REPRESENTATIONS OF FINITE SPECIAL LINEAR GROUPS IN NON-DEFINING CHARACTERISTIC

ALEXANDER S. KLESHCHEV AND PHAM HUU TIEP

1. Introduction

1.1. Background. Let q = p^{f} be a power of a prime number p, and SL_{n}(q) = SL_{n}(𝔽_{q}) be the special linear group over the field 𝔽_{q} with q elements. Let 𝔽 be an algebraically closed field of characteristic ℓ ≥ 0. We are interested in parametrizing irreducible representations of SL_{n}(q) over 𝔽. If ℓ = p, this is well known: a parametrization is given by the so-called q-restricted dominant weights, see for example [25, Theorem 43].

Assume from now on that ℓ ≠ p. Then it seems natural to proceed through the general linear group GL_{n}(q) = GL_{n}(𝔽_{q}). Indeed, a natural classification of irreducible 𝔽GL_{n}(q)-modules is available [7, 8, 9, 17], and one might try to study irreducible 𝔽SL_{n}(q)-modules by restricting from GL_{n}(q) to SL_{n}(q). A parametrization of complex irreducible characters of SL_{n}(q) along these lines was obtained in [22] (see also [3]).

Restricting irreducible modules from a finite group G to a normal subgroup S is the subject of Clifford theory. The first key general fact is that the restriction V ↓_{S} of an irreducible 𝔽G-module V is completely reducible. The situation is especially nice when G/S is cyclic, e.g. GL_{n}(q)/SL_{n}(q) ≅ C_{q−1}. In this case the restriction V ↓_{S} is also multiplicity-free. Clifford theory is most effective if G/S is a cyclic group of order prime to ℓ. So the problem of classifying irreducible 𝔽SL_{n}(q)-modules is easy if ℓ |̸(q − 1). An additional consideration shows that the problem is still easy if ℓ |̸ gcd(n, q − 1).

However, in the difficult case where ℓ| gcd(n, q − 1) there are no general tools to describe the number of irreducible summands in the restriction V ↓_{SL_{n}(q)} of an irreducible 𝔽GL_{n}(q)-module V. The main results of this paper are a precise description of this number and a corresponding parametrization of irreducible 𝔽SL_{n}(q)-modules.

We note that even a classification of irreducible 𝔽GL_{n}(q)-modules which are irreducible on restriction to SL_{n}(q), which is of course a special case of our problem, seems to have been unknown. This question is important for Aschbacher-Scott program [1, 23] on classifying maximal subgroups in finite classical groups and has been our original motivation, see [21]. Although representations of SL_{n}(q) in non-defining characteristic were studied in [15] using a different approach, it is not clear how to use [15] to describe irreducible restrictions from GL_{n}(q) to SL_{n}(q).

As another application of our results, we classify for the first time complex representations of SL_{n}(q) whose reductions modulo ℓ are irreducible, relying on a similar result for GL_{n}(q) from [19] (as above, we are assuming ℓ |̸q).

1991 Mathematics Subject Classification: 20C20, 20E28, 20G40.

Research supported by the NSF (grants DMS-0654147 and DMS-0600967). This work was completed when both authors visited MSRI, whom we thank for support.

For a finite group G, let IBr(G) (resp. Irr(G)) denote the set of isomorphism classes of irreducible 𝔽G-modules (resp. ℂG-modules) or the set of irreducible ℓ-Brauer characters (resp. complex characters), depending on the context. As a final application, we exhibit an explicit subset of Irr(SL_{n}(q)) of size |IBr(SL_{n}(q))| and a partial order on IBr(SL_{n}(q)) such that the corresponding decomposition submatrix is lower unitriangular. Combining this with a recent result of Bonnafé [3], we get a parametrization of irreducible 𝔽SL_{n}(q)-modules. This also yields an ordinary basic set for each ℓ-block of SL_{n}(q).

We outline an analogy with the alternating and symmetric groups A_{n}◁S_{n}. There are two cases to consider when one studies irreducible 𝔽A_{n}-modules via restrictions of irreducible 𝔽S_{n}-modules. The case ℓ = 2, where Clifford theory is of little help, was treated in [2] using double covers of S_{n}. In the case ℓ ≠ 2, Clifford theory tells us that an irreducible 𝔽S_{n}-module V splits into two components or remains irreducible on restriction to A_{n}, depending on whether V is isomorphic to V ⊗ sgn or not, where sgn is the sign representation of S_{n}. The first explicit description of V ⊗ sgn in terms of partitions has been found in [20, 14]. As for the problem of irreducible reductions modulo ℓ, we refer the reader to [12] and references therein (for alternating groups a complete classification is still only conjectural).

1.2. Statement of the main results. For a partition λ = (λ_{1}, λ_{2}, …), denote |λ| := λ_{1} + λ_{2} + …, and write λ′ for the transposed partition. Set Δ(λ) := gcd(λ_{1}, λ_{2}, …). For a multipartition \underline{λ} = (λ^{(1)}, λ^{(2)}, …, λ^{(a)}) (which means that each λ^{(i)} is a partition) we write \underline{λ}′ for the transposed multipartition ((λ^{(1)})′, …, (λ^{(a)})′), and set

Δ(\underline{λ}) := gcd(Δ(λ^{(1)}), …, Δ(λ^{(a)})).

For σ ∈ 𝔽̄_{q} we denote by [σ] the set of all roots of the minimal polynomial of σ over 𝔽_{q}; in particular, #[σ] = deg(σ). We say that σ_{1} and σ_{2} are conjugate if [σ_{1}] = [σ_{2}]. The order of σ, denoted |σ|, is its multiplicative order, and σ is an ℓ- (resp. ℓ′-) element if |σ| is a power of ℓ (resp. prime to ℓ). If ℓ = 0, all elements are ℓ′-elements.

We state a classification of irreducible 𝔽GL_{n}(q)-modules, referring the reader to §2.3 for details and references. An n-admissible tuple is a tuple

(([σ_{1}], λ^{(1)}), …, ([σ_{a}], λ^{(a)}))  (1.1)

of pairs, where σ_{1}, …, σ_{a} ∈ 𝔽̄_{q} are ℓ′-elements, and λ^{(1)}, …, λ^{(a)} are partitions, such that [σ_{i}] ≠ [σ_{j}] for all i ≠ j and ∑_{i=1}^{a} deg(σ_{i}) · |λ^{(i)}| = n. An equivalence class of the n-admissible tuple (1.1) up to a permutation of the pairs ([σ_{1}], λ^{(1)}), …, ([σ_{a}], λ^{(a)}) is called an n-admissible symbol and denoted

𝔰 = [([σ_{1}], λ^{(1)}), …, ([σ_{a}], λ^{(a)})]  (1.2)

The set 𝔏 of n-admissible symbols is the labeling set for irreducible 𝔽GL_{n}(q)- modules. The module corresponding to the symbol (1.2) is

L(𝔰) = L(σ_{1}, λ^{(1)}) ∘ … ∘ L(σ_{a}, λ^{(a)}),

and {L(𝔰) | 𝔰 ∈ 𝔏} is a complete set of representatives of irreducible 𝔽GL_{n}(q)-modules.

The subgroup O_{ℓ′}(𝔽_{q}^{×}) (consisting of the ℓ′-elements in the multiplicative group 𝔽_{q}^{×}) acts on the set 𝔏 of n-admissible symbols via

τ · [([σ_{1}], λ^{(1)}), …, ([σ_{a}], λ^{(a)})] = [([τσ_{1}], λ^{(1)}), …, ([τσ_{a}], λ^{(a)})]

for τ ∈ O_{ℓ′}(𝔽_{q}^{×}). The order of the stabilizer group in O_{ℓ′}(𝔽_{q}^{×}) of a symbol 𝔰 ∈ 𝔏 is called the ℓ′-branching number of 𝔰 and is denoted κ_{ℓ′}(𝔰). Next, if 𝔰 is of the form (1.2), then the ℓ-branching number κ_{ℓ}(𝔰) is the ℓ-part of gcd (n, q − 1, Δ(\underline{λ}′)) = gcd (q − 1, Δ(\underline{λ}′)), where \underline{λ} = (λ^{(1)}, λ^{(2)}, …, λ^{(a)}).

Theorem 1.1. Let V = L(𝔰) be the irreducible 𝔽GL_{n}(q)-module corresponding to 𝔰 ∈ 𝔏. Then V ↓_{SL_{n}(q)} is a sum of κ_{ℓ′}(𝔰) · κ_{ℓ}(𝔰) irreducible summands.

By Lemma 4.1, for 𝔰 and 𝔰′ in the same O_{ℓ′}(𝔽_{q}^{×})-orbit on 𝔏, the restrictions L(𝔰)↓_{SL_{n}(q)} and L(𝔰′)↓_{SL_{n}(q)} are isomorphic. Moreover, by Theorem 1.1, we have a decomposition into non-isomorphic irreducible components:

L(𝔰)↓_{SL_{n}(q)} = ⨁_{j=1}^{κ_{ℓ′}(𝔰)κ_{ℓ}(𝔰)} L(𝔰)_{j}

From Lemma 3.8 we now get:

Corollary 1.2. The set {L(𝔰)_{j}}, where 𝔰 runs through the O_{ℓ′}(𝔽_{q}^{×})-orbit representatives on 𝔏 and j runs through the integers between 1 and κ_{ℓ′}(𝔰)κ_{ℓ}(𝔰), is a complete set of representatives of the irreducible 𝔽SL_{n}(q)-modules.

Note that our labeling of the summands in the corollary is not canonical. This problem will be resolved with the use of Theorem 1.4 below.

To state the result on reductions modulo ℓ, we need more notation. Given two partitions λ = (λ_{1}, λ_{2}, …) and μ = (μ_{1}, μ_{2}, …) and k ∈ ℤ_{>0}, define new partitions λ + μ := (λ_{1} + μ_{1}, λ_{2} + μ_{2}, …) and by kλ := (kλ_{1}, kλ_{2}, …).

Assume until the end of the section that ℓ > 0 and as usual ℓ |̸q. All representations in characteristic 0 will be over ℂ, and the corresponding index will be used to distinguish from the modular representations. Thus the labeling set for irreducible ℂGL_{n}(q)-modules is the set 𝔏_{ℂ} of n-admissible symbols of form (1.2), where the σ_{i} need not be ℓ′-elements, but [(σ_{i})_{ℓ′}] ≠ [(σ_{j})_{ℓ′}] whenever i ≠ j. The irreducible ℂGL_{n}(q)-module corresponding to the n-admissible symbol (1.2) is

L_{ℂ}(𝔰) = L_{ℂ}(σ_{1}, λ^{(1)}) ∘ … ∘ L_{ℂ}(σ_{a}, λ^{(a)}).

Given 𝔰 ∈ 𝔏_{ℂ} as in (1.2), we define 𝔰^{⋆} ∈ 𝔏 as follows. For each i, let s_{i} (resp. u_{i}) be the ℓ′-part (resp. the ℓ-part) of σ_{i}, and let d_{i} := deg(s_{i}), k_{i} := deg(σ_{i})/ deg(s_{i}). Permuting if necessary, we may assumethat s_{1}, …, s_{m} form a complete set of representatives of the conjugacy classes [s_{i}], 1 ≤ i ≤ a (one for each). For 1 ≤ j ≤ m, set

{i_{1}, …, i_{b_{j}}} = {i | [s_{i}] = [s_{j}]},

and δ^{(j)} := (k_{i_{1}}(λ^{(i_{1})})′ + k_{i_{2}}(λ^{(i_{2})})′ + … + k_{i_{b_{j}}} (λ^{(i_{b_{j}})})′)′. Now

𝔰^{⋆} := [([s_{1}], δ^{(1)}), …, ([s_{m}], δ^{(m)})].

Finally, we say that the symbol 𝔰 is critical, if ℓ = 2, q ≡ 3(mod 4), k_{i} > 1 for all i, and there exists j such that k_{j} = 2, d_{j} is odd, and |u_{j}| ≥ 8.

Theorem 1.3. Let V_{ℂ} = L_{ℂ}(𝔰) be an irreducible ℂGL_{n}(q)-module and W_{ℂ} be an irreducible constituent of (V_{ℂ})↓_{SL_{n}(q)}. Then reduction modulo ℓ of W_{ℂ} is irreducible if and only if the following conditions hold:

(1) reduction modulo ℓ of V_{ℂ} is irreducible (so V_{ℂ} is known [19, (4.33)]);

(2) κ_{ℓ′}(𝔰) = κ_{ℓ′}(𝔰^{⋆}); and

(3) 𝔰 is not critical. In this case the reduction is isomorphic to an irreducible constituent of L(𝔰^{⋆})↓_{SL_{n}(q)}.

We refer the reader to Example 5.11 for an illustration.

Next, we show that a certain submatrix of the ℓ-decomposition matrix of SL_{n}(q) is unitriangular. To be more precise, in §6 we construct an explicit injective map Θ : IBr(SL_{n}(q)) → Irr(SL_{n}(q)) and a partial order ⊵ on IBr(SL_{n}(q)) such that:

Theorem 1.4. For any φ ∈ IBr(SL_{n}(q)),

Θ(φ)(mod ℓ) = ∑_{ψ∈IBr(SL_{n}(q))} d_{φ,ψ}ψ

with d_{φ,φ} = 1, d_{φ,ψ} ∈ ℤ_{≥0}, and d_{φ,ψ} = 0 unless ψ ⊵ φ.

Note that Bonnafé [3] has obtained a canonical parametrization of the summands L_{ℂ}(𝔰)_{j} of L_{ℂ}(𝔰)↓_{SL_{n}(q)}, which is independent of 𝔰 and dependent only on a fixed choice of a Gelfand-Graev representation of SL_{n}(q), thus obtaining a parametrization of irreducible ℂSL_{n}(q)-modules. Combining this with Theorem 1.4 yields a parametrization of irreducible 𝔽SL_{n}(q)-modules compatible with branching from GL_{n}(q), see Proposition 6.3 for more details. Acknowledgements. We are grateful to Cédric Bonnafé for useful explanations.

2. Preliminaries

2.1. Notation. Throughout the paper all groups are assumed to be finite. We use the following notation in addition to the one introduced in §1:

λ ⊢ k means that λ is a partition of k; ⊵ denotes the dominance order on partitions; m|λ means that m divides all parts of the partition λ (for m ∈ ℤ, λ ⊢ k); \underline{λ} ⊢ \underline{k} means that \underline{λ} = (λ^{(1)}, …, λ^{(a)}) a multipartition of \underline{k} = (k_{1}, …, k_{a}),

i.e. each λ^{(i)} is a partition of k_{i}; g^{G} the conjugacy class of an element g in a group G; \overline{V} = V (mod ℓ) reduction modulo ℓ of a ℂG-module V; ε a fixed generator of the cyclic group 𝔽_{q}^{×};

GL_{n} denotes GL_{n}(q) and SL_{n} denotes SL_{n}(q). If X ≤ Y are groups, U an 𝔽X-module, and V an 𝔽Y -module, then V ↓_{X} is V restricted to X, and U↑^{Y} is U induced to Y. Denote by κ_{X}^{Y}(V) the number of irreducible components of V ↓_{X} (sometimes called a branching number). We also use the notation κ_{X}(V), κ_{X}, κ(V) or just κ when the missing ingredients are clear from the context.

Finally, let R_{n} = R_{n}(q) be the subgroup such that SL_{n} ≤ R_{n} ≤ GL_{n} and

R_{n}/SL_{n} = O_{ℓ′}(GL_{n}/SL_{n}).

2.2. Conjugacy classes in GL_{n}. If σ ∈ 𝔽̄_{q}^{×} is an element of degree d over 𝔽_{q} then {1, σ, …, σ^{d−1}} is an 𝔽_{q}-basis of 𝔽_{q}(σ) = 𝔽_{q^{d}}. The left multiplication by an element of 𝔽_{q^{d}} is 𝔽_{q}-linear. In this way we get a group embedding ι^{σ} : 𝔽_{q^{d}}^{×} → GL_{d}(q). This obviously generalizes to an embedding

ι_{k}^{σ} : GL_{k}(q^{d}) → GL_{kd}(q) (k ∈ ℤ_{>0}).

Semisimple conjugacy classes of GL_{n} are represented (uniquely up to block permutation) by the block-diagonal matrices of the form

s = diag((ι^{σ_{1}}(σ_{1}))^{k_{1}}, …, (ι^{σ_{a}}(σ_{a}))^{k_{a}}),  (2.1)

where elements σ_{1}, …, σ_{a} ∈ 𝔽̄_{q}^{×} of degrees d_{1}, …, d_{a} respectively are not conjugate to each other, and n = k_{1}d_{1} + ⋯ + k_{a}d_{a}. Let S be a set of representatives of the semisimple classes of GL_{n} which are of the form (2.1) and S_{ℓ′} be the subset of the ℓ-regular elements in S. For s ∈ S set

\underline{k}(s) := (k_{1}, …, k_{a}).

The centralizer C_{GL_{n}(q)}(s) equals the (not necessarily split) Levi subgroup

GL_{k_{1}}(q^{d_{1}}) × ⋯ × GL_{k_{a}}(q^{d_{a}}),

embedded into GL_{k_{1}d_{1}}(q) × ⋯ × GL_{k_{a}d_{a}}(q) < GL_{n}(q) via ι_{k_{1}}^{σ_{1}} × ⋯ × ι_{k_{a}}^{σ_{a}}.

The unipotent conjugacy classes of GL_{n}(q) are represented (uniquely) by the Jordan matrices

J(λ) := diag(J(λ_{1}), …, J(λ_{m})) (λ ⊢ n),

where J(λ_{i}) is the Jordan block of size λ_{i}. If we write λ in the form λ = (1^{r_{1}}, 2^{r_{2}}, …), then by [24] there exists N = N(λ) ∈ ℤ_{≥0} such that

|C_{GL_{n}(q)}(J(λ))| = q^{N} ∏_{i≥1} |GL_{r_{i}}(q)|.  (2.2)

By the Jordan decomposition, the conjugacy classes of GL_{n} are labeled by pairs (s, \underline{λ}), where s ∈ S and \underline{λ} is a multipartition of \underline{k}(s). A canonical representative of the corresponding conjugacy class looks like su = us for the unipotent element u ∈ GL_{n} chosen as follows. For each i = 1, …, a, let u_{i} be the Jordan matrix matrix J(λ^{(i)}) ∈ GL_{k_{i}}(q^{d_{i}}), and let

u := diag(ι_{k_{1}}^{σ_{1}}(u_{1}), …, ι_{k_{a}}^{σ_{a}}(u_{a})) ∈ GL_{k_{1}d_{1}}(q) × ⋯ × GL_{k_{a}d_{a}}(q) < GL_{n}(q).

Note that the element su is ℓ-regular if and only if s is ℓ-regular.

2.3. Representation theory of GL_{n}. If n = kd, then to every, not necessarily ℓ-regular, element σ of degree d over 𝔽_{q} and λ ⊢ k we associate the irreducible 𝔽GL_{n}-module denoted L(σ, λ), see [4, (3.5.3)] (or [17], where it is denoted D_{𝔽}(σ, λ)). Let ∘ denote the Harish-Chandra induction.

For s ∈ S as in (2.1) and \underline{λ} = (λ^{(1)}, …, λ^{(a)}) ⊢ \underline{k}(s), define

L(s, \underline{λ}) := L(σ_{1}, λ^{(1)}) ∘ ⋯ ∘ L(σ_{a}, λ^{(a)}).

Theorem 2.1. The set \{L(s, \underline{λ}) | s ∈ S_{ℓ′}, \underline{λ} ⊢ \underline{k}(s)\}, is a complete set of representatives of isomorphism classes of the irreducible 𝔽GL_{n}-modules.

To connect with the notation used in the Introduction, define an nadmissible symbol

𝔰 := [([σ_{1}], λ^{(1)}), …, ([σ_{a}], λ^{(a)})].

Then L(s, \underline{λ}) = L(𝔰), cf. §1.2. This is the James’ classification of irreducible 𝔽GL_{n}-modules (see [17] and [4, 4.4b]). Even though this classification suggests that the modules L(σ, λ) for ℓ-singular σ are redundant, it is sometimes convenient to use them. So we will not assume that σ is ℓ-regular, unless otherwise stated.

If 𝔽 = ℂ, we write L_{ℂ}(σ, λ) instead of L(σ, λ), L_{ℂ}(𝔰) instead of L(𝔰), etc. in order to distinguish from the case of positive characteristic. The Specht module S_{𝔽}(σ, λ) can be constructed in any characteristic, and S_{𝔽}(σ, λ) is a reduction modulo ℓ of L_{ℂ}(σ, λ) = S_{ℂ}(σ, λ), see [17].

2.4. Conjugacy classes in the intermediate subgroup. We will make use of the following general:

Lemma 2.2. Let S ◁G and S ≤ R ≤ G be such that G/S is cyclic. For any g ∈ G, set c = (G : C_{G}(g)S) and d = (G : R). Then |g^{G}|/|g^{R}| = gcd(c, d). Moreover, if G/R is an ℓ-group and g is an ℓ′-element, then g ∈ R.

Proof. Denote C := C_{G}(g) and D := C ∩ R. Then C ∩ S = D ∩ S, hence

\frac{|DS|}{|CS|} = (\frac{|D||S|}{|D ∩ S|}) : (\frac{|C||S|}{|C ∩ S|}) = \frac{|D|}{|C|}.  (2.3)

Let G/S = ⟨a⟩ be of order m, and π : G → ⟨a⟩ be a surjection with kernel S. Then π(R) = ⟨a^{d}⟩ and π(CS) = ⟨a^{c}⟩, hence π(CS ∩ R) = ⟨a^{lcm(c,d)}⟩. It is easy to see that CS ∩ R = DS, so |π(DS)| = m/ lcm(c, d). Now we have

\frac{|g^{G}|}{|g^{R}|} = \frac{|G|}{|C|} : \frac{|R|}{|D|} = \frac{|G|}{|R|} \frac{|D|}{|C|} = \frac{|G|}{|R|} \frac{|DS|}{|CS|} = \frac{|G|}{|CS|} \frac{|DS|}{|R|} = c\frac{m/ lcm(c, d)}{m/d},

which is gcd(c, d), as required. □

To apply Lemma 2.2, we need the following fact.

Lemma 2.3. Let u ∈ GL_{n}(q) be a unipotent element whose conjugacy class corresponds to a partition λ. Then det maps C_{GL_{n}(q)}(u) onto the subgroup ⟨ε^{Δ(λ)}⟩ of 𝔽_{q}^{×}.

Proof. Write λ = (1^{r_{1}}, 2^{r_{2}}, …) and C := C_{GL_{n}(q)}(u). Then u is conjugate to the matrix ⨁_{i:r_{i}>0} I_{r_{i}} ⊗ J(i), where I_{r} is the identity r × r matrix. Hence C contains the subgroup D := ∏_{i:r_{i}>0} GL_{r_{i}}(q) ⊗ I_{i}. Let P be a Sylow psubgroup of C. Using (2.2), we conclude that C = P D. Note that det(P) is trivial, so det(C) = det(D). Now, for each i, det maps the subgroup GL_{r_{i}}(q) ⊗ I_{i} onto ⟨ε^{i}⟩. So it maps D onto ∏_{i:r_{i}>0}⟨ε^{i}⟩ = ⟨ε^{Δ(λ)}⟩. □

Proposition 2.4. Let g = su be a standard representative of an ℓ-regular conjugacy class in GL_{n} corresponding to the pair (s, \underline{λ}), where s ∈ S_{ℓ′} and \underline{λ} = (λ^{(1)}, …, λ^{(a)}) is a multipartition of \underline{k}(s). Then g ∈ R_{n} and

\frac{|g^{GL_{n}}|}{|g^{R_{n}}|} = gcd \{(GL_{n} : R_{n}), Δ(\underline{λ})\}.

Proof. In view of Lemma 2.2, we need to know the image of C := C_{GL_{n}(q)}(g) under the determinant map det : GL_{n}(q) → 𝔽_{q}^{×}. Now

C = C_{GL_{n}(q)}(su) = C_{C_{GLn(q)}(s)}(u) = ∏_{i=1}^{a} C_{GL_{k_{i}}(q^{d_{i}})}(u_{i}),  (2.4)

where u_{i} are as in §2.2.

For positive integers m, d and an element σ of degree d over 𝔽_{q}, we consider the composition map det_{d}:

det_{d} : GL_{m}(q^{d}) ⟶^{ι_{m}^{σ}} GL_{md}(q) ⟶^{det} 𝔽_{q}^{×}.

Observe that det_{d} is also the composition

det_{d} : GL_{m}(q^{d}) ⟶^{det} 𝔽_{q^{d}}^{×} ⟶^{N_{𝔽_{q^{d}} /𝔽q}} 𝔽_{q}^{×}.

(In particular, det_{d} does not depend on σ). Fix a generator ε_{d} of the group 𝔽_{q^{d}}^{×} such that ε = N_{𝔽_{q^{d_{i}}} /𝔽_{q}}(ε_{d}). By Lemma 2.3, det maps C_{GL_{k_{i}}(q^{d_{i}})}(u_{i}) onto

⟨ε_{d_{i}}^{Δ(λ^{(i)})}⟩, and N_{𝔽_{q^{d_{i}}} /𝔽_{q}} maps the latter onto ⟨ε^{Δ(λ^{(i)})}⟩. So the image of C

under det is ⟨ε^{Δ(\underline{λ})}⟩. The result now follows from Lemma 2.2. □

3. Clifford Theory

3.1. Known results. Let S ◁ G be a normal subgroup and V ∈ IBr(G). Clifford theory (see e.g. [13, III.2], [5, §49], [6, §11A]) gives information on the restriction V ↓_{S}. For example, it is known that V ↓_{S} is completely reducible, so we can write

V ↓_{S} = ⨁_{i=1}^{t} V_{i}, (V_{1}, …, V_{t} ∈ IBr(S)).

Moreover, the irreducible S-modules V_{1}, …, V_{t} are G-conjugate to each other (but not necessarily pairwise non-isomorphic in general). Let I := Stab_{G}(V_{1}) be the inertia group of V_{1} in G. Then I ≥ S and there is an 𝔽I-module Ṽ_{1} such that V_{1} is an irreducible component of Ṽ_{1}↓_{S} and V = Ṽ_{1}↑^{G}.

If G/S is an ℓ-group and an irreducible 𝔽S-module appears as a component of the restrictions V ↓_{S} and V′↓_{S} of the irreducible 𝔽G-modules V and V′, then V ≅ V′. If G/S is cyclic, V_{1}, …, V_{t} are pairwise non-isomorphic, Ṽ_{1} = V_{1} as vector spaces, and κ_{S}^{G}(V) = t = (G : I).

Lemma 3.1. Let S ≤ H be normal subgroups of a group G, V ∈ IBr(G), and U be an irreducible component of V ↓_{H}. Then κ_{S}^{G}(V) = κ_{H}^{G}(V) · κ_{S}^{H}(U).

Proof. Write V ↓_{S} = ⊕_{i=1}^{t}V_{i}, where V_{1}, …, V_{t} ∈ IBr(S) are G-conjugate, and κ_{S}^{G}(V) = t. Similarly, write V ↓_{H} = ⊕_{j=1}^{m}U_{j}, where U = U_{1}, …, U_{m} ∈ IBr(H) are G-conjugate, and κ_{H}^{G}(V) = m. We may assume that V_{1} is a constituent of U_{1}↓_{S}. Then U_{1}↓_{S} = ⊕_{j=1}^{a}U_{1,j}, where V_{1} ≃ U_{1,1}, …, U_{1,a} ∈ IBr(S) are H-conjugate, and a = (H : H_{1}) for H_{1} = Stab_{H}(V_{1}). For any i there is g ∈ G such that V_{i} ≃ ^{g}V_{1}. Hence

Stab_{H}(V_{i}) = H ∩ g^{−1} Stab_{G}(V_{1})g = g^{−1}(H ∩ Stab_{G}(V_{1}))g = g^{−1}H_{1}g

since H ◁ G. Thus κ_{S}^{H}(U_{j}) = a = κ_{S}^{H}(U) for all j, and so t = ma. □

3.2. Number of constituents for cyclic extensions. We need more results in spirit of Clifford theory, especially for the case of cyclic extensions, i.e. when G/S is cyclic. Given a G/S-module we consider it as a G-module via inflation without further comment.

Lemma 3.2. Let S ◁ G with G/S a cyclic ℓ′-group, and V ∈ IBr(G). (i) We have κ_{S}(V) = ♯{L ∈ IBr(G/S) | V ≃ V ⊗ L}.

(ii) If L ∈ IBr(G/S) and V ≃ V ⊗ L then κ_{S}(V) ≥ (G : Ker(L)).

Proof. (i) Let η := ♯{L ∈ IBr(G/S) | V ≃ V ⊗L}. If L ∈ IBr(G/I), we have

V ⊗ L = (Ṽ_{1}↑_{I}^{G}) ⊗ L ≃ (Ṽ_{1} ⊗ (L↓_{I}))↑^{G} ≃ Ṽ_{1}↑^{G} = V,

and so η ≥ κ_{S}(V). Conversely, assume that V ≃ V ⊗ L for some L ∈ IBr(G/S). To see that η ≤ κ_{S}(V), it suffices to show that I acts trivially on L. Again we have Ṽ_{1}↑^{G} ≃ V ≃ V ⊗ L ≃ (Ṽ_{1} ⊗ (L↓_{I}))↑^{G}. As a subspace Ṽ_{1} equals the V_{1}-isotypic component of the S-module V. It follows that Ṽ_{1} ≃ Ṽ_{1} ⊗ (L↓_{I}), so Ṽ_{1}^{∗} ⊗ Ṽ_{1} ≃ Ṽ_{1}^{∗} ⊗ Ṽ_{1} ⊗ (L↓_{I}). As S is irreducible on Ṽ_{1} and trivial on L, taking S-fixed points, we see that 1_{I} ≃ L↓_{I}, as required.

(ii) Arguing as in (i), we see that L↓_{I} is trivial. So κ_{S}(V) = (G : I) ≥ (G : Ker(L)). □

Lemma 3.2 provides information on κ(V) for cyclic ℓ′-extensions. For an arbitrary cyclic extension the problem breaks into two parts; the first (easier) part for a cyclic ℓ′-extension, and the second part for a cyclic ℓ-extension:

Lemma 3.3. Let r be a prime, S ◁ G with G/S cyclic, V ∈ IBr(G), and S ≤ A, B ≤ G be such that A/S = O_{r}(G/S) and B/S = O_{r′}(G/S). Also, let U (resp. W) be an irreducible constituent of V ↓_{A} (resp. V ↓_{B}). Then

(i) κ_{S}^{G}(V) = κ_{A}^{G}(V) · κ_{B}^{G}(V);

(ii) κ_{A}^{G}(V) = κ_{S}^{B}(W), κ_{B}^{G}(V) = κ_{S}^{A}(U).

Proof. By Lemma 3.1,

t := κ_{S}^{G}(V) = κ_{A}^{G}(V)κ_{S}^{A}(U) = κ_{B}^{G}(V)κ_{S}^{B}(W).

Write V ↓_{S} = ⊕_{i=1}^{t}V_{i} for V_{1}, …, V_{t} ∈ IBr(S), and let I = Stab_{G}(V_{1}), A_{1} = Stab_{A}(V_{1}), B_{1} = Stab_{B}(V_{1}). Then G/S = A/S × B/S and I/S ≥ A_{1}/S × B_{1}/S. Conversely, assume x ∈ I and write xS = yS · zS, where yS is the rpart of xS and zS is the r′-part of xS. Then yS and zS are powers of xS and so y and z stabilize V_{1}, hence y ∈ A_{1} and z ∈ B_{1}. Thus I/S = A_{1}/S ×B_{1}/S. So

t = (G : I) = (A : A_{1}) · (B : B_{1}) = κ_{S}^{A}(U)κ_{S}^{B}(W).

Hence κ_{S}^{B}(W) = κ_{A}^{G}(V), κ_{S}^{A}(U) = κ_{B}^{G}(V), and κ_{S}^{G}(V) = κ_{A}^{G}(V) · κ_{B}^{G}(V). □

Next we link branching numbers κ_{S}^{G} to those for intermediate extensions S ◁ H and H ◁ G.

Lemma 3.4. Let S◁G with G/S a cyclic ℓ-group, S ≤ H ≤ G, V ∈ IBr(G). Then κ_{H}^{G}(V) = min{κ_{S}^{G}(V), |G/H|}.

Proof. Let U ∈ IBr(H) an irreducible constituent of V ↓_{H} and W ∈ IBr(S) be an irreducible constituent of U↓_{S}. Let I := Stab_{G}(W). Then V = W̃ ↑^{G} for W̃ ∈ IBr(I) with W̃ ↓_{S} = W. As G/S is a cyclic ℓ-group, either I ≥ H or H > I. If I ≥ H then U′ := W̃ ↓_{H} is also irreducible and I-invariant. Conversely, if g ∈ G stabilizes U′ then g stabilizes U′↓_{S} = W, so g ∈ I. Thus Stab_{G}(U′) = I, whence

κ_{H}^{G}(V) = (G : I) = κ_{S}^{G}(V) ≤ |G/H|,

and we are done in the case I ≥ H.

If H > I then U′ := W̃ ↑^{H} is an irreducible component of V ↓_{H}. Hence κ_{H}^{G}(V) = dim V/ dim U′ = |G/H| < (G : I), and we are done again. □

Corollary 3.5. Let S ◁ G with G/S cyclic, S ≤ H ≤ B ≤ G such that B/H = O_{ℓ}(G/H), and V ∈ IBr(G). Then

κ_{H}^{G}(V) = κ_{B}^{G}(V) · min{(κ_{S}^{G}(V))_{ℓ}, |G/H|_{ℓ}}.

Proof. Consider A ≥ H with A/H = O_{ℓ′}(G/H), and R, T ≥ S with R/S = O_{ℓ′}(G/S), T/S = O_{ℓ}(G/S). By Lemma 3.3, κ_{H}^{G}(V) = κ_{A}^{G}(V) · κ_{B}^{G}(V) so we just need to compute κ_{A}^{G}(V). As

RH/H ≃ R/(R ∩ H) ≃ (R/S)/((R ∩ H)/S)

is an ℓ′-group, RH/H ≤ O_{ℓ′}(G/H) = A/H, whence R ≤ A. By Lemma 3.4, κ_{A}^{G}(V) = min{κ_{R}^{G}(V), |G/A|}. Clearly |G/A| = |G/H|_{ℓ}. By Lemma 3.3, κ_{S}^{G}(V) = κ_{R}^{G}(V) · κ_{T}^{G}(V). As the first factor is an ℓ-power and the second factor is coprime to ℓ, we have κ_{R}^{G}(V) = (κ_{S}^{G}(V))_{ℓ}, and we are done. □

Corollary 3.5 shows that if we know κ_{S}^{G}(V), then, to determine κ_{H}^{G}(V) for H ≥ S, it suffices to know κ_{B}^{G}(V), which can be found using Lemma 3.2.

Lemma 3.6. Let G = G_{1}×G_{2}, V ∈ IBr(G), H_{i}◁G_{i} and G_{i}/H_{i} = ⟨x̄_{i}⟩ ≃ C_{n} for i = 1, 2. Let the subgroup K ≤ G be such that G ≥ K ≥ H_{1} × H_{2} and K/(H_{1} ×H_{2}) = ⟨x̄_{1}x̄_{2}⟩. Then κ_{K}(V) = gcd(s_{1}, s_{2}), where s_{1} := κ_{H_{1}×G_{2}}(V), s_{2} := κ_{G_{1}×H_{2}}(V).

Proof. Throughout the proof we assume i ∈ {1, 2}. Write V = U^{1} ⊗ U^{2} for U^{i} ∈ IBr(G_{i}). By assumption, U^{i}↓_{H_{i}} = U_{1}^{i} ⊕ … ⊕ U_{s_{i}}^{i} for U_{1}^{i}, …, U_{s_{i}}^{i} ∈ IBr(H_{i}) and s_{i} = (G_{i} : I_{i}) where I_{i} := Stab_{G_{i}}(U_{1}^{i}). Clearly, J := Stab_{K}(U_{1}^{1}⊗ U_{1}^{2}) contains H_{1} × H_{2}. Choose x_{i} ∈ G_{i} such that x̄_{i} = x_{i}H_{i}. Then K = ⟨H_{1}, H_{2}, x_{1}x_{2}⟩ and G/K ≃ C_{n}. Assume that (x_{1}x_{2})^{m} ∈ J for some integer m. Then

U_{1}^{1} ⊗ U_{1}^{2} ≃ (x_{1}^{m}x_{2}^{m})(U_{1}^{1} ⊗ U_{1}^{2}) ≃ (x_{1}^{m}U_{1}^{1}) ⊗ (x_{2}^{m}U_{1}^{2})

as H_{1}×H_{2}-modules. By the irreducibility of the U_{1}^{i}, we see that x_{i}^{m}U_{1}^{i} ≃ U_{1}^{i}, i.e. x_{i}^{m} ∈ I_{i}. But G_{i}/I_{i} is cyclic of order s_{i}, hence s_{i}|m. Setting r = lcm(s_{1}, s_{2}), we see that r|m. Thus (x_{1}x_{2})^{m} ∈ J only if r|m. The converse is obviously true. Therefore, J/(H_{1} × H_{2}) = ⟨(x̄ȳ)^{r}⟩ and (K : J) = r.

Now choose T to be an irreducible constituent of V ↓_{K} that lies over U_{1}^{1} ⊗ U_{1}^{2}. As (K : J) = r, we have κ_{H_{1}×H_{2}}(T) = r. Hence

dim(T) = r dim(U_{1}^{1}) dim(U_{1}^{2}),

so dim(V)/ dim(T) = s_{1}s_{2}/r = gcd(s_{1}, s_{2}), and κ_{K}(V) = gcd(s_{1}, s_{2}). □

Lemma 3.7. Let S ◁G with G/S cyclic, S ≤ A ≤ G with A/S = O_{ℓ}(G/S), and U ∈ IBr(S). Assume that U is an S-composition factor of V_{i}↓_{S} for some V_{i} ∈ IBr(G), i = 1, 2. Then V_{2} = V_{1} ⊗ L for some L ∈ IBr(G/A).

Proof. By assumption, there are W_{i} ∈ IBr(A) such that U is a direct summand of W_{i}↓_{A} and W_{i} is a direct summand of V_{i}↓_{A}. Since A/S is an ℓ-group, W_{1} ≅ W_{2} ≅ W. Let I := Stab_{G}(W). By Clifford theory, we have V_{i} = M_{i}↑^{G} for some M_{i} ∈ IBr(I) and W = M_{i}↓_{A} (i = 1, 2). It follows that M_{2} ≃ M_{1}⊗N for some N ∈ IBr(I/A). But G/A is cyclic, hence N is also G/A-invariant and it extends to an irreducible G/A-module L. Thus

V_{2} ≃ (M_{1} ⊗ N)↑^{G} = (M_{1} ⊗ (L↓_{I}))↑^{G} ≃ (M_{1}↑^{G}) ⊗ L = V_{1} ⊗ L,

as required. □

We now address the question of describing irreducible 𝔽S-modules in terms of irreducible 𝔽G-modules and branching from G to S.

Lemma 3.8. Let S ◁ G with G/S cyclic, and S ≤ A, B ≤ G with A/S = O_{ℓ}(G/S), B/S = O_{ℓ′}(G/S). Introduce the equivalence relation ∼ on IBr(G) by setting V ∼ V′ if and only if V ≃ V′ ⊗ L for some L ∈ IBr(G/A), and let V_{1}, …, V_{m} be a complete set of representatives of the equivalence classes. Decompose

V_{i}↓_{S} = ⨁_{j=1}^{k_{i}} V_{ij} (1 ≤ i ≤ m)

where V_{ij} are irreducible 𝔽S-modules. Then

{V_{ij} | 1 ≤ i ≤ m, 1 ≤ j ≤ k_{i}}

is a complete set of representatives of the isomorphism classes of the irreducible 𝔽S-modules.

Proof. We only have to check that V_{ij} ≄ V_{kl} if i ≠ k, which follows immediately from Lemma 3.7. □

3.3. Clifford theory and blocks.

Lemma 3.9. Let S ◁ G with G/S an ℓ-group, V an 𝔽G-module, and

(i) V ↓_{S} has a filtration 0 = V^{0} < V^{1} < …, where each V_{i} := V^{i}/V^{i−1} is a G-conjugate of V_{1} = V^{1};

(ii) All composition factors of V_{1} belong to the same 𝔽S-block. Then all composition factors of V belong to the same 𝔽G-block.

Proof. Pick an S-composition factor U_{1} of V_{1} and W_{1} ∈ IBr(G) such that U_{1} is a submodule of W_{1}↓_{S}. Let B be the G-block containing W_{1}, and b be the S-block containing U_{1}. Then B covers b, as well as any G-conjugate of b, cf. [13, Lemma IV.4.10]. By the assumption (ii), every composition factor of V ↓_{S} belongs to some G-conjugate of b.

Now consider any composition factor M of V and any composition factor N of M↓_{S}. Then we may assume that N is a composition factor of some V_{i}. But V_{i} = ^{z}V_{1} for some z ∈ G. Hence N belongs to the S-block b^{z}. Now b^{z} is also covered by B. In fact, since G/S is an ℓ-group, B is the unique G-block that covers b^{z}, cf. [13, Lemma V.3.5]. Hence M belongs to B. □

Lemma 3.10. Let S ◁ G with G/S cyclic, and let B be an ℓ-block of S. Suppose that every ordinary character ρ belonging to B is G-invariant. Then every Brauer character φ in B is extendible to G.

Proof. By [11, Theorem 61.6], φ = ∑_{ρ∈B∩Irr(S)} a_{ρ}ρ̄ for some a_{ρ} ∈ ℤ, and ρ̄ denotes the restriction of irreducible characters ρ to ℓ′-classes of S. It follows that φ is G-invariant. Since G/S is cyclic, φ is extendible to G. □

Lemma 3.11. Let S ◁ G with G/S a cyclic ℓ-group, and B be an ℓ-block of S. Suppose there is b ∈ ℤ_{≥0} such that (G : Stab_{G}(ρ)) ≤ ℓ^{b} for any complex irreducible character ρ ∈ B. Then (G : Stab_{G}(ψ)) ≤ ℓ^{b} for any ψ ∈ B ∩ IBr(S).

Proof. We may assume that b ≤ a, where ℓ^{a} := |G/S|. Let J be the unique subgroup of index ℓ^{b} in G containing S. By assumption, any complex ρ ∈ B is J-invariant. Now Lemma 3.10 applied to J ◁ G implies that every ψ ∈ B ∩ IBr(S) is J-invariant. □

Lemma 3.12. Let S ◁ G with G/S a cyclic ℓ-group, and B be an ℓ-block of G. Suppose there is b ∈ ℤ_{≥0} such that κ_{S}^{G}(χ) ≤ ℓ^{b} for any complex irreducible character χ ∈ B. Then κ_{S}^{G}(φ) ≤ ℓ^{b} for any φ ∈ B ∩ IBr(G).

Proof. Let φ ∈ B ∩ IBr(G), ψ ∈ IBr(S) be an irreducible component of φ↓_{S}, and B_{1} be the ℓ-block of S containing ψ. As G/S is cyclic, by Lemma 3.11 it suffices to show that (G : Stab_{G}(ρ)) ≤ ℓ^{b} for any complex irreducible ρ ∈ B_{1}. Consider any such ρ. As G/S is an ℓ-group, B is the unique block of G that covers B_{1}. So there is some irreducible χ ∈ B such that ρ is an irreducible component of χ↓_{S}. As G/S is cyclic, (G : Stab_{G}(ρ)) = κ_{S}^{G}(χ), and so we are done. □

3.4. ℓ-elementary groups. A finite group is ℓ-elementary if it is a direct product of an ℓ-group and a cyclic ℓ′-group. It is easy to see that any subgroup and any quotient group of an ℓ-elementary group are also ℓ-elementary. In this subsection we explain why many results in Clifford theory valid for cyclic extensions, are also valid for ℓ-elementary extensions. This material, even though not needed in this paper, will be useful in other situations of Clifford theory.

Lemma 3.13. Any irreducible projective representation over the field 𝔽 of an ℓ-elementary group H has degree 1.

Proof. We can realize our projective representation as a linear representation Ψ of a covering group T of H. Writing Z := Z(T), we have T/Z ≃ H. As Ψ↓_{O_{ℓ}(Z)} is trivial, we may assume that Z is an ℓ′-group. Now O_{ℓ′}(T) ≥ Z and O_{ℓ′}(T)/Z ≃ O_{ℓ′}(H) is cyclic. Hence O_{ℓ′}(T) is abelian. As H is nilpotent, so is T. Hence T = O_{ℓ′}(T) × O_{ℓ}(T), and deg(Ψ) = 1. □

Lemma 3.14. Let S ◁ G with G/S being ℓ-elementary, and V ∈ IBr(G). Then V ↓_{S} is multiplicity free.

Proof. Let U be an irreducible component of V ↓_{S} and I := Stab_{G}(U). Assume first that G = I. By [6, 11.20], V = X ⊗ Y, where X is a projective representation of G of dimension equal to dim U and Y is a projective representation of G/S. By Lemma 3.13, dim Y = 1, so V ↓_{S} = U.

In the general case we have V = Ũ↑^{G} for some Ũ ∈ IBr(I) lying above U. As U is I-invariant and I/S is ℓ-elementary, the above argument shows that Ũ↓_{S} = U. In particular, dim V/ dim U = (G : I), and the claim follows. □

Now Lemmas 3.13 and 3.14 allow us to repeat the proof of Lemma 3.3. So the conclusions of that key lemma also hold for ℓ-elementary extensions.

4. Main Results

4.1. Computing κ_{ℓ′}. The irreducible 𝔽GL_{n}-modules which factor through SL_{n} are among the L(τ, (n)) with τ ∈ O_{ℓ′}(𝔽_{q}^{×}), see [17]. The following statement is needed in view of Lemma 3.2, cf. also [16, Lemma 2.9]:

Lemma 4.1. Let 𝔰 be an n-admissible symbol and τ ∈ O_{ℓ′}(𝔽_{q}^{×}). Then

L(𝔰) ⊗ L(τ, (n)) ≃ L(τ · 𝔰).

Proof. It suffices to prove that L(σ, λ)⊗L(τ, (n)) ≃ L(στ, λ), where deg(σ) = d, λ ⊢ k := n/d. We use notation and results from [17]. First of all, the values of the Brauer character of the irreducible 𝔽GL_{d}-module M_{𝔽}(σ, (1)) are given explicitly in [17, (3.1)]—they involve a certain function ⟨x, y⟩_{d} : 𝔽_{q^{d}}^{×} → ℂ^{×} with the property

⟨x_{1}x_{2}, y⟩_{d} = ⟨x_{1}, y⟩_{d} · ⟨x_{2}, y⟩_{d}.

In particular, ⟨στ, y⟩_{d} = ⟨σ, y⟩_{d} · ⟨τ, y⟩_{d}, and ⟨τ, y⟩_{d}^{q−1} = 1. These two equalities imply that the Brauer character of M_{𝔽}(στ, (1)) is just the product of the Brauer characters of M_{𝔽}(σ, (1)) and L(τ, (n)), whence

M_{𝔽}(σ, (1)) ⊗ L(τ, (n)) ≃ M_{𝔽}(στ, (1)).

This identity, together with the construction of M_{𝔽}(σ, (1^{k})) in [17], implies that

M_{𝔽}(σ, (1^{k})) ⊗ L(τ, (n)) ≃ M_{𝔽}(στ, (1^{k})).

Next, the submodule S_{𝔽}(σ, λ) of M_{𝔽}(σ, (1^{k})) is defined using some idempotent depending only on λ, see [17, (7.7)]. It follows that

S_{𝔽}(σ, λ) ⊗ L(τ, (n)) ≃ S_{𝔽}(στ, λ).

Each S_{𝔽}(σ, λ) has a unique maximal submodule, the quotient by which is exactly L(σ, λ). Hence L(σ, λ) ⊗ L(τ, (n)) ≃ L(στ, λ). □

4.2. Lower bound for κ_{ℓ}. Throughout §4.2, σ denotes an element of 𝔽̄_{q}^{×} of degree d|n and k = n/d.

Proposition 4.2. Let m ∈ ℤ_{≥1}, σ ∈ \overline{𝔽}_{q}^{×}, and SL_{n} ≤ R ≤ GL_{n}. If κ_{R}(L_{ℂ}(σ, λ)) ≥ m for all λ ⊢ k, then κ_{R}(L(σ, λ)) ≥ m for all λ ⊢ k.

Proof. By [17, 8.2], composition factors of S_{𝔽}(σ, λ) are of the form L(σ, μ) for μ ⊵ λ, and exactly one of them is L(σ, λ). We apply induction on the dominance order ⊵ on partitions of k. If λ = (k), then L(σ, λ) = S_{𝔽}(σ, λ), and the result follows, as S_{𝔽}(σ, λ) is a reduction modulo ℓ of L_{ℂ}(σ, λ).

For the induction step, we assume that for each μ ⊵ λ and μ ≠ λ, we have L(σ, μ)↓_{R} = ⊕_{i=1}^{m(μ)}L(σ, μ, i) for some m(μ) ≥ m and L(σ, μ, i) ∈ IBr(R). Since GL_{n}/R is cyclic, the inertia group of L(σ, μ, i) has index m(μ) in GL_{n}. By our assumption, L_{ℂ}(σ, λ)↓_{R} = ⊕_{j=1}^{t}V_{ℂ}(j) for some integer t ≥ m and some irreducible ℂR-modules V_{ℂ}(j), 1 ≤ j ≤ t, which are GL_{n}-conjugate. Let V (j) denote an ℓ-modular reduction of V_{ℂ}(j). Now in the Grothendieck group of 𝔽GL_{n}-modules we have

S_{𝔽}(σ, λ) = L(σ, λ) + ∑_{μ ⊵ λ, μ≠λ} d_{λμ}L(σ, μ)

for some integers d_{λμ} ≥ 0. Thus over R we have

∑_{j=1}^{t} V (j) = L(σ, λ)↓_{R} + ∑_{μ ⊵ λ, μ≠λ} d_{λμ} ∑_{i=1}^{m(μ)} L(σ, μ, i).  (4.1)

Assume for a contradiction that L(σ, λ)↓_{R} = L_{1}⊕…⊕L_{s}, where L_{1}, …, L_{s} are distinct GL_{n}-conjugates of L_{1} ∈ IBr(R), and s < m. We may assume that L_{1} is a composition factor of V (1). For any j > 1, V_{ℂ}(j) = ^{g}V_{ℂ}(1) for some g ∈ GL_{n}, and so ^{g}L_{1} is a subquotient of V (j). Thus the left-hand side of (4.1) contains at least t conjugates of L_{1}. Since t ≥ m > s, (4.1) now implies that some GL_{n}-conjugate ^{g}L_{1} of L_{1} must be isomorphic to some L(σ, μ, i). So the inertia group of ^{g}L_{1} has index m(μ) ≥ m > s in GL_{n}, a contradiction. □

Corollary 4.3. Let λ ⊢ k, 1 ≠ α ∈ 𝔽_{q}^{×}, and assume that σα is conjugate to σ. Then κ_{R}(L(σ, λ)) ≥ |α| for R := Ker(L_{ℂ}(α, (n))).

Proof. By Lemma 4.1, we have L_{ℂ}(σ, λ)⊗L_{ℂ}(α, (n)) ≃ L_{ℂ}(σα, λ) ≃ L_{ℂ}(σ, λ). Note also that |α| = (GL_{n} : R). By Lemma 3.2(ii), κ_{R}(L_{ℂ}(σ, λ)) ≥ |α|, and this is true for all λ ⊢ k. Now the statement follows from Proposition 4.2. □

Lemma 4.4. Let r, c, d be positive integers and ℓ^{c} be the ℓ-part of r − 1. Then the ℓ-part of \frac{r^{ℓ^{d}}−1}{r−1} is ℓ^{d}, unless c = 1, ℓ = 2 and r ≡ 3 (mod 4).

Proof. We only explain the case c ≥ 2. As \frac{r^{ℓ^{d}}−1}{r−1} = ∏_{i=1}^{d} \frac{r^{ℓ^{i}}−1}{r^{ℓ^{i−1}}−1}, it suffices

to see that (\frac{r^{ℓ^{i}}−1}{r^{ℓ^{i−1}}−1})_{ℓ} = ℓ for 1 ≤ i ≤ d. For such i we have r^{ℓ^{i−1}} = 1 + Aℓ^{b}

for some integer b ≥ 2 and (A, ℓ) = 1. So r^{ℓ^{i}} = (1+Aℓ^{b})^{ℓ} = 1+Aℓ^{b+1} +Bℓ^{2b} for some B ∈ ℤ, whence the ℓ-part of r^{ℓ^{i}} − 1 is ℓ^{b+1} (using 2b > b + 1). It

follows that (\frac{r^{ℓ^{i}}−1}{r^{ℓ^{i−1}}−1})_{ℓ} = \frac{ℓ^{b+1}}{ℓ^{b}} = ℓ, as required. □

Lemma 4.5. Assume that σ is an ℓ′-element and ℓ^{c}| gcd(n, q − 1) for some c ≥ 1. Then for any λ ⊢ k with ℓ^{c}|λ′, we have κ_{R_{n}}(L(σ, λ)) ≥ ℓ^{c}.

Proof. As ℓ^{c}|λ′, we have ℓ^{c}|k, and so ℓ^{c}d|n, where d := deg(σ). Let P := O_{ℓ}(𝔽_{q^{dℓ^{c}}}^{×}) and Q := O_{ℓ}(𝔽_{q^{d}}^{×}). Note that Q < P via 𝔽_{q^{d}} ⊂ 𝔽_{q^{dℓ^{c}}}, and ℓ^{c} divides |P/Q|.

First, assume that c = 1. Then there is τ ∈ P \ Q such that Q := ⟨τ^{ℓ}⟩. Note that τ has degree ℓ over 𝔽_{q^{d}}, whence 𝔽_{q^{d}}(τ) = 𝔽_{q^{ℓd}}. The ℓ′-part (resp. ℓ-part) of στ is σ (resp. τ). In particular, σ and τ are powers of στ. Therefore

𝔽_{q}(στ) = 𝔽_{q}(σ, τ) = 𝔽_{q}(σ)(τ) = 𝔽_{q^{d}}(τ) = 𝔽_{q^{ℓd}},

i.e. deg(στ) = ℓd. Set α := (στ)^{q^{d}−1}. We claim that α ∈ 𝔽_{q}^{×} and |α| = ℓ. Indeed, since σ^{q^{d}−1} = 1, we have α = τ^{q^{d}−1}. Now α^{ℓ} = (τ^{ℓ})^{q^{d}−1} = 1 since τ^{ℓ} ∈ Q; in particular, α ∈ 𝔽_{q}^{×}. Also, α ≠ 1, as otherwise τ ∈ 𝔽_{q^{d}}^{×}, a contradiction.

We have shown that στ has degree ℓd and (στ)^{q^{d}} = α(στ) with 1 ≠ α ∈ 𝔽_{q}^{×}. By Corollary 4.3, κ_{K}(L(στ, ν)) ≥ ℓ for K := Ker(L_{ℂ}(α, (n))) ≥ R_{n} and all ν ⊢ n/ℓd. In particular, κ_{K}(L(στ, μ)) ≥ ℓ, where μ ⊢ n/ℓd is chosen so that μ′ = (1/ℓ)λ′. Since L(στ, μ) ≃ L(σ, λ) by [9], the claim follows.

Now let c ≥ 2. Observe that ℓ^{c} = |P/Q| or (\frac{q^{dℓ^{c}}−1}{q^{d}−1})_{ℓ} = ℓ^{c} by Lemma 4.4.

Now if P = ⟨τ⟩, then Q = ⟨τ^{ℓ^{c}}⟩ and τ^{ℓ^{c−1}} ∉ Q. Note τ has degree ℓ^{c} over 𝔽_{q^{d}}, so 𝔽_{q^{d}}(τ) = 𝔽_{q^{dℓ^{c}}}. As the ℓ′-part and the ℓ-part of στ are σ and τ, respectively, σ and τ are powers of στ. So

𝔽_{q}(στ) = 𝔽_{q}(σ, τ) = 𝔽_{q}(σ)(τ) = 𝔽_{q^{d}}(τ) = 𝔽_{q^{dℓ^{c}}} ,

i.e. deg(στ) = dℓ^{c}. Set α := (στ)^{q^{d}−1} = τ^{q^{d}−1}. We claim α ∈ 𝔽_{q}^{×} has order ℓ^{c}. Indeed, α^{ℓ^{c}} = (τ^{ℓ^{c}})^{q^{d}−1} = 1 since τ^{ℓ^{c}} ∈ Q, and α ∈ 𝔽_{q}^{×} as ℓ^{c}|(q − 1). On the other hand, α^{ℓ^{c−1}} ≠ 1, as otherwise τ^{ℓ^{c−1}} ∈ 𝔽_{q^{d}}^{×} and so τ^{ℓ^{c−1}} ∈ Q, a

contradiction. Thus deg(στ) = dℓ^{c} and (στ)^{q^{d}} = α(στ) for α ∈ 𝔽_{q}^{×} of order ℓ^{c}. Now we finish as in the case c = 1. □

Theorem 4.6. Let ℓ^{c}| gcd(n, q − 1) and V = L(σ_{1}, λ^{(1)}) ∘ … L(σ_{a}, λ^{(a)}) be an irreducible 𝔽GL_{n}-module, where σ_{1}, …, σ_{a} are ℓ′-elements and ℓ^{c}|(λ^{(i)})′ for all i = 1, 2, …, a. Then κ_{R_{n}}(V) ≥ ℓ^{c}.

Proof. We apply induction on a. The case a = 1 is Lemma 4.5. For the induction step, assume a ≥ 2, and let λ_{i} ⊢ k_{i}, deg(σ_{i}) = d_{i}. Set r := k_{1}d_{1}, s := n − r, A := GL_{r}, B := GL_{s}, A_{1} := R_{r}, B_{1} := R_{s}, and

U := L(σ_{1}, λ^{(1)}) ∈ IBr(A), W := (L(σ_{2}, λ^{(2)}) ∘ … ∘ L(σ_{a}, λ^{(a)})) ∈ IBr(B).

As ℓ^{c}|(λ^{(i)})′ for all i, we have ℓ^{c}| gcd(r, q − 1) and ℓ^{c}| gcd(s, q − 1). By the inductive assumption,

u := κ_{A_{1}×B}(U ⊗ W) = ℓ^{α} ≥ ℓ^{c} and v := κ_{A×B_{1}}(U ⊗ W) = ℓ^{β} ≥ ℓ^{c}.

Choose x ∈ GL_{r}, y ∈ GL_{s} such that ⟨det(x)⟩ = O_{ℓ}(𝔽_{q}^{×}) and det(y) = det(x)^{−1}. Then x̄ := xA_{1} generates A/A_{1}, ȳ := yB_{1} generates B/B_{1}, and A/A_{1} ≃ B/B_{1}.

Next, consider the standard parabolic subgroup P = QL < GL_{n} with Levi subgroup L := GL_{r} × GL_{s}. Note that K := L ∩ R_{n} = ⟨A_{1}, B_{1}, xy⟩, i.e. K/(A_{1} × B_{1}) = ⟨x̄ȳ⟩. By Lemma 3.6,

κ_{K}(U ⊗ W) = gcd(u, v) = gcd(ℓ^{α}, ℓ^{β}) ≥ ℓ^{c}.

Now V = U ∘ W and (R_{n} : QK) = (GL_{n} : P), so

V ↓_{R_{n}} ≃ (infl_{K}^{QK}(U ⊗ W))↑^{R_{n}}.

Hence κ_{R_{n}}(V) ≥ κ_{K}(U ⊗ W) ≥ ℓ^{c}. □

Theorem 4.7. Let V = L(σ_{1}, λ^{(1)})∘…∘L(σ_{a}, λ^{(a)}) be an irreducible 𝔽GL_{n}- representation, where the σ_{i} are ℓ′-elements. Set \underline{λ} = (λ^{(1)}, …, λ^{(a)}). Then κ_{R_{n}}(V) = gcd \{(GL_{n} : R_{n}), Δ(\underline{λ}′)\}.

Proof. Note that gcd \{(GL_{n} : R_{n}), Δ(\underline{λ}′)\} is some power ℓ^{c(V)}, which divides gcd(n, q − 1), and all (λ^{(i)})′. By Theorem 4.6, we have κ_{R_{n}}(V) ≥ ℓ^{c(V)}.

To see that κ_{R_{n}}(V) = ℓ^{c(V)} for each V, we apply a counting argument. First apply Lemma 3.8 with S = B = R_{n} and G = A = GL_{n} to conclude that the set of all irreducible summands of the restriction W ↓_{R_{n}}, as W runs over the isomorphism classes [W] of irreducible 𝔽GL_{n}-modules, is exactly the set of representatives of the isomorphism classes of the irreducible 𝔽R_{n}- modules. Hence it suffices to show that the number of ℓ-regular conjugacy classes in R_{n} equals ∑_{[W]} ℓ^{c(W)}.

In order to compute the number of ℓ-regular conjugacy classes in R_{n}, we recall first that such classes in GL_{n} are parametrized by the pairs (s, \underline{μ}) where s ∈ S_{ℓ′} and \underline{μ} ⊢ \underline{k}(s). Moreover, by Proposition 2.4, such class splits into exactly gcd ((GL_{n} : R_{n}), Δ(\underline{μ})) R_{n}-conjugacy classes, which is precisely c(W) for the irreducible 𝔽GL_{n}-module W = L(s, \underline{μ}′). We conclude that the total number of ℓ-regular conjugacy classes in R_{n} equals ∑_{[W]} ℓ^{c(W)}. □

4.3. Proof of Theorem 1.1. We first apply Lemma 3.3 to the case G = GL_{n} and S = SL_{n}. In particular B = R_{n}. We conclude that κ_{S}(V) = κ_{A}(V) · κ_{B}(V). But by Lemmas 4.1 and 3.2(i), κ_{A}(V) = κ_{ℓ′}(𝔰). On the other hand, by Theorem 4.7, κ_{B}(V) = κ_{ℓ}(𝔰). The proof is complete.

5. An application: irreducible reductions modulo ℓ

Here we classify complex representations of SL_{n}(q) irreducible modulo ℓ.

5.1. A canonical composition factor. We will show that reduction modulo ℓ of L_{ℂ}(𝔰) has a canonical composition factor L(𝔰^{⋆}). The proof of the following lemma is an easy exercise.

Lemma 5.1. Let α^{(i)}, β^{(i)} ⊢ n_{i} with α^{(i)} ⊵ β^{(i)} for i = 1, …, m. Then: (i) α^{(1)} + … + α^{(m)} ⊵ β^{(1)} + … + β^{(m)}; (ii) if α^{(1)} + … + α^{(m)} = β^{(1)} + … + β^{(m)} then α^{(i)} = β^{(i)} for all i. □

Lemma 5.2. Any composition factor of V_{ℂ} := L_{ℂ}(σ, α^{(1)})∘…∘L_{ℂ}(σ, α^{(m)}) is of the form L_{ℂ}(σ, λ) with λ′ ⊴ ∑_{i=1}^{m}(α^{(i)})′. Moreover, L_{ℂ}(σ, λ) is a multiplicity one composition factor of V_{ℂ} if ∑_{i=1}^{m}(α^{(i)})′ = λ′.

Proof. By [10], [18], or [4], we have

L_{ℂ}(σ, α) ∘ L_{ℂ}(σ, β) = ∑_{γ⊢|α|+|β|} a_{αβγ}L_{ℂ}(σ, γ),

where a_{αβγ} are Littlewood-Richardson coefficients. Now the result follows by induction on m using associativity of Harish-Chandra induction and wellknown properties of Littlewood-Richardson coefficients. □

Lemma 5.3. If σ be an ℓ′-element, then every composition factor of V := L(σ, λ^{(1)}) ∘ … ∘ L(σ, λ^{(m)}) is L(σ, γ), with multiplicity one, or L(σ, η) with η ⊵ γ and η ≠ γ, where ∑_{i=1}^{m}(λ^{(i)})′ = γ′.

Proof. Let k = deg(σ). The unitriangularity of the submatrix Δ(σ, k) (see [10]) of the decomposition matrix of GL_{kd} implies that in the Grothendieck group of 𝔽GL_{kd}-modules we have

L(σ, λ^{(i)}) = \overline{L_{ℂ}(σ, λ^{(i)})} + ∑_{α ⊵ λ^{(i)}, α≠λ^{(i)}} x_{iα}\overline{L_{ℂ}(σ, α)}  (x_{iα} ∈ ℤ).

Hence

V = \overline{V}_{ℂ} + ∑ y_{α^{(1)},...,α^{(m)}} \overline{L_{ℂ}(σ, α^{(1)}) ∘ … ∘ L_{ℂ}(σ, α^{(m)})},  (5.1)

where the sum is over all α^{(i)} ⊵ λ^{(i)} such that α^{(j)} ≠ λ^{(j)} for at least one j, V_{ℂ} = L_{ℂ}(σ, λ^{(1)}) ∘ … ∘ L_{ℂ}(σ, λ^{(m)}) and y_{α^{(1)},...,α^{(m)}} ∈ ℤ.

Assume that L(σ, η) is a composition factor of some summand

\overline{L_{ℂ}(σ, α^{(1)}) ∘ … ∘ L_{ℂ}(σ, α^{(m)})}

in the summation in (5.1). Then by Lemma 5.2, L(σ, η) is a composition factor of some \overline{L_{ℂ}(σ, β)}, where ∑_{i=1}^{m}(α^{(i)})′ ⊵ β′. Then η ⊵ β, whence β′ ⊵ η′. But (λ^{(i)})′ ⊵ (α^{(i)})′ for each i (since α^{(i)} ⊵ λ^{(i)}). So by Lemma 5.1(i),

∑_{i=1}^{m}(λ^{(i)})′ ⊵ ∑_{i=1}^{m}(α^{(i)})′ ⊵ β′ ⊵ η′.

Since ∑_{i=1}^{m}(λ^{(i)})′ = γ′ by the choice of γ, we get η ⊵ γ. In particular, if η = γ, then Lemma 5.1(ii) implies that (λ^{(i)})′ = (α^{(i)})′, and so λ^{(i)} = α^{(i)} for all i, a contradiction.

On the other hand, by Lemma 5.2, L_{ℂ}(σ, γ) is a multiplicity one composition factor of V_{ℂ}, with other composition factors of the form L_{ℂ}(σ, μ) for μ ⊵ γ, μ ≠ γ. Finally, \overline{L_{ℂ}(σ, μ)} contains L(σ, μ) as a multiplicity one composition factor, with other factors of the form L(σ, η) for η ⊵ μ, η ≠ μ. So the right hand of (5.1) contains L(σ, γ) with multiplicity one, and all other composition factors are of the form L(σ, η) with η ⊵ γ and η ≠ γ, as stated. □

Theorem 5.4. Let V = L_{ℂ}(𝔰) be an irreducible ℂGL_{n}-module. Then V (mod ℓ) has L(𝔰^{⋆}) as a composition factor with multiplicity one.

Proof. In the notation of §1.2, assume that the ℓ′-parts s_{i} of σ_{i} are conjugate to s_{1} precisely when i ∈ {j_{1} = 1, j_{2}, …, j_{b}}. By [10], the Brauer character of

L_{ℂ}(σ_{j_{1}}, λ^{(j_{1})}) ∘ … ∘ L_{ℂ}(σ_{j_{b}}, λ^{(j_{b})})(mod ℓ)

is a sum of the Brauer character of W_{1} := L(s_{1}, μ^{(j_{1})}) ∘ … ∘ L(s_{1}, μ^{(j_{b})}) with some other Brauer character ψ, where (μ^{(i)})′ = k_{i}(λ^{(i)})′. By Lemma 5.3, W_{1} has L(s_{1}, δ^{(1)}) as a multiplicity one composition factor (see §1.2). On the other hand, using [9],

\overline{L_{ℂ}(σ_{j_{i}}, λ^{(j_{i})})} = L(σ_{j_{i}}, λ^{(j_{i})}) + ∑_{β^{(j_{i})} ⊵ λ^{(j_{i})}, β^{(j_{i})}≠λ^{(j_{i})}} x_{β^{(j_{i})}}^{i}L(σ_{j_{i}}, β^{(j_{i})})

= L(s_{1}, μ^{(j_{i})}) + ∑_{β^{(j_{i})} ⊵ λ^{(j_{i})}, β^{(j_{i})}≠λ^{(j_{i})}} x_{β^{(j_{i})}}^{i}L(s_{1}, ν^{(j_{i})}),

where x_{β^{(j_{i})}}^{i} ∈ ℤ and (ν^{(j_{i})})′ = k_{j_{i}}(β^{(j_{i})})′ ⊴ (μ^{(j_{i})})′. Hence, by Lemma 5.3,

any composition factor of ψ is of the form L(s_{1}, γ), where γ ⊵ δ^{(1)} but γ ≠ δ^{(1)}. Now the result comes from repeating this argument for all other conjugate classes of s_{i}. □

5.2. The James-Mathas Theorem. ℂGL_{n}-modules irreducible modulo ℓ have been classified by James and Mathas. For reader’s convenience, we reformulate the result here adapting to the notation of §1.2.

Theorem 5.5. [19, Theorem (4.33)] The reduction L_{ℂ}(𝔰)(mod ℓ) is irreducible if and only the following two conditions hold:

(JM1) If i ≠ j and deg(σ_{i}) = deg(σ_{j}), then [s_{i}] ≠ [s_{j}];

(JM2) For all i and all nodes (a, c) and (b, c) in the Young diagram of λ^{(i)}, the corresponding hook lengths h_{ac} and h_{bc} satisfy the condition

(\frac{q^{h_{ac}f} − 1}{q^{f} − 1})_{ℓ} = (\frac{q^{h_{bc}f} − 1}{q^{f} − 1})_{ℓ}

where f := deg(σ_{i}).

A partition λ will be called a JM-partition if it satisfies the condition (JM2) for some power q^{f} > 1. (Empty partition is a JM-partition).

Lemma 5.6. Let ℓ|(q − 1) and λ, λ^{(1)}, …, λ^{(m)}, μ, μ^{(1)}, …, μ^{(n)} be JMpartitions, with all the λ^{(i)}, μ^{(j)} non-empty. Then:

(i) if λ′ ≡ μ′(mod ℓ), then λ = μ; (ii) if ℓ|λ′, then λ is empty; (iii) if 0 ≤ a_{1} < a_{2} < … < a_{m}, 0 ≤ b_{1} < b_{2} < … < b_{n} are integers and ∑_{i=1}^{m} ℓ^{a_{i}}(λ^{(i)})′ = ∑_{j=1}^{n} ℓ^{b_{j}}(μ^{(j)})′, then m = n and a_{i} = b_{i}, λ^{(i)} = μ^{(i)} for all i.

Proof. (i) Adding empty columns to λ and μ, we may assume that both λ and μ have c columns and the cth column of both is empty. We prove by induction on i = 0, 1, …, c, that λ′_{c−i} = μ′_{c−i}. The induction base is clear. So suppose λ′_{j} = μ′_{j} for c − k + 1 ≤ j ≤ c, and prove that λ′_{c−k} = μ′_{c−k}. By assumption λ′_{c−k} ≡ μ′_{c−k}(mod ℓ), and we may assume λ′_{c−k} ≥ μ′_{c−k}. So λ′_{c−k} ≥ μ′_{c−k} + ℓ. Then we have

λ′_{c−k+1} = μ′_{c−k+1} ≤ μ′_{c−k} ≤ λ′_{c−k} − ℓ.

Hence, setting b := λ′_{c−k} and a := b − ℓ + 1, we have h_{b,c−k} = 1 and h_{a,c−k+1} = ℓ. Now, λ satisfies the condition (JM2) for some Q := q^{f} > 1. As ℓ|(q − 1), we have ℓ|(Q − 1), so (Q^{h_{b,c−k}} − 1)/(Q − 1) = 1, but (Q^{h_{a,c−k}} − 1)/(Q − 1) = (Q^{ℓ} − 1)/(Q − 1) is divisible by ℓ, contradicting (JM2).

(ii) follows from (i) by taking μ = ∅.

(iii) Adding empty partitions, we may assume that m = n and a_{i} = b_{i} = i − 1. The equality ∑_{i=1}^{n} ℓ^{i−1}(λ^{(i)})′ = ∑_{i=1}^{n} ℓ^{i−1}(μ^{(i)})′ now implies (λ^{(1)})′ ≡ (μ^{(1)})′( mod ℓ). By (i), we get λ^{(1)} = μ^{(1)}. Now we have ∑_{i=1}^{n−1} ℓ^{i−1}(λ^{(i+1)})′ = ∑_{i=1}^{n−1} ℓ^{i−1}(μ^{(i+1)})′, and the claim follows by induction on n. □

5.3. Relations between branching numbers.

Lemma 5.7. Let σ, σ′ ∈ \overline{𝔽}_{q}^{×}, σ_{ℓ} = u, σ_{ℓ′} = s, σ′_{ℓ} = u′, σ′_{ℓ′} = s′, and [στ] = [σ′] for some τ ∈ 𝔽_{q}^{×}. Then deg(σ) = deg(σ′). Moreover:

(i) if τ is an ℓ-element then [s] = [s′]; (ii) if τ is an ℓ′-element then [sτ] = [s′] and deg(s) = deg(s′).

Proof. The first statement is clear. By assumption, στ = (σ′)^{q^{i}} for some i ≥ 0. Then in (i) we have that τu(u′)^{−q^{i}} = (s′)^{q^{i}}s^{−1} is both an ℓ-element and an ℓ′-element and so it must be 1. Hence s = (s′)^{q^{i}}, i.e. [s] = [s′]. In (ii) we have that u(u′)^{−q^{i}} = (s′)^{q^{i}}s^{−1}τ^{−1} is both an ℓ-element and an ℓ′-element and so it must be 1. It follows that sτ = (s′)^{q^{i}}, i.e. [sτ] = [s′]. □

Lemma 5.8. If σ ∈ \overline{𝔽}_{q}^{×}, s = σ_{ℓ′}, deg(σ) = kd, and deg(s) = d, then the order of the group I = {τ ∈ O_{ℓ}(𝔽_{q}^{×}) | [σ] = [στ]} divides gcd(k, q − 1)_{ℓ}. In fact, |I| = gcd(k, q − 1)_{ℓ}, except for the case where ℓ = 2, q^{d} ≡ 3(mod 4), k = 2, and the ℓ-part u of σ has order ≥ 8. In the exceptional case |I| = 1.

Proof. We may assume that ℓ|(q−1). For τ ∈ I we have suτ = στ = (σ)^{q^{j}} = s^{q^{j}}u^{q^{j}} for some j ≥ 0. So s^{1−q^{j}} = u^{q^{j}−1}τ^{−1} = 1. As deg(s) = d, we must have d|j. Hence, using the facts that τ^{q} = τ and deg(σ) = kd, we get

τ^{k} = τ^{1+q^{j}+...+q^{(k−1)j}} = (σ^{q^{j}−1})^{1+q^{j}+...+q^{(k−1)j}} = σ^{q^{kj}−1} = 1.

Now |I| divides N := gcd(k, q − 1)_{ℓ}, as I is a cyclic ℓ-subgroup of 𝔽_{q}^{×}.

The second statement is now obvious if k = 1, so we may assume by [9] that k = ℓ^{a} and N = ℓ^{b} for some integers a ≥ b ≥ 1. Suppose first that ℓ ≠ 2 if q^{d} ≡ 3(mod 4). Then by Lemma 4.4, for any integers c, m ≥ 1, (q^{mdℓ^{c}} − 1)/(q^{md} − 1) has the ℓ-part equal to ℓ^{c}. Set

τ = σ^{q^{dℓ^{a−b}}−1} = u^{q^{dℓ^{a−b}}−1}.

In particular, τ is an ℓ-element. Since deg(σ) = dℓ^{a} and the ℓ-part of (q^{dℓ^{a}} − 1)/(q^{dℓ^{a−b}} − 1) is ℓ^{b}, we see that τ^{ℓ^{b}} = 1. But ℓ^{b}|(q − 1), hence τ^{q−1} = 1 and so τ ∈ I. On the other hand, if τ^{ℓ^{b−1}} = 1, then, since the ℓ-part of (q^{dℓ^{a−1}} − 1)/(q^{dℓ^{a−b}} − 1) is ℓ^{b−1}, we must have

σ^{q^{dℓ^{a−1}}−1} = τ^{(q^{dℓ^{a−1}}−1)/(q^{dℓ^{a−b}}−1)} = 1,

contrary to the equality deg(σ) = ℓ^{a}d. Thus |τ| = ℓ^{b} = N and so |I| = N.

Now let ℓ = 2 and q^{d} ≡ 3( mod 4); in particular, N = 2. First we consider the case where either a ≥ 2, or a = 1 but |u| < 8 (hence |u| = 4). Set

τ = σ^{q^{2^{a−1}d}−1} = u^{q^{2^{a−1}d}−1}.

Then τ ≠ 1 since deg(σ) = 2^{a}d. On the other hand, if a ≥ 2, then the 2-part of (q^{2^{a}d} − 1)/(q^{2^{a−1}d} − 1) is 2 and so τ^{2} = 1, whence τ ∈ I and |I| = N. If a = 1 and |u| = 4, then τ^{2} = u^{2(q^{d}−1)} = 1, and again |I| = N.

Finally, if a = 1 and I ∋ τ ≠ 1, then τ = −1 and −σ = (σ)^{q^{j}} for some j, 1 ≤ j < 2d. As above, d|j, so j = d. Hence −1 = (σ)^{q^{d}−1} = u^{q^{d}−1}. Since q^{d} ≡ 3(mod 4) and u is a 2-element, we conclude that |u| = 4. □

Let V_{ℂ} = L_{ℂ}(𝔰) be an arbitrary irreducible ℂGL_{n}-module. By Theorem 1.1, κ_{SL_{n}}^{GL_{n}}(V_{ℂ}) = {τ ∈ 𝔽_{q}^{×} | τ · 𝔰 = 𝔰}. Denote the ℓ-part (resp. ℓ′-part) of this number by κ_{ℓ}(V_{ℂ}) (resp. κ_{ℓ′}(V_{ℂ})). Thus κ_{SL_{n}}^{GL_{n}}(V_{ℂ}) = κ_{ℓ}(V_{ℂ})κ_{ℓ′}(V_{ℂ}).

Proposition 5.9. Let V_{ℂ} = L_{ℂ}(𝔰). Then:

(i) κ_{ℓ}(V_{ℂ}) divides κ_{ℓ}(𝔰^{⋆}).

(ii) Assume in addition that V_{ℂ}(mod ℓ) is irreducible. If 𝔰 is not critical, then κ_{ℓ}(V_{ℂ}) = κ_{ℓ}(𝔰^{⋆}). If 𝔰 is critical, then κ_{ℓ}(V_{ℂ}) = 1 and κ_{ℓ}(𝔰^{⋆}) = 2.

Proof. (i) Let τ be a generator of the group I := {σ ∈ O_{ℓ}(𝔽_{q}^{×}) | σ · 𝔰 = 𝔰}. Then κ_{ℓ}(V_{ℂ}) = |τ| = ℓ^{e} for some e. We adopt the notation σ_{i} = s_{i}u_{i}, k_{i}, d_{i}, as in §1.2. By Lemma 5.7(i), if [σ_{i}τ] = [σ_{j}] then [s_{i}] = [s_{j}]. So the multiplication by τ preserves the set S_{1} of all (σ_{i}, λ^{(i)}) with [s_{i}] = [s_{1}]. Consider one τ-orbit O ⊂ S_{1}, say τ permutes

(s_{1}u_{1}, λ^{(1)}), (s_{1}u_{2}, λ^{(1)}), …, (s_{1}u_{b}, λ^{(1)})

cyclically (after a suitable relabeling of the (σ_{i}, λ^{(i)})), with b = ℓ^{c} for some 0 ≤ c ≤ e. Then τ^{b} preserves (s_{1}u_{1}, λ^{(1)}). By Lemma 5.8, |τ^{b}| = ℓ^{e−c} divides gcd(k_{1}, q − 1)_{ℓ}. Defining γ_{O} := (bk_{1}(λ^{(1)})′)′, we see that ℓ^{e} = |τ| divides (γ_{O})′. Notice that, in the definition of 𝔰^{⋆}, (δ^{(1)})′ is just the sum of all the partitions (γ_{O})′, when O runs over all orbits inside S_{1}. Hence ℓ^{e}|(δ^{(1)})′. Similarly ℓ^{e}|(δ^{(j)})′ for all j, and of course ℓ^{e} divides n and q − 1. So ℓ^{e} divides κ_{ℓ}(𝔰^{⋆}).

(ii) is clear if ℓ |̸(q − 1), so let ℓ|(q − 1). We apply Theorem 5.5 to V_{ℂ}. Note that each orbit O as above has size 1, for, if [σ_{i}τ] = [σ_{j}], we saw that deg(σ_{i}) = deg(σ_{j}), and [s_{i}] = [s_{j}], whence i = j by (JM1). So the multiplication by τ preserves each ([σ_{i}], λ^{(i)}). Moreover, if [s_{i}] = [s_{j}] for some i ≠ j, then (JM1) implies that k_{i}d_{i} ≠ k_{j}d_{j}, and so k_{i} ≠ k_{j} as d_{i} = d_{j}. So, if

S_{1} = {(σ_{j_{1}}, λ^{(j_{1})}), …, (σ_{j_{b}}, λ^{(j_{b})})},

then we may assume that k_{j_{1}} < k_{j_{2}} < … < k_{j_{b}}. On the other hand, by Lemma 5.6(ii), we have ℓ |̸(λ^{(i)})′ for all i. As the k_{i} are ℓ-powers, the exact ℓ-power dividing (δ^{(1)})′ = ∑_{i=1}^{b} k_{j_{i}}(λ^{(j_{i})})′ is gcd(k_{j_{1}}, …, k_{j_{b}}). A similar result for all δ^{(j)} yields

κ_{ℓ}(𝔰^{⋆}) = gcd(q − 1, k_{1}, …, k_{a})_{ℓ}.  (5.2)

Let J_{i} := {σ ∈ O_{ℓ}(𝔽_{q}^{×}) | [σσ_{i}] = [σ_{i}]}. As noted above, τ ∈ J_{i} for each i, so I = ∩_{i=1}^{a}J_{i}. Assume ℓ ≠ 2 if q ≡ 3(mod 4). By Lemma 5.8, J_{i} is cyclic of order gcd(q − 1, k_{i})_{ℓ}, so |I| = gcd(q − 1, k_{1}, …, k_{a})_{ℓ}, as required. Finally, let ℓ = 2 and q ≡ 3(mod 4). Then |I| ≤ κ_{ℓ}(𝔰^{⋆}) ≤ 2. So |I| ≠ κ_{ℓ}(𝔰^{⋆}) if and only if |I| = 1 and κ_{ℓ}(𝔰^{⋆}) = 2. By (5.2), κ_{ℓ}(𝔰^{⋆}) = 2 if and only if all k_{i} ≥ 2. On the other hand, since I = ∩_{i=1}^{a}J_{i}, and each J_{i} is a subgroup of the cyclic group O_{2}(𝔽_{q}^{×}), we have |I| = 1 exactly when |J_{j}| = 1 for some j. As 2|k_{j}, Lemma 5.8 implies that |J_{j}| = 1 precisely when k_{j} = 2, d_{j} is odd, and |u_{j}| ≥ 8. We have shown that |I| ≠ κ_{ℓ}(𝔰^{⋆}) precisely when 𝔰 is critical. □

Proposition 5.10. Let V_{ℂ} = L_{ℂ}(𝔰). Then κ_{ℓ′}(V_{ℂ}) divides κ_{ℓ′}(𝔰^{⋆}).

Proof. Let J := {ν ∈ O_{ℓ′}(𝔽_{q}^{×}) | ν · 𝔰 = 𝔰} = ⟨ρ⟩. We adopt the notation of §1.2. By Lemma 5.7(ii), if [σ_{i}ρ] = [σ_{j}], then [s_{i}ρ] = [s_{j}] and k_{i} = k_{j}. So the multiplication by ρ sends the set S_{1} = {(σ_{i}, λ^{(i)}) | [s_{i}] = [s_{1}]} to some S_{j} = {(σ_{i}, λ^{(i)}) | [s_{i}] = [s_{j}]}, where [s_{j}] = [s_{1}ρ]. If S_{1} = {(σ_{i_{1}}, λ^{(i_{1})}), …, (σ_{i_{b}}, λ^{(i_{b})})}, then S_{j} = {(ρσ_{i_{1}}, λ^{(i_{1})}), …, (ρσ_{i_{b}}, λ^{(i_{b})})}, and

(δ^{(1)})′ = ∑_{v=1}^{b} k_{i_{v}}(λ^{(i_{v})})′, (δ^{(j)})′ = ∑_{v=1}^{b} k_{i_{v}}(λ^{(i_{v})})′.

So ρ sends the component (s_{1}, δ^{(1)}) of 𝔰^{⋆} to the component (s_{j}, δ^{(j)}) of 𝔰^{⋆}. Thus J stabilizes 𝔰^{⋆}, and so κ_{ℓ′}(V_{ℂ}) = |J| divides κ_{ℓ′}(𝔰^{⋆}). □

5.4. Proof of Theorem 1.3. Set G = GL_{n}, S = SL_{n}, and V_{ℂ}↓_{S} = ∑_{i=1}^{t} W_{ℂ}^{i}, a sum of t = κ_{ℓ′}(V_{ℂ}) · κ_{ℓ}(V_{ℂ}) irreducibles. By Theorem 5.4, V := L(𝔰^{⋆}) is a composition factor of V_{ℂ}(mod ℓ). By Theorem 1.1, V ↓_{S} is a sum of t′ := κ_{ℓ′}(𝔰^{⋆}) · κ_{ℓ}(𝔰^{⋆}) irreducibles. By Propositions 5.10, 5.9(i), t|t′.

If W_{ℂ}(mod ℓ) is irreducible then W_{ℂ}^{i}(mod ℓ) are irreducible for all i. So (V_{ℂ}(mod ℓ))↓_{S} has exactly t composition factors, and t ≥ t′. As t|t′, we have t = t′ and V_{ℂ}(mod ℓ) = V is irreducible. Now t = t′ implies κ_{ℓ′}(V_{ℂ}) = κ_{ℓ′}(𝔰^{⋆}) and κ_{ℓ}(V_{ℂ}) = κ_{ℓ}(𝔰^{⋆}), whence 𝔰 is not critical by Proposition 5.9(ii).

Conversely, if V_{ℂ}(mod ℓ) is irreducible, κ_{ℓ′}(𝔰) = κ_{ℓ′}(𝔰^{⋆}), and 𝔰 is not critical, then κ_{ℓ}(𝔰) = κ_{ℓ}(𝔰^{⋆}) by Proposition 5.9(ii). So V_{ℂ}(mod ℓ) = V, and t = t′. Hence W_{ℂ}(mod ℓ) is irreducible.

Example 5.11. (i) Let V_{ℂ} = L_{ℂ}(𝔰), W_{ℂ} be an irreducible constituent of V_{ℂ}↓_{SL_{n}}, and all σ_{i} be ℓ′-elements. Then V_{ℂ}(mod ℓ) is irreducible over GL_{n} if and only if W_{ℂ}( mod ℓ) is irreducible over SL_{n}. Indeed, in this case 𝔰 = 𝔰^{⋆} and that 𝔰 is not critical. So the claim follows fromTheorem 1.3.

(ii) The conditions (ii), (iii) of Theorem 1.3 cannot be relaxed. In the examples below W_{ℂ} = V_{ℂ}↓_{SL_{n}} is irreducible, but W_{ℂ}(mod ℓ) is reducible.

(a) Let q ≡ 3(mod 4), ℓ = 2, 2|n, and d be an odd divisor of n/2. Pick s ∈ \overline{𝔽}_{q}^{×} to be of order (q^{d} − 1)/2. Then deg(s) = d. Also, choose u ∈ 𝔽_{q^{2d}}^{×} of order ≥ 8. Set 𝔰 = [([su], (n/2d))]. Then L_{ℂ}(𝔰)(mod ℓ) is irreducible, κ_{ℓ′}(𝔰) = κ_{ℓ′}(𝔰^{⋆}) = 1 (as can be seen by direct computation), but 𝔰 is critical.

(b) Let r < ℓ be primes with ℓr|(q − 1). Pick an element ε ∈ 𝔽_{q}^{×} of order r, and r distinct ℓ-elements 1 = t_{1}, t_{2}, …, t_{r} ∈ 𝔽_{q}^{×}. Define

𝔰 = [([εt_{1}], (n/r)), ([ε^{2}t_{2}], (n/r)), …, ([ε^{r}t_{r}], (n/r))].

Then L_{ℂ}(𝔰)( mod ℓ) is irreducible, 𝔰 is not critical, but κ_{ℓ′}(𝔰) = 1 (by Lemma 5.7) and κ_{ℓ′}(𝔰^{⋆}) = r.

6. A unitriangular decomposition submatrix

We will need the following partial converse to Lemma 5.8:

Lemma 6.1. Let d ∈ ℕ and ℓ^{a}|(q − 1) for some a ∈ ℤ_{≥0}. Then there exists an ℓ-element u = u(a, d) ∈ \overline{𝔽}_{q}^{×}, depending only on a, d, such that, for any ℓ′-element s ∈ \overline{𝔽}_{q}^{×} of degree d, we have deg(su) = ℓ^{a}d and the order of I := {t ∈ O_{ℓ}(𝔽_{q}^{×}) | [su] = [sut]} is ℓ^{a}.

Proof. Taking u = 1 in the case a = 0, we may assume that a ≥ 1. Let P := O_{ℓ}(𝔽_{q^{dℓ^{a}}}^{×}) and Q := O_{ℓ}(𝔽_{q^{d}}^{×}). Note that Q < P via 𝔽_{q^{d}} ⊂ 𝔽_{q^{dℓ^{a}}}, P is cyclic, and ℓ^{a} divides |P/Q|. We will distinguish two cases.

Case 1: Either ℓ ≠ 2, or ℓ = 2 but q^{d} ≡ 1(mod 4). Then |P/Q| = ℓ^{a} by Lemma 4.4, and we choose u to be a generator of P.

Case 2: ℓ = 2 and q^{d} ≡ 3(mod 4). Then Q = ⟨−1⟩ ≃ C_{2}, ℓ^{a} = 2, and we choose u ∈ P such that u^{2} = −1.

In either case, we have u ∈ P with u^{ℓ^{a}} ∈ Q but u^{ℓ^{a−1}} ∉ Q. In particular,

u^{(q^{d}−1)ℓ^{a}} = 1, but u^{(q^{d}−1)ℓ^{a−1}} ≠ 1.  (6.1)

We show that deg(su) = dℓ^{a} for any s as in the assumption. Note that e := deg(su) is divisible by d = deg(s), as s is the ℓ′-part of su. Write e = kd for some k ∈ ℕ. Then k|ℓ^{a} since s, u ∈ 𝔽_{q^{dℓ^{a}}}. If k < ℓ^{a} then k|ℓ^{a−1}, so

1 = (su)^{q^{dℓ^{a−1}}−1} = u^{q^{dℓ^{a−1}}−1}.  (6.2)

In Case 1, the ℓ-part of \frac{q^{dℓ^{a−1}}−1}{q^{d}−1} is ℓ^{a−1} by Lemma 4.4. As u is an ℓ-element,

(6.2) implies u^{(q^{d}−1)ℓ^{a−1}} = 1, contrary to (6.1). In Case 2, we have a = 1, so (6.2) yields 1 = u^{q^{d}−1}, again contradicting (6.1). Thus k = ℓ^{a} as needed.

Set t := (su)^{q^{d}−1} = u^{q^{d}−1}. Then sut = (su)^{q^{d}} and so [sut] = [su]. By (6.1), t^{ℓ^{a}} = u^{(q^{d}−1)ℓ^{a}} = 1. But ℓ^{a}|(q − 1), whence t ∈ O_{ℓ}(𝔽_{q}^{×}) and so t ∈ I. Moreover, t^{ℓ^{a−1}} = u^{(q^{d}−1)ℓ^{a−1}} ≠ 1 by (6.1). Thus |t| = ℓ^{a}. On the other hand, |I| divides gcd(ℓ^{a}, q − 1)_{ℓ} = ℓ^{a} by Lemma 5.8. So |I| = ℓ^{a}. □

Let 𝔏/O_{ℓ′}(𝔽_{q}) be the set of O_{ℓ′}(𝔽_{q}^{×})-orbits on 𝔏 or a set of the orbit representatives depending on the context. We define a relation ⊵ on 𝔏/O_{ℓ′}(𝔽_{q}). First, for 𝔰 = [(σ_{1}, λ^{(1)}), …, (σ_{a}, λ^{(a)})] and 𝔱 = [(τ_{1}, μ^{(1)}), …, (τ_{b}, μ^{(b)})] in 𝔏, we set 𝔰 ⊵ 𝔱 if a = b, and, after a suitable relabeling of the (σ_{i}, λ^{(i)}), there is ν ∈ O_{ℓ′}(𝔽_{q}^{×}) such that [σ_{i}ν] = [τ_{i}] and λ^{(i)} ⊵ μ^{(i)} for all i. This relation factors through the action of O_{ℓ′}(𝔽_{q}^{×}) to give a relation ⊵ on 𝔏/O_{ℓ′}(𝔽_{q}).

Next, we define a relation ⊵ on IBr(SL_{n}) as follows. By Corollary 1.2, IBr(SL_{n}) = {L(𝔰)_{j} | 𝔰 ∈ 𝔏/O_{ℓ′}(𝔽_{q}), 1 ≤ j ≤ κ_{ℓ′}(𝔰)κ_{ℓ}(𝔰)}. For each 𝔰 ∈ 𝔏/O_{ℓ′}(𝔽_{q}) fix an arbitrary linear order ⊵ on the set

{L(𝔰)_{j} | 1 ≤ j ≤ κ_{ℓ′}(𝔰)κ_{ℓ}(𝔰)}.

If 𝔰, 𝔱 ∈ 𝔏/O_{ℓ′}(𝔽_{q}), 𝔰 ≠ 𝔱, set L(𝔰)_{i} ⊵ L(𝔱)_{j} if and only if 𝔰 ⊵ 𝔱.

Lemma 6.2. ⊵ is a partial order on 𝔏/O_{ℓ′}(𝔽_{q}) and on IBr(SL_{n}).

Proof. First, consider ⊵ on 𝔏/O_{ℓ′}(𝔽_{q}). Clearly, it is reflexive and transitive. Assume in the above notation that 𝔰 ⊵ 𝔱 and 𝔱 ⊵ 𝔰. Then a = b, and, after a suitable relabeling, [σ_{i}ν] = [τ_{i}] and λ^{(i)} ⊵ μ^{(i)} for all i and for some ν ∈ O_{ℓ′}(𝔽_{q}^{×}). Also, there is π ∈ S_{a} such that μ^{(π(i))} ⊵ λ^{(i)} for all i. Thus

λ^{(1)} + … + λ^{(a)} ⊵ μ^{(1)} + … + μ^{(a)} = μ^{(π(1))} + … + μ^{(π(a))} ⊵ λ^{(1)} + … + λ^{(a)}.

By Lemma 5.1(ii), λ^{(i)} = μ^{(i)} for all i, i.e. 𝔰 and 𝔱 belong to the same O_{ℓ′}(𝔽_{q}^{×})-orbit. Thus ⊵ is anti-symmetric. Now, ⊵ on IBr(SL_{n}) is clearly reflexive and transitive. If L(𝔰)_{i} ⊵ L(𝔱)_{j} and L(𝔱)_{j} ⊵ L(𝔰)_{i}, we may assume 𝔰 ≠ 𝔱. Then 𝔰 ⊵ 𝔱 and 𝔱 ⊵ 𝔰, and so 𝔰 = 𝔱, a contradiction. □

Proposition 6.3. Let G = GL_{n}, S = SL_{n}. For any V ∈ IBr(G) there is V_{ℂ} ∈ Irr(G) such that:

(i) V is a composition factor of V_{ℂ}(mod ℓ); (ii) κ_{S}^{G}(V) = κ_{S}^{G}(V_{ℂ}); (iii) If W is an irreducible constituent of V ↓_{S} then there is a unique irreducible consitutuent W_{ℂ} of (V_{ℂ})↓_{S} such that W is a multiplicity one composition factor of W_{ℂ}(mod ℓ) and all other composition factors U of W_{ℂ}(mod ℓ) satisfy U ⊵ W.

Proof. Write V = L(𝔰), κ_{ℓ}(𝔰) = ℓ^{c} for c ∈ ℤ_{≥0}, and κ_{ℓ′}(𝔰) = |J|, where J := {ν ∈ O_{ℓ′}(𝔽_{q}^{×}) | ν · 𝔰 = 𝔰} = ⟨ρ⟩. As ℓ^{c} divides each (λ^{(i)})′, we may choose μ^{(i)} with (λ^{(i)})′ = ℓ^{c}(μ^{(i)})′. Next, consider a ρ-orbit

([σ_{j_{1}}], λ^{(j_{1})}) ↦ ([σ_{j_{2}}], λ^{(j_{2})}) ↦ … ↦ ([σ_{j_{b}}], λ^{(j_{b})}) ↦ ([σ_{j_{1}}], λ^{(j_{1})}). (6.3)

By Lemma 5.7, the σ_{j_{i}} all have the same degree, say d. Also, because of (6.3) we may assume that σ_{j_{i}} = σ_{j_{1}}ρ^{i−1} for 1 ≤ i ≤ b. Let u := u(c, d) be as in Lemma 6.1. Set τ_{j_{m}} = σ_{j_{m}}u(c, d) for 1 ≤ m ≤ b. Do the same for all ρ-orbits on the set of components of 𝔰, and set V_{ℂ} := L_{ℂ}(𝔱), where

𝔱 := [([τ_{1}], μ^{(1)}), ([τ_{2}], μ^{(2)}), …, ([τ_{a}], μ^{(a)})].

As (τ_{i})_{ℓ′} = σ_{i} and deg(τ_{i}) = ℓ^{c} deg(σ_{i}) for all i, (i) holds by Theorem 5.4.

Next we show that κ_{ℓ′}(V_{ℂ}) = κ_{ℓ′}(𝔰). If I := {ν ∈ O_{ℓ′}(𝔽_{q}^{×}) | ν · 𝔱 = 𝔱} then κ_{ℓ′}(V_{ℂ}) = |I|. If ν ∈ I sends ([τ_{i}], μ^{(i)}) to ([τ_{j}], μ^{(j)}) then μ^{(i)} = μ^{(j)} and [νσ_{i}] = [σ_{j}] by Lemma 5.7(ii). Hence λ^{(i)} = λ^{(j)} ν sends ([σ_{i}], λ^{(i)}) to ([σ_{j}], λ^{(j)}), i.e. ν ∈ J. Thus I ⊂ J. To get the opposite inclusion we show that ρ ∈ I. This is obvious if ℓ |̸(q − 1) as then 𝔱 = 𝔰. So assume ℓ|(q − 1) and consider a ρ-orbit as in (6.3). Then λ^{(j_{1})} = … = λ^{(j_{b})}, and so μ^{(j_{1})} = … = μ^{(j_{b})}. Next, it is clear that ρ sends τ_{j_{i}} to τ_{j_{i+1}} for 1 ≤ i ≤ b−1. It remains to show [σ_{j_{1}}ρ^{b}u(c, d)] = [σ_{j_{1}}u(c, d)]. Denote σ := σ_{j_{1}}, β := ρ^{b}, u := u(c, d), k := |u|. Then [σβ] = [σ] by (6.3), whence σβ = σ^{q^{e}} and so β = σ^{q^{e}−1} for some e ∈ ℤ_{≥0}. Therefore

β^{k} = β^{1+q^{e}+...+q^{(k−1)e}} = σ^{q^{ke}−1} = (σu)^{q^{ke}−1}.

The last equality holds because ℓ|(q − 1) and so k divides q^{ke} − 1 by Lemma 4.4. Thus [σu] = [(σu)^{q^{ke}}] = [σuβ^{k}], i.e. β^{k} preserves [σu]. But β is an ℓ′-element and k is an ℓ-power, hence β also preserves [σu], as required.

Now we show that κ_{ℓ}(V_{ℂ}) = κ_{ℓ}(𝔰). Recall that [(τ_{i})_{ℓ′}] = [σ_{i}] ≠ [σ_{j}] = [(τ_{j})_{ℓ′}] whenever i ≠ j. So by Lemma 5.7(i), γ ∈ O_{ℓ}(𝔽_{q}^{×}) preserves 𝔱 precisely when γ preserves each [τ_{i}], which by Lemma 6.1 happens exactly when |γ| divides ℓ^{c}, whence the claim. In particular, we have established (ii).

By [10], in the Grothendieck group of 𝔽G-modules, we have

\overline{L_{ℂ}(τ_{i}, μ^{(i)})} = L(τ_{i}, μ^{(i)}) + ∑_{ν^{(i)} ⊵ μ^{(i)}, ν^{(i)}≠μ^{(i)}} x_{ν^{(i)}}^{i}L(τ_{i}, ν^{(i)})

for x_{ν^{(i)}}^{i} ∈ ℤ. As [(τ_{i})_{ℓ′}] = [σ_{i}] and deg(τ_{i}) = ℓ^{c} deg(σ_{i}), by [9] we have

\overline{L_{ℂ}(τ_{i}, μ^{(i)})} = L(σ_{i}, λ^{(i)}) + ∑_{ν^{(i)} ⊵ μ^{(i)}, ν^{(i)}≠μ^{(i)}} x_{ν^{(i)}}^{i}L(σ_{i}, η^{(i)}),

where (η^{(i)})′ = ℓ^{c}(ν^{(i)})′. As ν^{(i)} ⊵ μ^{(i)}, we have (λ^{(i)})′ = ℓ^{c}(μ^{(i)})′ ⊵ ℓ^{c}(ν^{(i)})′ = (η^{(i)})′, i.e. η^{(i)} ⊵ λ^{(i)}. As everything is true for every 1 ≤ i ≤ a, we have

V̄_{ℂ} = V + ∑_{η^{(1)},...,η^{(a)}} y_{η^{(1)},...,η^{(a)}}L(σ_{1}, η^{(1)}) ∘ … ∘ L(σ_{a}, η^{(a)}),  (6.4)

for some y_{η^{(1)},...,η^{(a)}} ∈ ℤ. Moreover, η^{(i)} ⊵ λ^{(i)} for all i and η^{(j)} ≠ λ^{(j)} for at least one j. Furthermore, each V′ := L(σ_{1}, η^{(1)}) ∘ … ∘ L(σ_{a}, η^{(a)}) ∈ IBr(G). By definition of ⊵, for any irreducible constituent W′ of V′↓_{S} and for any irreducible constituent W of V ↓_{S}, we have W′ ⊵ W and W′ ≄ W.

Also, by (ii), we may write (V_{ℂ})↓_{S} = ∑_{i=1}^{κ} W_{ℂ}^{i} and V ↓_{S} = ∑_{i=1}^{κ} W^{i}, where W_{ℂ}^{i} ∈ Irr(S), W^{i} ∈ IBr(S), and κ = κ_{S}^{G}(V). We may assume that W^{1} is a composition factor of W_{ℂ}^{1}( mod ℓ). Since W_{ℂ}^{1} and W^{1} have the same inertia group in G, we may relabel the W_{ℂ}^{i} so that W_{ℂ}^{i} = ^{g_{i}}W_{ℂ}^{1} whenever W^{i} = ^{g_{i}}W^{1}. Hence W^{i} is a composition factor of W_{ℂ}^{i}( mod ℓ) for all i. Now (iii) follows from (6.4) and the following remarks. □

Proof of Theorem 1.4. Let W be an irreducible 𝔽SL_{n}-module. Then W is a constituent of V ↓_{SL_{n}} for some V ∈ IBr(GL_{n}). Now we apply Proposition 6.3 to W and define Θ(W) = W_{ℂ}. □

Taking intersection with any ℓ-block B of SL_{n}, we also get a block version of Theorem 1.4: The decomposition submatrix of B, with respect to Θ(B ∩ Irr(SL_{n}) and the order ⊵ on B ∩ IBr(SL_{n}), is lower unitriangular. In particular, Θ(B ∩ Irr(SL_{n})) yields an ordinary basic set for B.

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Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

E-mail address: klesh@uoregon.edu

Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

Since August 2008: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA

E-mail address: tiep@math.arizona.edu