Block components of the Lie module for the symmetric group

Roger M. Bryant and Karin Erdmann

Let F be a field of prime characteristic p and let B be a non-principal block of the group algebra F S_{r} of the symmetric group S_{r}. The block component Lie(r)_{B} of the Lie module Lie(r) is projective, by a result of Erdmann and Tan, although Lie(r) itself is projective only when p ∤ r. Write r = p^{m}k, where p ∤ k, and let S_{k}^{∗} be the diagonal of a Young subgroup of S_{r} isomorphic to S_{k} × ⋯ × S_{k}. We show that p^{m} Lie(r)_{B} ≅ (Lie(k)↑_{S_{k}^{∗}}^{S_{r}})_{B}. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of Lie(r)_{B}. Corresponding results are obtained, when F is infinite, for the rth Lie power L^{r}(E) of the natural module E for the general linear group GL_{n}(F).

1. Introduction and summary of results

Let r be a positive integer and let S_{r} denote the symmetric group of degree r. For any field F the Lie module Lie(r) is the F S_{r}-module given by the right ideal ω_{r}F S_{r} of the group algebra F S_{r} where ω_{r} is the ‘Dynkin–Specht–Wever’ element defined by ω_{1} = 1 and, for r ⩾ 2,

ω_{r} = (1 − c_{r})(1 − c_{r−1}) ⋯ (1 − c_{2}),  (1.1)

where c_{i} is the i-cycle (1 i i − 1 … 2).

If F has prime characteristic p and p ∤ r then Lie(r) is a direct summand of F S_{r} because ω_{r}^{2} = rω_{r} (see, for example, [?, Section 3]); so in this case Lie(r) is projective. However, if char F = p and p | r then Lie(r) is not projective (because its dimension is not then divisible by the order of a Sylow p-subgroup of S_{r}), but it was shown recently that every non-principal block component of Lie(r) is projective (see [?]). Here we show that each such component can be described in a surprisingly simple way in terms of Lie(k), where k is the p′-part of r.

Supported by EPSRC Standard Research Grant EP/G025487/1.

MSC2000: primary 20C30, 20G43; secondary 20C20.

Keywords: Lie module, symmetric group, Lie power, Schur algebra, block.

The Lie module occurs naturally in a number of contexts in algebra, algebraic topology and elsewhere (see [?] for a fuller discussion). Here we shall only be concerned with the connection with free Lie algebras, where our results on the Lie module give new insight into the module structure of the homogeneous components.

Let G be a group and V an F G-module. Let L^{r}(V) denote the homogeneous component of degree r in the free Lie algebra L(V) freely generated by any basis of V. (Here L(V) may be regarded as the Lie subalgebra generated by V in the tensor algebra or free associative algebra on V: see Section 2.1.) The vector space L^{r}(V) is called the rth ‘Lie power’ of V and it inherits the structure of an F G-module.

Suppose that F is infinite, let n be a positive integer, and let E denote the natural n-dimensional module over F for the general linear group GL_{n}(F). Then L^{r}(E), as a module for GL_{n}(F), is a homogeneous polynomial module of degree r. In other words it is a module for the Schur algebra S_{F} (n, r) (see [?]). In the case where n ⩾ r the Schur functor f_{r} maps S_{F} (n, r)-modules to F S_{r}-modules and we have f_{r}(L^{r}(E)) ≅ Lie(r) (see Section 2.4).

Recall that if B is the set of blocks of an algebra Γ and V is a Γ-module then we may write V as a direct sum of ‘block components’ V = ⨁_{B∈B} V_{B} where V_{B} ∈ B for all B. Our main results concern the block components of Lie(r) and L^{r}(E) when F has prime characteristic p. The basic results are for Lie(r), and the results for L^{r}(E) are obtained from these by means of the Schur functor. To state the results we write r = p^{m}k where m ⩾ 0, k ⩾ 1, and p ∤ k.

Let S_{k}^{∗} be a subgroup of S_{r} such that S_{k}^{∗} ≅ S_{k} and S_{k}^{∗} is the ‘diagonal’ of a Young subgroup of S_{r} isomorphic to S_{k} × ⋯ × S_{k} (with p^{m} factors). (See Section 2.4 for more details.) Since S_{k}^{∗} ≅ S_{k} we may regard Lie(k) as an F S_{k}^{∗}-module and, since p ∤ k, this module is projective. Thus the induced module Lie(k)↑_{S_{k}^{∗}}^{S_{r}} is also projective. It was proved in [?, Theorem 3.1] that if B is a non-principal block of F S_{r} then Lie(r)_{B} is projective. Here we shall prove (Theorem 3.2) that

Lie(r)_{B} ≅ \frac{1}{p^{m}} (Lie(k)↑_{S_{k}^{∗}}^{S_{r}})_{B}  (1.2)

when B satisfies the condition B̃ ≠ ∅ (see Section 2.2): this condition is satisfied when B is non-principal. (The notation U ≅ (1/q) V used in (1.2) means that q U ≅ V, where q U denotes U ⊕ ⋯ ⊕ U with q summands.) The special case where k = 1 is of particular interest: it yields

Lie(p^{m})_{B} ≅ \frac{1}{p^{m}} (F S_{p^{m}})_{B}.

An important part of the proof of (1.2) is played by the Decomposition Theorem of [?]. This will be described in Section 2.5.

Since (1.2) holds for each non-principal block B, a comparison of dimensions gives a weaker result (Corollary 3.3) for the principal block B_{0}:

dim Lie(r)_{B_{0}} = \frac{1}{p^{m}} dim (Lie(k)↑_{S_{k}^{∗}}^{S_{r}})_{B_{0}}.

The projective indecomposable F S_{r}-modules may be labelled P^{λ} where λ ranges over the p-regular partitions of r (see Section 2.2). For any F S_{r}-block B we write λ ∈ B when P^{λ} ∈ B. Since Lie(r)_{B} is projective when B is non-principal, there are non-negative integers m_{λ} such that

Lie(r)_{B} ≅ ⨁_{λ∈B} m_{λ} P^{λ}.

In Theorem 3.4 we prove that

m_{λ} = \frac{1}{r} ∑_{d|k} μ(d) β^{λ}(τ^{k/d}),  (1.3)

where μ is the Möbius function, τ is an element of S_{r} of cycle type (k, k, …, k), and β^{λ} denotes the Brauer character of D^{λ}, the irreducible F S_{r}-module isomorphic to the head of P^{λ}.

Now suppose that F is infinite and n is a positive integer. We have observed that L^{r}(E) is an S_{F} (n, r)-module. Similarly, L^{k}(E^{⊗p^{m}}) is an S_{F} (n, r)-module, and (by the argument in [?, Section 3.1]) it is isomorphic to a direct summand of E^{⊗r}. It is a consequence of [?, Theorem 3.2] that if B is a block of S_{F} (n, r) satisfying the condition B̃ ≠ ∅ (see Section 2.3) then L^{r}(E)_{B} is isomorphic to a direct sum of summands of E^{⊗r}. Here we shall prove (Theorem 3.6) that

L^{r}(E)_{B} ≅ \frac{1}{p^{m}} L^{k}(E^{⊗p^{m}})_{B}.

The indecomposable summands of E^{⊗r} are ‘tilting’ modules T (λ) where λ is a p-regular partition of r (see Section 2.3). For any S_{F} (n, r)-block B we write λ ∈ B when T (λ) ∈ B. In Theorem 3.7 we prove that if B̃ ≠ ∅ then

L^{r}(E)_{B} ≅ ⨁_{λ∈B} m_{λ} T (λ),

where the multiplicities m_{λ} are given by (1.3). This extends [?, Section 3.3, Theorem], which gives the same result in the case where p ∤ r.

All modules over fields in this paper will be assumed to be finite-dimensional, and modules for algebras are right modules unless otherwise specified.

2. Preliminaries

2.1 The Lie module Let r and n be positive integers where n ⩾ r. Let Δ be the free associative ring (ℤalgebra) on free generators x_{1}, …, x_{n} and let L be the Lie subring of Δ generated by x_{1}, …, x_{n}. By [?, chap. II, §3, Théorème 1] , L is free on x_{1}, …, x_{n}. Let Δ_{r} denote the homogeneous component of Δ of degree r. Then S_{r} has a left action by ‘place permutations’ on Δ_{r} given by α(y_{1} ⋯ y_{r}) = y_{1α} ⋯ y_{rα} for all α ∈ S_{r} and all y_{1}, …, y_{r} ∈ {x_{1}, …, x_{n}}. (Note that we write multiplication in S_{r} from left to right.) Hence Δ_{r} is a left ℤS_{r}-module. Let ω_{r} be the element of ℤS_{r} given by (1.1). Then it is well known and easily verified that ω_{r}(y_{1} ⋯ y_{r}) = [y_{1}, …, y_{r}] where [y_{1}, …, y_{r}] denotes the left-normed Lie product [⋯ [[y_{1}, y_{2}], y_{3}], …, y_{r}].

The group S_{r} also has a right action on Δ by automorphisms, where x_{i}α = x_{iα} for i = 1, …, r and x_{i}α = x_{i} for i > r. Thus Δ_{r} becomes a (ℤS_{r}, ℤS_{r})- bimodule. Let Δ_{r}^{0} be the ℤ-subspace of Δ_{r} spanned by the monomials x_{1α} ⋯ x_{rα} with α ∈ S_{r}. Thus Δ_{r}^{0} is a sub-bimodule of Δ_{r} and the map ξ : ℤS_{r} → Δ_{r}^{0} defined by ξ(α) = x_{1α} ⋯ x_{rα} is an isomorphism of bimodules.

Let L_{r}^{0} = L ∩ Δ_{r}^{0}. Then L_{r}^{0} is spanned over ℤ by all elements of the form [x_{1α}, …, x_{rα}]. Also, by [?, chap. II, §3, Théorème 2], L_{r}^{0} is free of rank (r − 1)! as a ℤ-module. We have ω_{r}Δ_{r}^{0} = L_{r}^{0}. Thus the isomorphism ξ maps ω_{r}ℤS_{r} to L_{r}^{0}, and so ω_{r}ℤS_{r} is isomorphic to L_{r}^{0} as a right ℤS_{r}-module.

All of the above still applies if ℤ is replaced by R, where R is an arbitrary commutative ring with unity. Also (using subscripts to show coefficient rings) we have R ⊗_{ℤ} L_{r,ℤ}^{0} ≅ L_{r,R}^{0}. The Lie module Lie_{R}(r) is the RS_{r}-module defined by Lie_{R}(r) = ω_{r}RS_{r}. Thus Lie_{R}(r) ≅ L_{r,R}^{0}. It follows that Lie_{R}(r) ≅ R ⊗_{ℤ} Lie_{ℤ}(r) and Lie_{R}(r) is free of rank (r − 1)! as an R-module. If F is a field we have

dim Lie_{F} (r) = (r − 1)!  (2.1)

and, if F is understood, we write Lie_{F} (r) as Lie(r).

When K is a field of characteristic zero there is a formula for the character ψ_{r} of Lie_{K}(r). Let μ denote the Möbius function and let σ be an r-cycle of S_{r}. For each divisor d of r let C_{d} denote the conjugacy class of σ^{r/d} in S_{r}. Then, for g ∈ S_{r},

ψ_{r}(g) = \{ \begin{matrix}μ(d)(r − 1)!/|C_{d}| & if g ∈ C_{d}, \\ 0 & if g ∉ C_{d} for all d. \end{matrix}  (2.2)

(See, for example, [?, Section 3.2].) Hence, if θ is any class function on S_{r} with values in K and we write

(θ, ψ_{r})_{S_{r}} = \frac{1}{|S_{r}|} ∑_{g∈S_{r}} θ(g)ψ_{r}(g^{−1}),  (2.3)

we have

(θ, ψ_{r})_{S_{r}} = \frac{1}{r} ∑_{d|r} μ(d)θ(σ^{r/d}).  (2.4)

2.2 Representations of S_{r} By a partition of a non-negative integer r we mean, as usual, a finite sequence λ = (λ_{1}, …, λ_{s}) of integers satisfying λ_{1} ⩾ ⋯ ⩾ λ_{s} > 0 and λ_{1} + ⋯ + λ_{s} = r. Here λ_{1}, …, λ_{s} are the ‘parts’ of λ. We write Λ^{+}(r) for the set of all partitions of r. If r = 0 then Λ^{+}(r) contains only the empty partition, which we denote by ∅. Let p be a prime number. A partition λ is ‘p-regular’ if λ does not have p or more equal parts, and we write Λ_{p}^{+}(r) for the set of all p-regular partitions of r. The ‘p-core’ of a partition λ of r is the partition λ̃ of r′, for some r′ ⩽ r, obtained from (the diagram of) λ by the removal of as many ‘rim p-hooks’ as possible: see [?, Section 2.7]. We write C_{r} for the set of all p-cores of elements of Λ^{+}(r).

Let r be a positive integer and let F be a field of prime characteristic p. The irreducible F S_{r}-modules may be labelled (up to isomorphism) as D^{λ} with λ ∈ Λ_{p}^{+}(r), where D^{λ} is a quotient module of the Specht module S^{λ} (see [?, 7.1.14]). Here D^{(r)} is isomorphic to the trivial F S_{r}-module F because S^{(r)} ≅ F. For each λ we write P^{λ} for the projective cover of D^{λ}. Thus the projective indecomposable F S_{r}-modules are the P^{λ} with λ ∈ Λ_{p}^{+}(r). If F′ is an extension field of F then (using subscripts to show coefficient fields) we have D_{F′}^{λ} ≅ F′ ⊗_{F} D_{F}^{λ} and P_{F′}^{λ} ≅ F′ ⊗_{F} P_{F}^{λ}.

We recall a few general facts about blocks. If Γ is a finite-dimensional F - algebra we may write Γ uniquely as a finite direct sum of indecomposable two-sided ideals, Γ = ⨁_{B∈B} Γ_{B}. These ideals are the ‘blocks’ of Γ, but it is convenient also to refer to the labels B as the blocks. The identity element of Γ may be written as ∑_{B∈B} e_{B}, with e_{B} ∈ Γ_{B} for all B. The elements e_{B} are the ‘block idempotents’: they are primitive central idempotents of Γ (see, for example, [?]). Any Γ-module V satisfying V e_{B} = V is said to belong to B, and we write V ∈ B. Every Γ-module V may be written uniquely in the form V = ⨁_{B∈B} V_{B}, where V_{B} ∈ B for all B (indeed, V_{B} = V e_{B}). We call V_{B} the block component of V corresponding to B.

By the ‘Nakayama Conjecture’ (see [?, 6.1.21]), the blocks of F S_{r} may be labelled B(ν) with ν ∈ C_{r} in such a way that S^{λ} ∈ B(λ̃) for all λ ∈ Λ_{p}^{+}(r). Since D^{λ} is a quotient of S^{λ} and P^{λ} is indecomposable with D^{λ} as a quotient, we have D^{λ}, P^{λ} ∈ B(λ̃). We use the same notation for the blocks of F S_{r} for every field F of characteristic p. By consideration of composition factors we see that, for any F S_{r}-module V, any extension field F′ of F, and any ν, we have

(F′ ⊗_{F} V)_{B(ν)} ≅ F′ ⊗_{F} V_{B(ν)}.  (2.5)

If B is a block and B = B(ν) we write B̃ = ν. Also, for λ ∈ Λ_{p}^{+}(r), we write λ ∈ B if D^{λ} ∈ B (or equivalently P^{λ} ∈ B). The ‘principal block’ is the block B_{0} containing the trivial irreducible D^{(r)}. Thus B̃_{0} = (\overline{r}), where \overline{r} denotes the remainder on dividing r by p. If B̃ = ∅ then p | r and \overline{r} = 0 so that B = B_{0}. Hence if B is non-principal we have B̃ ≠ ∅.

If p ∤ r then Lie(r) is projective (see Section 1). But if p | r and B̃ ≠ ∅ then B ≠ B_{0} and so Lie(r)_{B} is projective by [?, Theorem 3.1]. Hence we have the following result.

Theorem 2.1 [?]. If B is a block of F S_{r} such that B̃ ≠ ∅ then Lie(r)_{B} is projective.

As is well known, Brauer characters of F S_{r}-modules have integer values: this follows, for example, from [?, Chapter 3, Lemma 6.13]. (Consequently Brauer characters of F S_{r}-modules are uniquely defined and do not depend upon choices of roots of unity.) We regard Brauer characters as maps from S_{r} to ℤ by assigning the value zero to p-singular elements of S_{r}. For each λ ∈ Λ_{p}^{+}(r) we write β^{λ} and ζ^{λ} for the Brauer characters of D^{λ} and P^{λ}, respectively. By the orthogonality relations for Brauer characters (see [?, Chapter 3, Theorem 6.10]) we have

(β^{λ}, ζ^{ρ})_{S_{r}} = \{ \begin{matrix}1 & if λ = ρ, \\ 0 & if λ ≠ ρ, \end{matrix}  (2.6)

where (β^{λ}, ζ^{ρ})_{S_{r}} is defined as in (2.3).

2.3 Polynomial representations of GL_{n}(F) Suppose now that F is an infinite field of prime characteristic p and let n be a positive integer. We refer to [?] and [?] for background concerning polynomial GL_{n}(F) modules and the Schur algebra S_{F} (n, r). Let E denote the natural GL_{n}(F)-module. Thus E^{⊗r} is an S_{F} (n, r)-module. The symmetric group S_{r} has a left action on E^{⊗r} by place permutations. Hence E^{⊗r} may be regarded as a left F S_{r}-module. This left action of S_{r} commutes with the right action of GL_{n}(F), so E^{⊗r} is an (F S_{r}, S_{F} (n, r))-bimodule. If q and t are positive integers such that r = qt and if V is an S_{F} (n, t)-module then V^{⊗q} and L^{q}(V) are S_{F} (n, r)-modules.

Let Λ^{+}(n, r) denote the set of all partitions of r with at most n parts and let Λ_{p}^{+}(n, r) denote the set of all p-regular partitions in Λ^{+}(n, r). The irreducible S_{F} (n, r)-modules may be labelled L(λ) with λ ∈ Λ^{+}(n, r). For each such λ there is also an indecomposable S_{F} (n, r)-module T (λ) called a ‘tilting module’, and (see [?, Section 1.3]) there are non-negative integers n_{λ} such that

E^{⊗r} ≅ ⨁_{λ∈Λ_{p}^{+}(n,r)} n_{λ} T (λ).  (2.7)

The main facts about the blocks of S_{F} (n, r) were obtained in [?] and summarised in [?]. When n ⩾ r the blocks may be labelled B(ν) with ν ∈ C_{r} in such a way that L(λ) ∈ B(λ̃). If B is a block and B = B(ν) we write B̃ = ν. When n < r, p-cores do not necessarily label unique blocks, but if L(λ) and L(ρ) are in the same block then λ̃ = ρ̃. Thus, for each block B, there is an element B̃ of C_{r} (where B̃ has at most n parts) with the property that λ̃ = B̃ whenever L(λ) ∈ B. For each ν ∈ C_{r} we write B(ν) for the set of blocks B such that B̃ = ν. (Thus B(ν) is empty if ν has more than n parts.) If V is an S_{F} (n, r)-module we write V_{B(ν)} for the direct sum of the block components V_{B} of V corresponding to blocks B in B(ν). For all n and all λ ∈ Λ^{+}(n, r), T (λ) is indecomposable and has L(λ) as a composition factor (see [?, Section 1.3]); thus T (λ) and L(λ) belong to the same block. For a block B and λ ∈ Λ^{+}(n, r) we write λ ∈ B if L(λ) ∈ B (or equivalently T (λ) ∈ B). We define the principal block to be the block B_{0} containing L(λ) where λ = (r). Thus B̃_{0} = (\overline{r}), with \overline{r} as before. As in the case of F S_{r}, if n ⩾ r and B is non-principal then B̃ ≠ ∅.

Let T denote the class of all S_{F} (n, r)-modules that are isomorphic to direct sums of tilting modules T (λ) where λ ∈ Λ_{p}^{+}(n, r). Thus E^{⊗r} ∈ T by (2.7). If p ∤ r then L^{r}(E) is isomorphic to a direct summand of E^{⊗r} (see Section 1) and so L^{r}(E) ∈ T . But if p | r and B̃ ≠ ∅ then L^{r}(E)_{B} ∈ T by [?, Theorem 3.2]. Hence we have the following result.

Theorem 2.2 [?]. If B is a block of S_{F} (n, r) such that B̃ ≠ ∅ then L^{r}(E)_{B} ∈ T .

Suppose now that n_{1} and n_{2} are positive integers with n_{1} ⩾ n_{2} and let d_{n_{1},n_{2}} denote the functor from the category of S_{F} (n_{1}, r)-modules to the category of S_{F} (n_{2}, r)-modules described in [?, Section 6.5]. This functor is exact (in particular it preserves direct sums) and we call it ‘truncation’. Note that Λ^{+}(n_{2}, r) ⊆ Λ^{+}(n_{1}, r). We temporarily use subscripts to distinguish between modules for S_{F} (n_{1}, r) and S_{F} (n_{2}, r). Then, if λ ∈ Λ^{+}(n_{1}, r) and M(λ) denotes either L(λ) or T (λ), we have

d_{n_{1},n_{2}}(M_{n_{1}}(λ)) ≅ \{ \begin{matrix}M_{n_{2}}(λ) & if λ ∈ Λ^{+}(n_{2}, r), \\ 0 & otherwise. \end{matrix}  (2.8)

(For the case of L(λ) see [?, Section 6.5] and for T (λ) see [?, Section 1.7].)

Write d = d_{n_{1},n_{2}} and use the same notation for arbitrary r. Then, if q and t are positive integers and V is an S_{F} (n_{1}, t)-module, it is easy to check that d(V^{⊗q}) ≅ d(V)^{⊗q} and d(L^{q}(V)) ≅ L^{q}(d(V)). Furthermore, for all b ∈ F S_{t}, we have d(bE_{n_{1}}^{⊗t}) ≅ bE_{n_{2}}^{⊗t}. Also, if V is an S_{F} (n_{1}, r)-module and ν ∈ C_{r}, it follows from (2.8) that

d(V_{B(ν)}) ≅ d(V)_{B(ν)}.  (2.9)

2.4 The Schur functor We continue with the notation of Section 2.3 but now assume that n ⩾ r. The Schur functor f_{r} is an exact functor from the category of S_{F} (n, r)-modules to the category of F S_{r}-modules (see [?, Chapter 6]). If U is an S_{F} (n, r)-module then f_{r}(U) may be thought of as the weight space of U corresponding to the weight (1, …, 1, 0, …, 0), with r coordinates equal to 1, and the action of S_{r} on f_{r}(U) comes by taking S_{r} as a group of permutation matrices in GL_{n}(F) (see, for example, [?, Section 1.2]). It is easily seen that f_{r}(E^{⊗r}) ≅ F S_{r}. More generally, for all b ∈ F S_{r}, we have

f_{r}(bE^{⊗r}) ≅ bF S_{r}.  (2.10)

Let {e_{1}, …, e_{n}} be the standard basis of E. Then f_{r}(L^{r}(E)) is the subspace of L^{r}(E) spanned by the left-normed Lie products [e_{1α}, …, e_{rα}] with α ∈ S_{r}. In the notation of Section 2.1, f_{r}(L^{r}(E)) ≅ L_{r,F}^{0} . Thus, since L_{r,F}^{0} ≅ Lie(r), we obtain

f_{r}(L^{r}(E)) ≅ Lie(r).  (2.11)
For all λ ∈ Λ_{p}^{+}(n, r) = Λ_{p}^{+}(r), we have (see [?, Section 1.3])
f_{r}(T (λ)) ≅ P^{λ}.  (2.12)

As observed in [?], f_{r} sends modules in the S_{F} (n, r)-block B(ν) to modules in the F S_{r}-block B(ν) labelled by the same p-core. Thus, if V is any S_{F} (n, r)-module, we have

f_{r}(V_{B(ν)}) ≅ f_{r}(V)_{B(ν)}.  (2.13)

Suppose that q and t are positive integers such that r = qt. Write {1, …, r} as the disjoint union of subsets Ω_{1}, …, Ω_{q} of cardinality t. Specifically, take Ω_{i} = {(i − 1)t + j : j = 1, …, t} for i = 1, …, q. The subgroup S_{t}^{(q)} of S_{r} consisting of those permutations fixing each Ω_{i} setwise is a Young subgroup isomorphic to S_{t} × ⋯ × S_{t}. For each α ∈ S_{q} let α^{∗} denote the element of S_{r} defined by ((i−1)t+j)α^{∗} = (iα−1)t+j for i = 1, …, q and j = 1, …, t. Thus Ω_{i}α^{∗} = Ω_{iα} for i = 1, …, q. The set {α^{∗} : α ∈ S_{q}} is a subgroup S_{q}^{∗} of S_{r} isomorphic to S_{q}. It is the ‘diagonal’ of the Young subgroup afforded by the sets Σ_{1}, …, Σ_{t}, where Σ_{j} = {(i − 1)t + j : i = 1, …, q} for j = 1, …, t. The subgroup S_{t}^{(q)}S_{q}^{∗} of S_{r} is isomorphic to the wreath product S_{t} wr S_{q}. In the case where q = k and t = p^{m} we have subgroups S_{p^{m}}^{(k)} and S_{k}^{∗}, where S_{k}^{∗} is the diagonal of S_{k} ×⋯×S_{k}, as considered in Section 1.

We use the above notation to state two useful results. Let V be an S_{F} (n, t)- module. Then f_{t}(V) is an F S_{t}-module, so f_{t}(V)^{⊗q} may be regarded as an F S_{t}^{(q)}- module. The first result we need is a special case of [?, Section 2.5, Lemma].

Lemma 2.3 [?]. In the above notation,

f_{r}(V^{⊗q}) ≅ (f_{t}(V)^{⊗q})↑_{S_{t}^{(q)}}^{S_{r}} .

The second result we need is part of [?, Corollary 3]. Observe that f_{t}(V)^{⊗q} may be regarded as an F S_{t}^{(q)}S_{q}^{∗}-module, by means of the action of S_{q}^{∗} that permutes the tensor factors. Also, Lie(q) may be regarded as an F S_{t}^{(q)}S_{q}^{∗}-module, by taking trivial action of S_{t}^{(q)}.

Lemma 2.4 [?]. In the above notation,

f_{r}(L^{q}(V)) ≅ (f_{t}(V)^{⊗q} ⊗ Lie(q))↑_{S_{t}^{(q)}S_{q}^{∗}}^{S_{r}} .

2.5 Decompositions We next summarise [?, Theorems 3.4 and 3.6]: these results are based on results from [?] and [?], particularly the Decomposition Theorem of [?]. If F is a field, G is a group and V is an F G-module, then we regard V^{⊗r} as an (F S_{r}, F G)- bimodule, where S_{r} acts by place permutations. Also we regard any field of prime characteristic p as an extension field of the field 𝔽_{p} with p elements.

Theorem 2.5 [?]. Let p be a prime number. For each positive integer t there is an idempotent b_{t} of 𝔽_{p}S_{t} such that the idempotents b_{t} have the following properties. Let F be a field of characteristic p, G a group and V an F G-module. Let r be a positive integer and write r = p^{m}k where m ⩾ 0, k ⩾ 1, and p ∤ k. Then

L^{r}(V) ≅ ⨁_{i=0}^{m} L^{p^{m−i}}(b_{p^{i}k}V^{⊗p^{i}k})  (2.14)

and

L^{k}(V^{⊗p^{m}}) ≅ ⨁_{i=0}^{m} p^{i} (b_{p^{i}k}V^{⊗p^{i}k})^{⊗p^{m−i}}.  (2.15)

Let F be any field of characteristic p and let r = qt. We take S_{t}^{(q)} and S_{q}^{∗} as in Section 2.4. Then (b_{t}F S_{t})^{⊗q} may be regarded either as an F S_{t}^{(q)}-module or an F S_{t}^{(q)}S_{q}^{∗}-module. As in Section 2.4 we regard Lie(q) as an F S_{t}^{(q)}S_{q}^{∗}-module. Thus we may define F S_{r}-modules X(q, t) and Y (q, t) by

X(q, t) = ((b_{t}F S_{t})^{⊗q} ⊗ Lie(q))↑_{S_{t}^{(q)}S_{q}^{∗}}^{S_{r}} , Y (q, t) = ((b_{t}F S_{t})^{⊗q})↑_{S_{t}^{(q)}}^{S_{r}} .

Now write r = qt = p^{m}k where m ⩾ 0 and p ∤ k. Assume that F is infinite and n ⩾ r. Then, by (2.10) and Lemmas 2.3 and 2.4, we have

f_{r}(L^{q}(b_{t}E^{⊗t})) ≅ X(q, t)  (2.16)

and

f_{r}((b_{t}E^{⊗t})^{⊗q}) ≅ Y (q, t).  (2.17)

By (2.10) and Lemma 2.4,

f_{r}(L^{k}(E^{⊗p^{m}})) ≅ ((F S_{p^{m}})^{⊗k} ⊗ Lie(k))↑_{S_{p^{m}}^{(k)}S_{k}^{∗}}^{S_{r}} .

Clearly (F S_{p^{m}})^{⊗k} is a transitive permutation module under the action of S_{p^{m}}^{(k)}S_{k}^{∗} and the stabiliser of the basis element 1 ⊗ ⋯ ⊗ 1 is S_{k}^{∗}. Thus (F S_{p^{m}})^{⊗k} is induced from a one-dimensional trivial module for F S_{k}^{∗} and (using [?, Proposition 3.3.3 (i)]) we obtain

(F S_{p^{m}})^{⊗k} ⊗ Lie(k) ≅ Lie(k)↑_{S_{k}^{∗}}^{S_{p^{m}}^{(k)}S_{k}^{∗}} .

Hence

f_{r}(L^{k}(E^{⊗p^{m}})) ≅ Lie(k)↑_{S_{k}^{∗}}^{S_{r}} .  (2.18)

Corollary 2.6. Let F be a field of prime characteristic p. Let r be a positive integer and write r = p^{m}k where m ⩾ 0, k ⩾ 1, and p ∤ k. Then

Lie(r) ≅ ⨁_{i=0}^{m} X(p^{m−i}, p^{i}k)  (2.19)

and

Lie(k)↑_{S_{k}^{∗}}^{S_{r}} ≅ ⨁_{i=0}^{m} p^{i} Y (p^{m−i}, p^{i}k).  (2.20)

Proof. We first assume that F is infinite. Take n ⩾ r and consider Theorem 2.5 in the case where G = GL_{n}(F) and V = E. We apply the Schur functor. By (2.14), with the use of (2.11) and (2.16), we obtain (2.19). Similarly, by (2.15), with (2.18) and (2.17), we obtain (2.20). Now let F be arbitrary and let F′ be an infinite extension field of F. Each of the modules in (2.19) and (2.20) taken over F′ is obtained by tensoring with F′ the corresponding module over F. Since (2.19) and (2.20) hold over F′ they hold over F by the Noether–Deuring theorem. □

3. Main results

Our main results depend on the following result, to be proved in Section 4. Recall from Section 2.2 that if B is a non-principal block of F S_{r} then B̃ ≠ ∅.

Theorem 3.1. Let F be a field of prime characteristic p. Let q and t be positive integers, where q is a power of p, and let r = qt. Let B be a block of F S_{r} such that B̃ ≠ ∅. Then, for the modules defined in Section 2.5,

X(q, t)_{B} ≅ \frac{1}{q} Y (q, t)_{B}.

In the rest of this section we show how Theorem 3.1 can be used to obtain our main results about Lie modules and Lie powers.

Theorem 3.2. Let F be a field of prime characteristic p. Let r be a positive integer and write r = p^{m}k where m ⩾ 0, k ⩾ 1, and p ∤ k. Let B be a block of F S_{r} such that B̃ ≠ ∅ and let S_{k}^{∗} be the diagonal of a Young subgroup S_{k} × ⋯ × S_{k} of S_{r}. Then

Lie(r)_{B} ≅ \frac{1}{p^{m}} (Lie(k)↑_{S_{k}^{∗}}^{S_{r}})_{B}.

Proof. By (2.19) and Theorem 3.1,

Lie(r)_{B} ≅ ⨁_{i=0}^{m} X(p^{m−i}, p^{i}k)_{B} ≅ ⨁_{i=0}^{m} \frac{1}{p^{m−i}} Y (p^{m−i}, p^{i}k)_{B}.

Thus the result follows by (2.20). □

Note that Lie(k)↑_{S_{k}^{∗}}^{S_{r}} is projective since Lie(k) is projective (see Section 1).

Corollary 3.3. If B_{0} is the principal block of F S_{r} then

dim Lie(r)_{B_{0}} = \frac{1}{p^{m}} dim (Lie(k)↑_{S_{k}^{∗}}^{S_{r}})_{B_{0}} .

Proof. For each non-principal block B of F S_{r} we have

dim Lie(r)_{B} = \frac{1}{p^{m}} dim (Lie(k)↑_{S_{k}^{∗}}^{S_{r}})_{B} ,

by Theorem 3.2. However, by (2.1),

dim Lie(k)↑_{S_{k}^{∗}}^{S_{r}} = (k − 1)! r!/k! = p^{m}(r − 1)! = p^{m} dim Lie(r).

The result follows. □

Theorem 3.4. In the notation of Theorem 3.2, we have

Lie(r)_{B} ≅ ⨁_{\begin{matrix}λ∈Λ_{p}^{+}(r) \\ λ∈B \end{matrix}} m_{λ} P^{λ},  (3.1)

where, for each λ,

m_{λ} = \frac{1}{r} ∑_{d|k} μ(d) β^{λ}(τ^{k/d}),  (3.2)

where τ is an element of S_{r} of cycle type (k, k, …, k) and β^{λ} denotes the Brauer character of D^{λ}.

Proof. By Theorem 2.1, Lie(r)_{B} is projective. Thus it satisfies (3.1) for suitable non-negative integers m_{λ}. It remains to prove (3.2). If F′ is an extension field of F then Lie_{F′}(r) ≅ F′ ⊗ Lie_{F} (r) and P_{F′}^{λ} ≅ F′ ⊗ P_{F}^{λ}. Also, block components are preserved under field extensions, by (2.5). Hence it suffices to prove the result for the field 𝔽_{p} and then, by the same argument, it suffices to prove the result for any chosen field F of characteristic p. We choose F so that there is a p-modular system (K, R, F) with the properties specified in [?, Chapter 3, Section 6]. Note, in particular, that K has characteristic 0 and contains sufficient roots of unity, K is the field of fractions of R, and F = R/(π) where (π) is the maximal ideal of R.

Since Lie(k)↑_{S_{k}^{∗}}^{S_{r}} is projective we have

Lie(k)↑_{S_{k}^{∗}}^{S_{r}} ≅ ⨁_{ρ∈Λ_{p}^{+}(r)} m′_{ρ} P^{ρ}

for suitable non-negative integers m′_{ρ}. Let λ ∈ Λ_{p}^{+}(r) where λ ∈ B. By Theorem 3.2 we have m_{λ} = (1/p^{m})m′_{λ}. Let ϕ denote the Brauer character of Lie(k)↑_{S_{k}^{∗}}^{S_{r}}. By the orthogonality relation (2.6) we have

m_{λ} = \frac{1}{p^{m}} m′_{λ} = \frac{1}{p^{m}} (β^{λ}, ϕ)_{S_{r}}.

By the description of the Lie module in Section 2.1, Lie(k) ≅ F ⊗_{R}Lie_{R}(k) and Lie_{K}(k) ≅ K ⊗_{R} Lie_{R}(k). Let M = Lie_{R}(k)↑_{S_{k}^{∗}}^{S_{r}}. Then Lie(k)↑_{S_{k}^{∗}}^{S_{r}} ≅ F ⊗_{R} M and Lie_{K}(k)↑_{S_{k}^{∗}}^{S_{r}} ≅ K ⊗_{R} M. The character of Lie_{K}(k)↑_{S_{k}^{∗}}^{S_{r}} is ψ_{k}↑_{S_{k}^{∗}}^{S_{r}}, where ψ_{k} denotes the character of Lie_{K}(k) as a KS_{k}^{∗}-module. By [?, Chapter 3, Lemma 6.4], ϕ and ψ_{k}↑_{S_{k}^{∗}}^{S_{r}} take the same value on p′-elements of S_{r}. Thus, by Frobenius reciprocity,

m_{λ} = \frac{1}{p^{m}} (β^{λ}, ψ_{k}↑_{S_{k}^{∗}}^{S_{r}})_{S_{r}} = \frac{1}{p^{m}} (β^{λ}↓_{S_{k}^{∗}}^{S_{r}}, ψ_{k})_{S_{k}^{∗}}.

Let τ be as in the statement of the theorem. Then τ is conjugate to, and therefore can be taken to be, an element σ^{∗} of S_{k}^{∗} corresponding to a k-cycle σ in S_{k}. Thus, by (2.4), we have

m_{λ} = \frac{1}{p^{m}} (β^{λ}↓_{S_{k}^{∗}}^{S_{r}}, ψ_{k})_{S_{k}^{∗}} = \frac{1}{p^{m}k} ∑_{d|k} μ(d) β^{λ}(τ^{k/d}).

The result follows. □

We now turn to Lie powers and, for the rest of this section, we assume that F is infinite. As before, let n be a positive integer and let E be the natural GL_{n}(F)-module. We first obtain an analogue of Theorem 3.1.

Theorem 3.5. Let F be an infinite field of prime characteristic p. Let q and t be positive integers, where q is a power of p, and let r = qt. Let B be a block of S_{F} (n, r) such that B̃ ≠ ∅. Then, for the idempotent b_{t} of Theorem 2.5,

L^{q}(b_{t}E^{⊗t})_{B} ≅ \frac{1}{q} ((b_{t}E^{⊗t})^{⊗q})_{B}.  (3.3)

Proof. Let T be as defined in Section 2.3. Thus, by Theorem 2.2, L^{r}(E)_{B} ∈ T . Hence, by (2.14), L^{q}(b_{t}E^{⊗t})_{B} ∈ T . Also, since (b_{t}E^{⊗t})^{⊗q} is a direct summand of E^{⊗r}, we have (b_{t}E^{⊗t})^{⊗q} ∈ T , by (2.7).

Suppose first that n ⩾ r. Then we may write B = B(ν) where ν ≠ ∅. By (2.16) and (2.13),

f_{r}(q L^{q}(b_{t}E^{⊗t})_{B(ν)}) ≅ q X(q, t)_{B(ν)}.

Similarly, by (2.17) and (2.13),

f_{r}(((b_{t}E^{⊗t})^{⊗q})_{B(ν)}) ≅ Y (q, t)_{B(ν)}.

Also, by Theorem 3.1, q X(q, t)_{B(ν)} ≅ Y (q, t)_{B(ν)}. It follows from (2.12) that if U, V ∈ T and f_{r}(U) ≅ f_{r}(V) then U ≅ V. Hence (3.3) holds when n ⩾ r.

Now suppose that n < r and let B̃ = ν. Thus B ∈ B(ν). Consider the S_{F} (r, r)-block B(ν). By the first case, there is an isomorphism of S_{F} (r, r)-modules,

L^{q}(b_{t}E^{⊗t})_{B(ν)} ≅ \frac{1}{q}((b_{t}E^{⊗t})^{⊗q})_{B(ν)}.  (3.4)

We apply truncation d_{r,n} to (3.4). By (2.9) and the other properties of d_{r,n} given in Section 2.3, we obtain (3.4) for S_{F} (n, r)-modules. Hence the corresponding block components are isomorphic for all S_{F} (n, r)-blocks in B(ν) and we obtain (3.3). □

We shall obtain our main results on Lie powers from Theorem 3.5.

Theorem 3.6. Let F be an infinite field of prime characteristic p. Let r be a positive integer and write r = p^{m}k where m ⩾ 0, k ⩾ 1, and p ∤ k. Let B be a block of S_{F} (n, r) such that B̃ ≠ ∅. Then

L^{r}(E)_{B} ≅ \frac{1}{p^{m}} L^{k}(E^{⊗p^{m}})_{B}.

Proof. This follows from Theorem 3.5 in the same way as Theorem 3.2 follows from Theorem 3.1, using Theorem 2.5 instead of Corollary 2.6. □

Theorem 3.7. In the notation of Theorem 3.6, we have

L^{r}(E)_{B} ≅ ⨁_{\begin{matrix}λ∈Λ_{p}^{+}(n,r) \\ λ∈B \end{matrix}} m_{λ} T (λ),

where m_{λ} is given by (3.2).

Proof. By Theorem 2.2, L^{r}(E)_{B} ∈ T . Thus L^{r}(E)_{B} is isomorphic to a direct sum of tilting modules T (λ) with λ ∈ Λ_{p}^{+}(n, r) and λ ∈ B. Let B̃ = ν. Then, for n ⩾ r, we have f_{r}(L^{r}(E)_{B(ν)}) ≅ Lie(r)_{B(ν)}, by (2.11) and (2.13), and f_{r}(T (λ)) ≅ P^{λ} for all λ ∈ Λ_{p}^{+}(n, r), by (2.12). Thus, for n ⩾ r, the result is given by Theorem 3.4. For n < r the result follows by truncation, as in the proof of Theorem 3.5. (Note that the effect of truncation on T (λ) is given by (2.8).) □

4. Proof of Theorem 3.1

Let F, p, r, q and t be as in the statement of Theorem 3.1, where r = qt and q is a power of p. Write r = qt = p^{m}k where m ⩾ 0 and p ∤ k. Also, within the symmetric group S_{r}, write A = S_{t}^{(q)} and H = AS_{q}^{∗} = S_{t}^{(q)}S_{q}^{∗}. Thus A ≅ S_{t} × ⋯ × S_{t} (with q factors). We regard (b_{t}F S_{t})^{⊗q} as an F A-module or an F H-module and Lie(q) as an F H-module. Write X = X(q, t) and Y = Y (q, t).

Thus X = ((b_{t}F S_{t})^{⊗q} ⊗ Lie(q))↑_{H}^{S_{r}} and Y = ((b_{t}F S_{t})^{⊗q})↑_{A}^{S_{r}}. If F′ is an extension field of F and B is a block of F S_{r} then, by (2.5), (X_{F′})_{B} ≅ F′ ⊗ (X_{F})_{B} and (Y_{F′})_{B} ≅ F′ ⊗ (Y_{F})_{B}. Hence it suffices to prove Theorem 3.1 for 𝔽_{p} and then, by the Noether–Deuring theorem, it suffices to prove the theorem for any chosen field F of characteristic p. We take F from the p-modular system (K, R, F) used in the proof of Theorem 3.4, where F = R/(π).

We state some standard facts associated with p-modular systems in order to establish terminology and notation. If G is any finite group then the natural epimorphism R → F yields an epimorphism RG → F G. If this epimorphism maps u to v, where u ∈ RG and v ∈ F G, we say that v ‘lifts’ to u. By an ‘RG-lattice’ we mean an RG-module that is free of finite rank as an R-module. If M is an RG-lattice we write \overline{M} = M/πM. Thus \overline{M} ≅ F ⊗_{R} M and \overline{M} has the structure of an F G-module. An F G-module V is said to be ‘liftable’ if there exists M such that \overline{M} ≅ V, in which case we say that V lifts to M. If M is an RG-lattice then K ⊗_{R} M is a KG-module. If U is any KG-module then there is an RG-lattice M such that U ≅ K ⊗_{R} M (see [?, Lemma 1.9.1]) and we say that \overline{M} is obtained from U by ‘modular reduction’.

By a standard result (see [?, Theorem 1.9.4]), each block idempotent e_{B} of F S_{r} can be lifted to an element ê_{B} of RS_{r} to obtain pairwise-orthogonal primitive central idempotents of RS_{r} summing to the identity. If M is an RS_{r}-lattice such that Mê_{B} = M we write M ∈ B. Every RS_{r}-lattice M may be written uniquely in the form M = ⨁_{B} M_{B} where, for each B, M_{B} is an RS_{r}-lattice belonging to B. Similar facts and notation apply to KS_{r}-modules, using the same idempotents ê_{B}. If U is a KS_{r}-module then, since K has characteristic zero, U_{B} is a direct sum of irreducible KS_{r}-modules belonging to B.

It is easily verified that if M is an RS_{r}-lattice and B is a block then

(K ⊗_{R} M)_{B} ≅ K ⊗_{R} M_{B} and \overline{M_{B}} ≅ \overline{M}_{B}.  (4.1)

Hence, if V is an F S_{r}-module that lifts to M then V_{B} lifts to M_{B}.

Let σ^{∗} be an element of S_{q}^{∗} corresponding to a q-cycle σ of S_{q} (thus σ^{∗} has cycle type (q, q, …, q) as an element of S_{r}) and recall that A = S_{t}^{(q)}.

Lemma 4.1. If g is an element of S_{q}^{∗} conjugate in S_{q}^{∗} to a non-trivial element of the cyclic subgroup ⟨σ^{∗}⟩ and if χ is the character of an irreducible KS_{r}-module U belonging to a block B such that B̃ ≠ ∅ then χ(ag) = 0 for all a ∈ A.

Proof. We refer to the notation Ω_{1}, …, Ω_{q} introduced in Section 2.4. Let g′ be a non-trivial element of ⟨σ^{∗}⟩. Then Ω_{i}g′ ≠ Ω_{i} for all i, since no non-trivial element of ⟨σ⟩ has fixed points in {1, …, q}. If a′ ∈ A then Ω_{i}a′g′ = Ω_{i}g′ ≠ Ω_{i} for all i. Thus a′g′ has no fixed points in {1, …, r}.

Let M be an RS_{r}-lattice such that U ≅ K ⊗_{R} M. Since U belongs to B it follows from (4.1) that M belongs to B. Let D be the defect group of the F S_{r}- block B (see [?, Section 6.1]). (Thus D is a p-group, determined up to conjugacy in S_{r}.) By [?, Corollary 6.1.3], D is also the defect group of B regarded as a block of RS_{r}. Thus, by [?, Proposition 6.1.2], M is projective relative to D.

Let B = B(ν) where ν ∈ C_{r}. Thus ν ≠ ∅ and so ν is a partition of r′ for some r′ satisfying 0 < r′ ⩽ r. It follows from [?, 6.2.45] that D can be taken to be a Sylow p-subgroup of a subgroup S_{r−r′} of S_{r} fixing r′ points of {1, …, r}. Hence every element of D fixes some point of {1, …, r}.

Let g be as in the statement of the lemma and let a ∈ A. Then ag is conjugate in H to an element a_{1}g_{1} where a_{1} ∈ A and g_{1} is a non-trivial element of ⟨σ^{∗}⟩. Since χ is constant on conjugacy classes we may therefore assume that g is a non-trivial element of ⟨σ^{∗}⟩. The p-part of ag has the form (ag)^{l}, where p ∤ l, and we have (ag)^{l} = a′g^{l} where a′ ∈ A and where g^{l} is non-trivial because g has order a power of p. Thus, as we saw above, a′g^{l} has no fixed points in {1, …, r}. It follows that the p-part of ag is not conjugate in S_{r} to an element of D. Therefore, by [?, Chapter 4, Theorem 7.4], we have χ(ag) = 0, as required. □

Recall that b_{t} is an idempotent of F S_{t}. Thus (by [?, Theorem 1.9.4]) b_{t} lifts to an idempotent b̂_{t} of RS_{t}. Hence b_{t}F S_{t} lifts to the RS_{t}-lattice b̂_{t}RS_{t} and (b_{t}F S_{t})^{⊗q} lifts to the RH-lattice (b̂_{t}RS_{t})^{⊗q}. Also, Lie(q) lifts to the RH-lattice Lie_{R}(q). It follows that X and Y lift to the RS_{r}-lattices X_{R} and Y_{R}, respectively, where

X_{R} = ((b̂_{t}RS_{t})^{⊗q} ⊗ Lie_{R}(q))↑_{H}^{S_{r}}, Y_{R} = ((b̂_{t}RS_{t})^{⊗q})↑_{A}^{S_{r}} .

We write X_{K} = K ⊗X_{R} and Y_{K} = K ⊗Y_{R}. Thus X and Y are modular reductions of X_{K} and Y_{K}, respectively. Also

X_{K} ≅ ((b̂_{t}KS_{t})^{⊗q} ⊗ Lie_{K}(q))↑_{H}^{S_{r}}, Y_{K} ≅ ((b̂_{t}KS_{t})^{⊗q})↑_{A}^{S_{r}} .

Lemma 4.2. For any block B such that B̃ ≠ ∅, we have q (X_{K})_{B} ≅ (Y_{K})_{B}.

Proof. Let ψ_{q} denote the character of the KH-module Lie_{K}(q) and let θ denote the character of the KH-module (b̂_{t}KS_{t})^{⊗q}. Thus the KS_{r}-modules X_{K} and Y_{K} have characters (ψ_{q}θ)↑_{H}^{S_{r}} and θ↑_{A}^{S_{r}}, respectively. In order to prove the lemma it suffices to show that the multiplicity of each irreducible KS_{r}-module U belonging to B is the same in q X_{K} as in Y_{K}. Let χ be the character of U. By the orthogonality relations for ordinary characters, it suffices to prove

q ((ψ_{q}θ)↑_{H}^{S_{r}}, χ)_{S_{r}} = (θ↑_{A}^{S_{r}}, χ)_{S_{r}}.  (4.2)

By Frobenius reciprocity, ((ψ_{q}θ)↑_{H}^{S_{r}}, χ)_{S_{r}} = (ψ_{q}θ, χ↓_{H}^{S_{r}})_{H}. Hence

((ψ_{q}θ)↑_{H}^{S_{r}}, χ)_{S_{r}} = \frac{1}{|H|} ∑_{h∈H} ψ_{q}(h)θ(h)χ(h^{−1}) = \frac{1}{|H|} ∑_{\begin{matrix}a∈A \\ g∈S_{q}^{∗} \end{matrix}} ψ_{q}(ag)θ(ag)χ(g^{−1}a^{−1}).

Note that ψ_{q}(ag) = ψ_{q}(g) and, by (2.2), ψ_{q}(g) = 0 unless g is conjugate in S_{q}^{∗} to an element of ⟨σ^{∗}⟩. Also, χ(g^{−1}a^{−1}) = χ(a_{1}^{−1}g^{−1}) where a_{1} = a^{g} ∈ A. Hence, by Lemma 4.1, χ(g^{−1}a^{−1}) = 0 if g is conjugate to a non-trivial element of ⟨σ^{∗}⟩. Thus

((ψ_{q}θ)↑_{H}^{S_{r}}, χ)_{S_{r}} = \frac{1}{|H|} ∑_{a∈A} ψ_{q}(1)θ(a)χ(a^{−1}).

However, ψ_{q}(1) = dim Lie_{K}(q) = (q − 1)! and |H| = |A|q!. Thus

q ((ψ_{q}θ)↑_{H}^{S_{r}}, χ)_{S_{r}} = \frac{1}{|A|} ∑_{a∈A} θ(a)χ(a^{−1}) = (θ↓_{A}^{H}, χ↓_{A}^{S_{r}})_{A} = (θ↑_{A}^{S_{r}}, χ)_{S_{r}}.

This gives (4.2). □

We can now prove Theorem 3.1. Let B be a block such that B̃ ≠ ∅. Since qX and Y lift to qX_{R} and Y_{R}, respectively, it follows by (4.1) that q X_{B} and Y_{B} lift to q (X_{R})_{B} and (Y_{R})_{B}, respectively. Also, by (4.1), K ⊗ q(X_{R})_{B} ≅ q(X_{K})_{B} and K ⊗ (Y_{R})_{B} ≅ (Y_{K})_{B}. Thus q X_{B} and Y_{B} are modular reductions of q (X_{K})_{B} and (Y_{K})_{B}, respectively. However, by Lemma 4.2, these two last modules are isomorphic. Therefore, by [?, Chapter 3, Lemma 6.4], q X_{B} and Y_{B} have the same Brauer character.

By Theorem 2.1, Lie(r)_{B} is projective. Therefore, by (2.19), X_{B} is projective. Since b_{t}F S_{t} is a projective F S_{t}-module, (b_{t}F S_{t})^{⊗q} is a projective F A-module. Hence Y is a projective F S_{r}-module, and so Y_{B} is projective. Thus q X_{B} and Y_{B} are projective modules with the same Brauer characters. Therefore, by [?, Corollary 5.3.6], these modules are isomorphic. This proves Theorem 3.1.

References

[1] D. J. Benson, Representations and cohomology I , Cambridge University Press, Cambridge, 1995.

[2] N. Bourbaki, Groupes et algèbres de Lie, chapitres 2 et 3, Hermann, Paris, 1972.

[3] R. M. Bryant, “Lie powers of infinite-dimensional modules”, Beiträge zur Algebra und Geometrie 50 (2009), 179–193.

[4] R. M. Bryant and M. Schocker, “The decomposition of Lie powers”, Proc. London Math. Soc. (3) 93 (2006), 175–196.

[5] R. M. Bryant and M. Schocker, “Factorisation of Lie resolvents”, J. Pure Appl. Algebra 208 (2007), 993–1002.

[6] S. Donkin, “On Schur algebras and related algebras, IV. The blocks of the Schur algebras”, J. Algebra 168 (1994), 400–429.

[7] S. Donkin and K. Erdmann, “Tilting modules, symmetric functions, and the module structure of the free Lie algebra”, J. Algebra 203 (1998), 69–90.

[8] K. Erdmann, “Symmetric groups and quasi-hereditary algebras”, pp. 123–161 in Finite dimensional algebras and related topics, edited by V. Dlab and L. L. Scott, Kluwer, Dordrecht, 1994.

[9] K. Erdmann and K. M. Tan, “The non-projective part of the Lie module for the symmetric group”, Arch. Math. (to appear). arXiv 1101.0254v1

[10] J. A. Green, Polynomial representations of GL_{n}, Lecture Notes in Mathematics 830, Springer, Berlin, 1980.

[11] G. James and A. Kerber, “The Representation Theory of the Symmetric Group”, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley, Reading, 1981.

[12] K. J. Lim and K. M. Tan, “The Schur functor on tensor powers”, preprint, 2011. arXiv 1102.4157v2

[13] H. Nagao and Y. Tsushima, Representations of finite groups, Academic Press, San Diego, 1989.

roger.bryant@manchester.ac.uk School of Mathematics, University of Manchester,

Manchester, M13 9PL, United Kingdom

erdmann@maths.ox.ac.uk  Mathematical Institute, University of Oxford,

Oxford, OX1 3LB, United Kingdom