}}{(p^{m−i}k)^{p^{s}−1}} )^{p^{i−s}} .

Proof. We have

\frac{a′_{i}}{a′_{i−s}} = \frac{(p^{m−i+s}k)^{p^{i−s}−1}}{(p^{m−i}k)^{p^{i}−1}} = \frac{(p^{s})^{p^{i−s}−1}}{(p^{m−i}k)^{p^{i}−p^{i−s}}} = p^{−s} (\frac{p^{s}}{(p^{m−i}k)^{p^{s}−1}} )^{p^{i−s}} .

□

Proof of Theorem 1.2. Note the similarity between (3.6) and (2.3). Lemma 3.5 is stronger than Lemma 2.3 and therefore gives an analogue of Corollary 2.4. We do not need an analogue of Lemma 2.5 because the first term on the right-hand side of (3.6) is 1. Thus, by the argument of the proof of Theorem 1.1, we obtain c_{r} → 1 as r → ∞ with r ∈ A. □

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(Roger M. Bryant) School of Mathematics, University of Manchester, Manchester M13 9PL, UK.

   E-mail address: roger.bryant@manchester.ac.uk

(Kay Jin Lim and Kai Meng Tan) Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076.

E-mail address, K. J. Lim: matlkj@nus.edu.sg

E-mail address, K. M. Tan: tankm@nus.edu.sg