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Uno's invariant conjecture for the general linear and unitary groups in nondefining characteristics
Journal of Algebra 284 (2005) 462–479 www.elsevier.com/locate/jalgebra
Uno’s invariant conjecture for the general linear and unitary groups in nondefining characteristics Jianbei An Department of Mathematics, University of Auckland, Auckland, New Zealand Received 5 November 2002 Available online 23 December 2004 Communicated by Michel Broué
Abstract Uno’s invariant conjecture, which is a combination of Dade’s invariant conjecture and the Isaacs– Navarro conjecture is proved for general linear and unitary groups in nondefining characteristics. 2004 Elsevier Inc. All rights reserved.
1. Introduction Recently, Isaacs and Navarro [5] proposed a new conjecture which is a significant generalization of the Alperin–McKay conjecture, and Uno [9] raised an alternating sum version of the Isaacs–Navarro conjecture which is a refinement of the Dade conjecture. Dade’s invariant conjecture was proved in [1] for the general linear and unitary groups G in nondefining characteristics. In this paper, we show that Uno’s invariant conjecture also holds for G in nondefining characteristics. It is mentioned by Uno that his projective conjecture implies the Isaacs–Navarro conjecture. The paper is organized as follows. In Section 2, we fix notation, state the Isaacs–Navarro conjecture, the Dade and Uno invariant conjectures in detail. In Section 3 we prove some lemmas and a key proposition, which is due to Paul Fong. In the last section we verify
E-mail address: [email protected]. 0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2004.09.037
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Uno’s invariant conjecture of the general linear and unitary groups in nondefining characteristics.
2. The conjectures Let G be a finite group and r a prime. Given an r-subgroup chain C : P0 < P1 < · · · < Pw
(2.1)
of G, define |C| = w, Ck : P0 < P1 < · · · < Pk , and N(C) = NG (C) = N(P0 ) ∩ N(P1 ) ∩ · · · ∩ N(Pw ).
(2.2)
Then C is said to be radical if it satisfies the following two conditions: (a) P0 = Or (G) and (b) Pk = Or (N(Ck )) for 1 k w, where Or (N(Ck )) is the largest normal r-subgroup of the normalizer N(Ck ). Denote by R = R(G) the set of all radical r-chains of G, Irr(G) the set of all irreducible ordinary characters of G and Blk(G) the set of r-blocks. Let E be an extension of G, F = E/G, C ∈ R(G), ψ ∈ Irr(NG (C)) and let NE (C, ψ) be the stabilizer of (C, ψ) in E. Then NF (C, ψ) = NE (C, ψ)/NG (C) is a subgroup of F . For a subgroup U F , denote by Irr(NG (C), B, d, U ) the set of characters ψ in Irr(NG (C)) such that d(ψ) = d, B(ψ)G = B and NF (C, ψ) = U , where d(ψ) = logr (|G|r ) − logr (ψ(1)r ) is the r-defect of ψ and B(ψ) is the block of NG (C) containing ψ. Let H G, ϕ ∈ Irr(H ) and let α(ϕ) = αr (ϕ) be the integer 0 < α(ϕ) (r − 1) such that the r -part (|H |/ϕ(1))r of |H |/ϕ(1) satisfies |H | α(ϕ) ≡ (mod r). ϕ(1) r Given an integer α 1, let Irr(H, [α]) be the subset of Irr(H ) consisting of characters ϕ such that α(ϕ) ≡ ±α (mod r), and let Irr(H, B, d, U, [α]) = Irr(H, B, d, U ) ∩ Irr(H, [α]) and k(H, B, d, U, [α]) = |Irr(H, B, d, U, [α])|. Note that if r = 2, then Irr(H, [α]) = Irr(H ) and we may always suppose r is odd and 1 α (r − 1)/2. Let B ∈ Blk(G) with a defect group D = D(B) and the Brauer correspondent b ∈ Blk(NG (D)). Then k NG (D), B, d(B), U, [α] k NG (D), B, d(B), [α] = U E
is the number of characters ϕ ∈ Irr(b) such that ϕ has height 0 and α(ϕ) ≡ ±α (mod r), where d(B) is the defect of B.
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Isaacs–Navarro Conjecture [5, Conjecture B]. In the notation above, k G, B, d(B), [α] = k NG (D), B, d(B), [α] . Uno’s Invariant Conjecture [9, Conjecture 3.2]. If Op (G) = 1 and if D(B) > 1, then for any integers d 0 and 1 α (r − 1)/2, (−1)|C|k NG (C), B, d, U, [α] = 0, (2.3) C∈R/G
where R/G is a set of representatives for the G-orbits of R. Set k(NG (C), B, d, U ) = |Irr(NG (C), B, d, U )|. In the notation above the Dade invariant conjecture is stated as follows. Dade’s Invariant Conjecture [3]. If Op (G) = 1 and D(B) > 1, then for any integer d 0, (−1)|C| k NG (C), B, d, U = 0. C∈R/G
Note that if r = 2 or 3, then Uno’s conjecture is equivalent to Dade’s, so in the verification of Uno’s conjecture, we may suppose r 5. The following proposition will be used in Section 4 and its proof is an application of the proofs of [5]. Proposition (2A). Uno’s invariant conjecture holds for all blocks with cyclic defect groups. Proof. Let B ∈ Blk(G) with D = x, and let Ω(D) D be the unique subgroup of order = NG (Ω(D)). Then (2.3) is equivalent to p and G B, d, U, [α] . k G, B, d, U, [α] = k G,
(2.4)
∈ Blk(G), and Lr the Let b ∈ Blk(NG (D)) be the Brauer correspondent of B, bG = B r-regular elements of L = CG (D). By [5, Theorem (2.2)] there is a bijection χ → ψ = ψχ of Irr(B) onto Irr(b) such that
χ(xy) = εχ ψ(xy),
(2.5)
for any y ∈ Lr , where εχ = ±1. As shown in the proof of [5, Theorem (2.1)] (2.5) implies that χ(1)r ≡ ±|G : NG (D)|r ψ(1)r (mod r) and in particular, α(χ) ≡ ±α(ψ)
(mod r).
(2.6)
and B by B. It follows that there exists a bijection Φ : χ → χ˜ of Irr(B) Replace G by G such that α(χ) ≡ ±α(χ˜ ) (mod r). onto Irr(B)
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Let Irr(B, [α]) = Irr(G, [α]) ∩ Irr(B), χ ∈ Irr(B, [α]) and ψ = ψχ ∈ Irr(b, [α]). If τ ∈ −1 −1 NF (B), then τ normalizes D and so x = x τ is also a generator of D. Thus y = y τ ∈ Lr and (cf. [2, Corollary 1.9]) χ τ (xy) = χ x y = εχ ψ x y = εχ ψ τ (xy).
(2.7)
Similarly, χ˜ τ and ψ τ satisfy (2.7) with χ replaced by Φ(χ) = χ˜ . It follows by the defini [α]) are NF (B)-isomorphic. tion of Φ that Φ(χ τ ) = Φ(χ)τ , so that Irr(B, [α]) and Irr(B, This implies (2.4). 2
3. A main proposition and several lemmas We will follow the notation of Section 2. The first lemma is a simple observation. Lemma (3A). (a) Suppose C, C ∈ R(G) such that NG (C) = NG (C ) and NE (C) = NE (C ). Then for any block B ∈ Blk(G), any integers d, α and any subgroup U F k NG (C), B, d, U, [α] = k NG C , B, d, U, [α] .
If moreover, (−1)|C| = −(−1)|C | , then we can omit the two chains C and C from the alternating sum (2.3). (b) Let H be a subgroup of G and ϕ a character of H . Then α IndG H (ϕ) = α(ϕ). Proof. The proof of (a) is trivial. Since IndG H (ϕ)(1) = |G : H |ϕ(1), it follows that α IndG H (ϕ) =
|G|r |H |r = = α(ϕ). |G : H |r ϕ(1)r ϕ(1)r
2
Let X and X be subsets of irreducible characters, τ act on both X and X . A map Φ : X → X is defect preserving if for any ξ ∈ X , d Φ(ξ ) = d(ξ ); Φ is α-preserving if α Φ(ξ ) ≡ α(ξ )
(mod r);
and Φ is compatible with τ if Φ ξ τ = Φ(ξ )τ .
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Next we show that some defect preserving bijections given in [1] are also α-preserving. Let H = H1 × H2 be a direct product of two finite groups H1 and H2 , and let πi be the natural projection from H onto Hi for i = 1, 2. If C ∈ R(H ) is given by (2.1), then in general, Or (Hi ) πi (P1 ) πi (P2 ) · · · πi (Pw ) is not a radical chain of Hi . Let Y = {πi (P ): 0 w} and suppose |Y| = u. Relabel the subgroups in Y such that Y = {Q0 < Q1 < · · · < Qu }. Thus Q0 < Q1 < · · · < Qu is a radical chain of Hi , which is denoted by πi (C). Lemma (3B). Let n be a nonnegative integer, Bi ∈ Blk(Hi ) and R(Bi ) the subfamily of R(Hi ) consisting of chains C such that Blk(NHi (C)) contains a Brauer correspondent of Bi . (a) Suppose Xi ⊆ Irr(Hi ) for i = 1, 2, and ψ is a defect and α-preserving bijection of X1 onto X2 . In addition, let Ki = Hi S(n), and let Xi S(n) be the subset of Irr(Ki ) consisting of all characters covering a character of (Xi )n , where S(n) is the symmetric group on n letters. Then ψ can be extended to a defect and α-preserving bijection Ψ of X1 S(n) onto X2 S(n). Moreover, if τ is an automorphism of Ki for i = 1, 2 such that (Hi )τ = Hi , ψ(η)τ = ψ(ητ ) and y τ = y for each η ∈ X1 and y ∈ S(n), then Ψ is compatible with τ . (b) Let B = B1 × B2 ∈ Blk(H ), and let L(Bi ) be an Hi -invariant subset of R(Bi ), and L(B) a subset of R(B) consisting of all chains C such that πi (C) ∈ L(Bi ) for i = 1, 2. Suppose U Out(H ) is generated by τ1 , . . . , τt and τj = π1 (τj ) × π2 (τj ), where each πi (τj ) is an automorphism of Hi . Let Ui = πi (τ1 ), . . . , πi (τt ) for i = 1, 2. Then
(−1)|C| k NH (C), B, d, U, [α]
C∈L(B)/H
=
d1 +d2 =d, α1 α2 =α
2 i=1
(−1)
|C|
k NHi (C), Bi , di , Ui , [αi ] .
C∈L(Bi )/Hi
Proof. (a) It suffices to check that the defect preserving bijection Ψ given in the proof of [1, (1A)(a)] is also α-preserving. Since Ψ is induced by induction Ind, (3B)(a) follows by Lemma (3A)(b). (b) Similarly, since the bijection ϕ given in the proof (b) of [1, (1A)] satisfies Lemma (3A)(a), the same proof can be applied here with some obvious modifications. 2 Let q be a power of a prime p distinct from r, ε = +1 or −1 and let e be the multiplicative order of εq modulo r, and a = logr (|q e − ε|r ). In addition, let GLε (n, q) = GL(n, q) or U(n, q) according as ε = +1 or −1.
Lemma (3C). Let L = GLε (m, q)×GLε (m , q δ ) be a regular subgroup of G = GLε (n, q), where n = m + m δ or m + 2m δ according as (ε, ε ) = (−1, +1) or (ε, ε ) = (−1, +1). If e | n − m, then the perfect isometry RLG is an α-preserving bijection, up to a sign, between Er (L, (s)) and Er (G, (s)), where s is a semisimple r -element such that CG (s) L.
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Proof. Write n = m + eu for some integer u. First notice that if et is even, then q et /2 ≡ ±1 (mod r). This is trivial when e is odd. If e = 2e is even, then ε = +1 (cf. [4, Proposition (4F)]), et/2 = e t and q e ≡ ±1 (mod r), so the claim is also clear. If χ ∈ Er (L, (s)), then RLG (χ)(1) = |G : L|p χ(1) and so α RLG (χ) =
|G|p |G|r = α(χ). (|G : L|p )r χ(1)r |L|p
But n(n − 1) = m(m − 1) + eu(eu + 2m − 1) and eu(eu + 2m − 1) is even, so |G|p = q n(n−1)/2 ≡ ±q m(m−1)/2 ≡ ±|L|p (mod r). 2 The following proposition is the key step in the verification of Uno’s conjecture for GLε (n, q), its proof is due to Paul Fong. Proposition (3D). Let G = GLε (n, q), and let µ be a partition of n, and χµ the unipotent character of G parametrized by µ. Suppose µ has e-core κ, e-weight w and e-quotient (µ(1) , µ(2), . . . , µ(e) ), so that n = ew + |κ|. If w(i) = |µ(i) | and m = |κ|, then e w α(χµ ) ≡ ±ew q e − ε r α(χκ ) α(ωµ(i) )
(mod r),
(3.1)
i=1
where ωµ(i) ∈ Irr(S(w(i) )) is parametrized by µ(i) and χκ is the unipotent character of GLε (m, q) parametrized by κ. Here α(χκ ) = 1 if κ is a empty set. We may suppose r is odd since (3.1) is trivial when r = 2. Rewrite (3.1) as e w |GLε (m, q)|r |G|r |S(w(i) )|r ≡ ±ew q e − ε r χµ (1)r χκ (1)r ωµ(i) (1)r
(mod r).
(3.2)
i=1
Let ∼ be the following equivalence relation on nonzero integers: a ∼ b (mod r) if ar = br and ar ≡ br (mod r). By convention the r-part ar of a is taken to be positive if a is negative. Note equivalences for ∼ can be multiplied and cancelled. Thus (3.2) will hold if we show w |GLε (m, q)| |G| ∼ ±ew q e − ε χµ (1) χκ (1)
e |S(w(i) )| i=1
ωµ(i) (1)
(mod r).
We analyze the various quantities occurring in (3.3). First, suppose ε = +1, so |G| = q
n(n−1)/2
n k=1
q −1 , k
n k d(µ) k=1 (q − 1) . χµ (1) = q h h∈Hµ (q − 1)
(3.3)
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Here d(µ) = i (i − 1)µi ; µ1 , µ2 , µ3 , . . . are the parts of µ in decreasing order; and Hµ is the multiset of hook lengths of µ. For any positive integers k and n, set [k]q =
qk − 1 , q −1
[n]!q =
n
[k]q .
k=1
If H is a multiset of positive integers and e is a positive integer, let He = {h ∈ H : e | h} and He = {h ∈ H : e h}. Set [H ]!q =
[h]q ,
[He ]!q =
h∈H
[h]q ,
[He ]!q =
h∈He
[h]q .
h∈He
By convention [0]q = [0]!q = [∅]!q = 1. In particular, |G| = q n(n−1)/2−d(µ)(q − 1)n [Hµ ]!q . χµ (1)
(3.4)
Lemma (3E). In the notation above, [Hµ ]!q
(q e − 1)w w ! ∼± e [H ] [Hµ(i) ]!q e κ q (q − 1)we e
(mod r),
i=1
q d(µ) ∼ ±q d(κ)
(mod r).
Proof. Let D be a set of β-numbers for µ. The hooks of µ correspond to pairs (x, y) of integers, where 0 x < y, x ∈ / D, and y ∈ D. The hook (x, y) then has length y − x. Suppose µ∗ is gotten from µ by removing the hook (a, b) of length e. Removing b from D and inserting a into D gives a set D ∗ of β-numbers for µ∗ . We compare hooks of D and D ∗ . A hook (x, y) of D with x = a and y = b remains a hook of D ∗ of the same length. A hook (x, b) of D with x < a corresponds to the hook (x, a) of D ∗ . Moreover, b − x ≡ a − x (mod e) so that [b − x]q ≡ [a − x]q (mod e). A hook (b, y) of D with y > b corresponds to the hook (a, y) of D ∗ . Moreover, y − a ≡ y − b (mod e) so that [y − a]q ≡ [y − b]q (mod r). Lastly, D has e hooks with no counterparts in D ∗ : Namely, (x, b) where x ∈ / D and a x b, and (a, y) where y ∈ D with a y b. These hooks of D have lengths 1, 2, . . . , e, and contribute a factor [1]q [2]q · · · [e − 1]q ≡ ±e(q − 1)−(e−1) (mod r) to [Hµ,e ]q , where Hµ,e = (Hµ )e . Thus [Hµ,e ]!q ∼ ±e(q − 1)−(e−1) [Hµ∗ ,e ]!q (mod r). Now κ arises from µ by removing a sequence of w hooks of length e. Since Hκ,e = Hκ , iteration gives [Hµ,e ]!q ∼ ±ew (q − 1)−w(e−1) [Hκ ]!q
(mod r).
(3.5)
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On the other hand, [Hµ,e ]!q
=
qe − 1 q −1
e q eh − 1 q e − 1 w = [Hµ(i) ]!q e . qe − 1 q −1
w e
i=1 h∈Hµ(i)
(3.6)
i=1
The first equation of (3E) now follows from (3.5) and (3.6). Let µ = (µ1 , µ2 , µ3 , . . .), where µ1 µ2 µ3 · · ·, and let µ = (µ1 , µ2 , µ3 , . . .) be the dual partition of µ, where µ1 µ2 µ3 · · ·. By [6, Chapter I, (1.5), (1.6)] d(µ) =
1 µj µj − 1 . (i − 1)µi = 2 i
(3.7)
j
The node (i, j ) of the Young diagram of µ has hook length hij (µ) = µi + µj − i − j + 1 and content cij (µ) = j − i. Thus (3.7) implies that
hij (µ) = d µ + d(µ) + |µ|,
i,j
cij (µ) = d µ − d(µ).
i,j
In particular, 2d(µ) =
i,j
hij (µ) −
cij (µ) − |µ|.
(3.8)
i,j
Let µ, µ∗ , D, D ∗ be as in the first paragraph of the proof. Recall that each hook in D ∗ corresponds to a hook in D of congruent length modulo e. In addition, D has e hooks with no counterparts in D ∗ , these hooks having lengths 1, 2, . . . , e. Thus i,j
hij (µ) −
i,j
1 hij µ∗ ≡ 1 + 2 + 3 + · · · + e ≡ e(e + 1) (mod e). 2
Moreover, i,j
cij (µ) −
i,j
1 cij µ∗ ≡ e(e + 1) (mod e). 2
Indeed, µ∗ arises from µ by removing the hook (a, b). The nodes in this hook have contents 1, 2, . . . , e (mod e) and sum to 12 e(e + 1) (mod e). Lastly, |µ| − |µ∗ | = e. It follows from (3.8) that 2d(µ) ≡ 2d(µ∗ ) (mod e). Iterating gives 2d(µ) ≡ 2d(κ) (mod e), or equivalently, q d(µ) ∼ ±q d(κ) (mod r). This completes the proof of (3E). 2
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Corollary (3F). The congruence (3.3) reduces to e e |S(w(i) )| [Hµ(i) ]!q e ∼ ± ωµ(i) (1) i=1
(mod r).
i=1
Proof. As shown in the proof of (3C) q n(n−1)/2 ∼ q m(m−1)/2 (mod r). Thus (3F) follows from (3.4) and (3E). 2 By Corollary (3F), to show (3.3) it suffices to show [Hµ(i) ]!q e ∼ ±
|S(w(i) )| ωµ(i) (1)
(mod r).
(3.9)
We change notation for the sake of simplicity. We suppose from now on that r is a prime dividing q − 1 and µ is a partition of w. It suffices to show [Hµ ]!q ∼
|S(w)| ωµ (1)
(mod r).
(3.10)
In other words, q e and µ(i) in (3.9) are now denoted as q and µ in (3.10). Let µ˜ and (µ1 , µ2 , . . . , µr ) be the r-core and r-quotient of µ. The hooks of µ of length divisible by r are in bijection with the hooks of µ1 , µ2 , . . . , µr , a hook of µ of length rm corresponding to a hook in the r-quotient of length m. We iterate this process: Let µ˜ i and (µi1 , µi2 , . . . , µir ) be the r-core and r-quotient of µi ; let µ˜ ij and (µij 1 , µij 2 , . . . , µij r ) be the r-core and r-quotient of µij ; and so on. The configurations {µ˜ i1 i2 ...ik : k 0} and {µi1 i2 ...ik : k 0} are called the r-core tower and the r-quotient tower of µ. Here µ˜ ∅ = µ˜ and µ∅ = µ. Since µ is determined by its r-core and r-quotient, it follows that µ is determined by its r-core tower. It will be convenient to set I = {1, 2, . . . , r} and to represent elements of the k-fold Cartesian product I k by i = (i1 , i2 , . . . , ik ). By convention I 0 = {∅}. Let Hµ be the multiset of hook lengths of µ as before, let Hµ,r = {h ∈ Hµ : r | h}, and let Hµ,r = {h ∈ Hµ : r h}. Set hµ =
h∈Hµ
h,
hµ,r =
h,
hµ,r =
h∈Hµ,r
h.
h∈Hµ,r
The hook-length formula gives ωµ (1) = |S(w)|/ hµ . So (3.10) reduces to [Hµ ]!q ∼ ±hµ
(mod r).
(3.11)
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If r does not divide h, then [h]q = (q h − 1)/(q − 1) = q h−1 + · · · + q + 1 ≡ h (mod r). This implies [Hµ,r ]!q =
[h]q ≡
h∈Hµ,r
h = hµ,r
(mod r).
h∈Hµ,r
On the other hand, qh − 1 q −1
[Hµ,r ]!q =
h∈Hµ,r
=
qr − 1 q −1
= [r]q
n1 rh q −1 qr − 1
n1
i∈I h∈Hµi
[Hµi ]!q r ,
i∈I
where n1 =
i∈I
|µi |. Thus
n [Hµi ]!q r [Hµ ]!q = [Hµ,r ]!q [Hµ,r ]!q ∼ [r]q 1 hµ,r
(mod r).
i∈I
Iteration gives n n n hµi,r [Hµ ]!q ∼ [r]q 1 [r]q r 2 [r]q r 2 3 · · · ×
(mod r),
(3.12)
k0 i∈I k
where nk =
i∈I k
|µi |.
Lemma (3G). [r]q ∼ r (mod r). Proof. Write q − 1 = r a t, where a 1 and gcd(r, t) = 1. Then
q −1= 1+r t r
a
r
r r ai i r t , −1= i i=1
whence [r]q =
r q r − 1 r a(i−1) i−1 = r t . q −1 i i=1
The summand for i = 1 is for 2 i < r are divisible by r 2 since r divides the summandsa(i−1) r r; ; the summand for i = r is divisible by r 2 the binomial coefficient i and r divides r 2 a(r−1) since r divides r . Thus (3G) holds. 2
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We apply (3G) to each of the terms [r]q , [r]q r , [r]q r 2 , . . . in (3.12) to get [Hµ ]!q ∼ r n1 +n2 +n3 +··· ×
hµi,r
(mod r).
(3.13)
k0 i∈I k
But hµ = hµ,r hµ,r n1 =r hµi hµ,r i∈I
= r n1
hµi ,r hµi ,r hµ,r
i∈I
=r
n1 +n2
hµi
i∈I 2
= ··· = r n1 +n2 +···
hµi
,r
hµ,r
i∈I
hµi ,r .
k0 i∈I k
This and (3.13) now imply (3.11). If ε = −1, then e is odd and we replace q by −q in the proof above. This gives a proof of (3.3).
4. The proof of Uno’s invariant conjecture Let V be the underlying space of G = GLε (n, q), H G and let CV (H ) and [V , H ] be the subspaces of V generated by the vectors of V fixed and moved by H , respectively. Let r 5 and let A(R) be the intersections of all the maximal normal abelian subgroups of an r-subgroup R G, and P (R) = Ωa (A(R)). A radical chain C : 1 < P1 < · · · < Pw of G is central radical if Pk = P (Pk ) for 1 k w. Denote by CR = CR(G) the set of all central radical chains of G. Let B ∈ Blk(G), CR(B) = CR ∩ R(B), and let CR∗ (B) be the subfamily of CR(B) consisting of chains C satisfying CV (P1 ) = CV (D), where P1 is the first nontrivial subgroup of C and D = D(B). Let Fq be the set of polynomials serving as elementary divisors for all semisimple elements of G (cf. [1, p. 379]), Fqk = Fqk (r) the subset of F = Fq consisting of all polynomials whose roots have multiplicative order r k . In addition, let Fq (r, a) = Fq0 ∪ Fq1 ∪ · · · ∪ Fqa and Fq (r) = k0 Fqk (r). Then Fq0 = {T − 1} and by [8, (1) and (2)], |Fq (r, a)| =
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1 + (r a − 1)/e and |Fqa+i | = (r a − r a−1 )/e for i 1 (note that the number e here is the e in [7] and [8]). Moreover, by [7, (1.5)], the induced degree δΓ = e or er i according as Γ ∈ Fq (r, a)\{T − 1} or Fqa+i , and each elementary divisor of an r-element is an element of Fq (r). We write
Fq (r, a) = ∆(0, j ): 0 j r a − 1 /e ,
∆(0, 0) = T − 1
and for i 1
Fqa+i (r) = ∆(i, j ): 1 j r a − r a−1 /e . A sequence (wi ) = (w0 , w1 , w2 , . . . , wt ) of nonnegative integers is called an r-weight sequence of u if t
wi r i = u.
(4.1)
i=0
Given such a sequence (wi ), let ΩG ((wi )) be the subset of Irr(B0 (G)) consisting of irreducible characters χy,µ of G = GLε (eu, q) such that a −1)/e 1+(r
a−1 (r a −r )/e
m0,j = w0 ,
j =0
mi,j = wi
j =1
for i 1 and µ = ,j µ ,j , where B0 (G) is the principal block of G, m ,j = m∆( ,j )(y) and each µ ,j is a partition of m ,j , except when = j = 0, in which case µ0,0 is a partition of em0,0 with the empty e-core. Thus ΩG (wi ) (disjoint), Irr B0 (G) = (wi )
where (wi ) runs over all r-weight sequences of u. Since χy,µ (1) = |G : CG (y)|p χµ (1), it follows that α(χy,µ ) =
|G|p α(χµ ) ≡ α(χµ ) |CG (y)|p
(mod r),
where χµ is the unipotent character of CG (y) parametrized by µ. Thus α(χy,µ ) ≡ α(χµ ) =
α(χµ ,j )
(mod r),
(4.2)
,j
where χµ0,0 is the unipotent character of GLε (em0,0 , q) parametrized by µ0,0 and for other j , j , χµ ,j is the unipotent character of GLε (m ,j , q er ) parametrized by µ ,j .
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β
An r-weight sequence (wi ) of (4.1) determines a unique partition λ = (λ0 0 , . . . , λt t ) of u, called an r-weight partition, where λi = r i and βi = wi for all i 0. β β β Let H = GLε (u, q e ) G = GLε (eu, q), λ = (λ0 0 , λ2 2 , . . . , λt t ) a partition of u with λ0 < λ1 < · · · < λt and R = Rλ a radical subgroup of H and G such that CG (R) = CH (R) =
t
β
Ki i ,
NH (R) =
i=0
NG (R) =
t
Ki S(βi ),
i=0 t τi , Ki S(βi ), i=0
where Ki = GLε (λi , q e ) GLε (eλi , q) and τi ∈ GLε (eλi , q) induces a field automorphism of order e on Ki . Thus P (Rλ ) = Rλ and each radical subgroup W of G or H satisfying P (W ) = W and CV (W ) = 0 is conjugate to some Rλ . If K G, denote by CR∗ (K) the subfamily of R(CK (Or (K))) consisting of the chains whose subgroups W satisfies P (W ) = W and CV (W ) = 0. Let CR∗K (λ) be the subfamily of CR∗ (K) consisting of all chains whose first subgroup P1 = Rλ , CR∗K (λ)e the subfamily of CR∗K (λ) consisting of the chains C with |C| even and CR∗K (λ)o = CR∗ (K)\CR∗K (λ)e . If C ∈ CR∗H (λ) is given by (2.1) with Rλ = Or (H ), then g(C) : Rλ < P2 < · · · < Pw is a chain of CR∗ (NH (Rλ )), |g(C)| = |C| − 1, NH (C) = NNH (Rλ ) (g(C)) and g is a bijection of CR∗H (λ) onto CR∗ (NH (Rλ )). We can identify CR∗H (λ) with CR∗ (NH (Rλ )). For K = H or G, let XK0 (λ) = ΩK ((β)) or ∅ according to whether or not λ is an r-weight partition, where β = (β0 , . . . , βt ). In addition, let XK+ (λ) = and XK− (λ) = XK0 (λ)
Irr B0 NK (C)
(4.3)
C∈CR∗K (λ)e /NK (Rλ )
C∈CR∗K (λ)o /NK (Rλ ) Irr(B0 (NK (C))).
Lemma (4A). Let XH0 = ∅ or ΩH ((δi )) according as u is not a power of r or u = r , and let XH+ =
Irr B0 NH (C) ,
C∈CR∗ (H )e /H
XH− = XH0
Irr B0 NH (C) .
C∈CR∗ (H )o /H
Then there exists a defect and α-preserving bijection Ψ of XH+ onto XH− which is also compatible with each τ ∈ Out(H ). Proof. If C0 (H ) ∈ R(H ) with |C0 (H )| = 0, then the decompositions XH0 (λ), Irr B0 (H ) = XH0 λ
CR∗ (H ) = C0 (H ) CR∗H (λ) λ
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475
are disjoint, where λ runs over the partitions of u with λ = (u). It then suffices to show that there exists a defect and α preserving bijection Ψ of XH+ (λ) onto XH− (λ) which is compatible with each τ ∈ Out(H ). But as shown in the proof of [1, (4C)] there is a bijection of XH+ (λ) onto XH− (λ) which is defect preserving and compatible with each τ ∈ Out(H ), so it suffices to show that Ψ is α-preserving. If u = 1, then XH+ (λ) = XH− (λ) = ∅, XH+ = XH− = XH0 = ΩH ((1, 0)) and Ψ is the identity map. Let Ki = GLε (λi , q e ) and Wi = GLε (λi , q e ) S(βi ), so that Ki < G. By induction, there is a defect and α preserving bijection ψ of XK+i onto XK−i , which is compatible with each τi ∈ Out(Ki ). By (3B)(a), ψ extends to a defect and α preserving bijection Ψi of XK+i S(βi ) and XK−i S(βi ), which is compatible with each τ ∈ Out(GLε (λi βi , q e )), where Wi GLε (λi βi , q e ) H . β But CR∗ (Wi ) = CR∗ (Ki i ), so as shown in the proof of [1, (1E)(a)] XK+i S(βi ) =
Irr B0 NWi (C)
C∈CR∗ (Wi )e /Wi
and XK−i S(βi ) = XK0 i S(βi ) C∈CR∗ (Wi )o /Wi Irr(B0 (NWi (C))). Let πWi be the projection of NH (Rλ ) onto Wi . Then
CR∗H (λ) = C ∈ CR∗ (H ): πWi (C) ∈ CR∗ (Wi ) C0 (H ) and so by Lemma (3B), there is a defect and α-preserving bijection Φ of XH+ (λ) =
t + i=0 XKi S(βi ) onto t i=0
0 XK−i S(βi ) = YH (λ)
Irr B0 NH (C) ,
C∈CR∗H (λ)o /NH (Rλ )
0 where YH (λ) = ti=0 XK0 i S(βi ). Thus it suffices to show that there is a defect and α 0 (λ). preserving bijection of XH0 (λ) onto YH 0 (λ) = ∅, and Φ is a required If some λi is not a power of r, then XK0 i = ∅, XH0 (λ) = YH + − bijection of XH (λ) onto XH (λ). Suppose each λi is a power of r, and we may suppose λi = r i for all i 0. Thus β = (β0 , β1 , . . . , βt ) is an r-weight sequence, XK0 i = ΩKi ((δj i )). It follows by [1, (4A)] 0 (λ) = that there is a defect preserving bijection ψ ∗ between XH0 (λ) = ΩH (β) and YH
t i=0 ΩKi ((δj i )) S(βi ), which is compatible with each τ ∈ Out(H ). As shown in the proof of [1, (4A)] the bijection ψ ∗ is given by induction Ind, so ψ ∗ is also α preserving by Lemma (3A)(b). This proves (4A). 2 Theorem (4B). Let B be an r-block of G = GLε (n, q) with a positive defect. If r and q are relatively prime, then B satisfies Uno’s invariant conjecture.
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Proof. (1) First, we reduce the proof to the subfamily CR of central radical chains. Namely, we show
(−1)|C| k N(C), B, d, U, [α] =
C∈R/G
(−1)|C| k N(C), B, d, U, [α] .
C∈CR/G
Let M be the complementary G-invariant subfamily R\CR of CR in R. It suffices to show that
(−1)|C| k N(C), B, d, U, [α] = 0.
(4.4)
C∈M/G
As shown in the proof of [1, (2D)], there is a bijection ϕ from M to itself satisfying NG (C) = NG (ϕ(C)), ϕ(ϕ(C)) = C, |ϕ(C)| = |C| ± 1 and NE (C) = NE (ϕ(C)) for an extension E of G. Thus (4.4) follows by Lemma (3A)(a). (2) Reduce the proof to the subfamily CR∗ , namely,
(−1)|C| k N(C), B, d, U, [α] =
C∈CR(B)/G
(−1)|C| k N(C), B, d, U, [α] .
C∈CR∗ (B)/G
(4.5) If we replace the initial step of the induction on the dimension in the proof of [1, (3A)] by Proposition (2A), then the same proof can be applied here with some obvious modifications. (3) Reduce the proof to blocks B with D = D(B) satisfying CV (D) = {0}. Let B be a block of G = GLε (V ) labelled by (s, κ), D = D(B), V0 = CV (D) and V+ = [V , D]. Then D = D0 × D+ and we may suppose s ∈ CG (D) = GLε (V0 ) × C+ , where D0 = 1V0 , D+ G+ = GLε (V+ ) and C+ = CG+ (D+ ). Thus s = s0 × s+ with s0 ∈ GLε (V0 ) and s+ ∈ C+ . Let B(0) and B(+) be blocks of G0 = GLε (V0 ) and G+ labelled by (s0 , κ) and (s+ , −), respectively. Then B(0) has defect 0 consisting of a unique irreducible character χ0 with α(χ0 ) = α0 . If L = G0 × G+ G, then RLG is a perfect isometry between Irr(B(0) × B(+)) and Irr(B) which is compatible with the outermorphisms of G (cf. [1, p. 381]). By Lemma (3C), RLG is α-preserving. Since RLG is also defect preserving and each character of Irr(B(0) × B(+)) has the form χ0 × χ+ for some χ+ ∈ Irr(B(+)) and since α(χ0 × χ+ ) = α0 α(χ+ ), it follows that k G, B, d, U, [α] = k G+ , B(+), d, U+ , [α+ ] , where α+ ≡ α/α0 (mod r). Suppose C ∈ CR∗ (B) is given by (2.1) with |C| 1. We may suppose Pt = 1V0 × P+ (t) for all t 1, where P+ (t) G+ . Define ϕ(C) : 1 < P+ (1) < P+ (2) < · · · < P+ (w).
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477
Then ϕ(C) ∈ CR∗ (B(+)), NG (C) = G0 × NG+ (ϕ(C)) and |C| = |ϕ(C)|. Applying the proof of [1, (3B)], we have k NG (C), B, d, U, [α] = k NG+ ϕ(C) , B(+), d, U+ , [α+ ] .
(4.6)
This implies the step (3). (4) Reduce the proof to the principal block B = B0 . The reduction [1, (3C)] is obtained by several defect preserving bijections and a cancellation. Each bijection is either given by a perfect isometry satisfying Lemma (3C) or by an induction IndH K of some subgroups K H G. By Lemma (3A)(b) and Lemma (3C), these bijections are also α-preserving. The last cancellation give in [1, p. 383] is given by a bijection satisfying Lemma (3A)(a), which is also α-preserving. Applying the modified proof of [1, (3C)], we get the proof of step (4). (5) Final calculation. Suppose B is the principal block of G = GLε (eu, q) for some u 1. The proof is similar the that of Lemma (4A). Note 0 XG (λ), Irr B0 (G) =
CR∗ (G) = C0 (G) CR∗G (λ),
λ
λ
where |C0 (G)| = 0 and λ runs over all partitions of u. So it suffices to show that there exists a defect and α preserving bijection Ψ of XG+ (λ) and XG− (λ) which is compatible with each τ ∈ Out(G). Given integer i 0, let Li = NGLε (eλi ,q) (Ki ) = τi , Ki for some τi ∈ GLε (eλi , q) inducing a field automorphism of order e on Ki . By (4A), there is a defect and α preserving bijection ψ of XK+i onto XK−i , which is compatible with each τ ∈ Out(Ki ). In particular, ψ(χ τi ) = ψ(χ)τi for any χ ∈ XK+i . Since CR∗ (Li ) = CR∗ (Ki ), it follows that ψ extends to a defect and α preserving bijection of XL+i and XL−i , where XL±i is defined similar to that of XK±i with XK0 i replaced by τi , XK0 i . Here τi , XK0 i denotes the irreducible characters of Li covering characters of XK0 i . Similar proof to that
of Lemma (4A) shows that there is a defect and α-preserving bijection Φ of XG+ (λ) = ti=0 XL+i S(βi ) onto t i=0
0 XL−i S(βi ) = YG (λ)
Irr B0 NG (C) ,
C∈CR∗G (λ)o /NG (Rλ )
0 (λ) = t τ , X 0 S(β ). It then suffices to show that there is a defect and α where YG i i=0 i Ki 0 preserving bijection of YG (λ) onto XG0 (λ). 0 (λ) = X 0 (λ) = ∅. Suppose λ is an r-weight If λ is not an r-weight partition, then YG G 0 partition, so that XG (λ) = ΩG ((β)) and we may suppose λi = r i . Thus (δ i ) is an r-weight sequence of r i , XK0 i = ΩKi ((δ i )) consists of irreducible characters ψ∆ of a+i ∗ a+i ∗ or Ki = GLe (r i , q e ) parametrized by (∆, 1) for ∆ ∈ (Fqa+i e ) , where (Fq e ) = Fq e
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Fq e (r, a) according as i > 0 or i = 0. If a character χ ∈ τi , ΩKi ((δ i )) is not an exteni sion of the trivial character, then χ has degree e|Ki |p /(q er − ε); if χ is one of the e extensions of the trivial character, then χ(1) = 1. In particular, τi , ΩK (δ i ) = r a − r a−1 /e i
or e + r a − 1 /e
according as i > 0 or i = 0. We my suppose
τ0 , ΩK0 (δ 0 ) = ξ1 , . . . , ξe , ξΓ | Γ ∈ Fq (r, a)\{T − 1}
and τi , ΩKi ((δ i )) = {ξΓ | Γ ∈ Fqa+i } for i > 0, where each ξi ∈ τ0 , ΩK0 ((δ 0 )) is an i extension of the trivial character and ξΓ (1) = e|Ki |p /(q er − ε) for Γ ∈ Fqa+i or Fq (r, a) according as i > 0 or i = 0. It follows that i
e|Ki |r (q er − ε)r α(ξΓ ) = ≡ ± qe − ε r , e(|Ki |p )r
α(ξi ) ≡ ±e q e − ε r
(mod r).
(4.7)
Each character ζ ∈ τi , ΩKi ((δ i )) S(βi ) has the form τ ,K S(β ) ζ = ζ (i, µi ) = Indτii ,Kii S(zii) ξ˜ ωµi , where ξ˜ is an extension of ξ ∈ (τi , ΩKi ((δ i )))βi to (τi , Ki )βi S(zi ) and zi is the type of ξ and ωµi is a irreducible character of the Young subgroup S(zi ). Thus if i > 0 and f = (r a − r a−1 )/e, then µi = (µi,1 , µi,2 , . . . , µi,f ) and each µi,j is a partition of the multiplicity ui,j = mξ∆(i,j) (ξ ) of ξ∆(i,j ) in ξ . If i = 0 and h = (r a − 1)/e, then µ0 = (µ∗0,1 , µ∗0,2 , . . . , µ∗0,e , µ0,1 , . . . , µ0,h ) and each µ∗0,j is a partition of u∗0,j = mξj (ξ ) and each µ0,j is a partition of u0,j = mξ∆(0,j) (ξ ). By Lemma (3A)(b) and (4.7), ±((q e − ε)r )βi α(ωµi ) α(ζ ) = α ξ˜ α(ωµi ) ≡ ±eu0,0 ((q e − ε)r )β0 α(ωµ0 )
if i > 0, if i = 0,
(mod r)
(4.8)
where u0,0 = u∗0,1 + u∗0,2 + · · · + u∗0,e . Let y be a semisimple element of G = GLε (eu, q) such that m∆(i,j ) (y) = ui,j or eu0,0 according as (i, j ) = (0, 0) or (i, j ) = (0, 0). Then y is a uniquely determined r-element, up to conjugacy. In addition, let γi,j = µi,j for all i, j except when i = j = 0, in which case let γ0,0 be the unique
partition of em0,0 with empty e-core and e-quotient (µ∗0,1 , µ∗0,2 , . . . , µ∗0,e ), and let γ = i,j γi,j . Then χy,γ ∈ ΩG ((β)). If χγ is the unipotent character of CG (y) parametrized by γ , then by Proposition (3D) and (4.8), α(χγ ) ≡ ±
t α ζ(i, µi ) (mod r). i=0
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479
0 If η = ti=0 ζ (i, µi ), then η ∈ YG (λ) = ti=0 τi , ΩKi ((δ i )) S(βi ). Define ϕ(η) = χy,γ . Then by (4.2) α(ϕ(η)) ≡ ±α(η) (mod r). As shown in the proof of [1, (4D)(a)] ϕ is also 0 (λ) onto Ω ((β)), which is also compatible with the a defect preserving bijection of YG G outer automorphisms of G. This completes the proof. 2
Acknowledgments The author would like to thank Paul Fong for Proposition (3D) and its proof, and thank Jorn Olsson for valuable suggestions. The author would also like to thank the Marsden Fund for their support.
References [1] J. An, Dade’s invariant conjecture for the general linear and unitary groups in nondefining characteristics, Trans. Amer. Math. Soc. 353 (2001) 365–390. [2] E.C. Dade, Blocks with cyclic defect groups, Ann. of Math. 84 (1966) 20–48. [3] E.C. Dade, Counting characters in blocks, in: II. 9. Representation Theory of Finite Groups, Columbus, OH, 1995, in: Ohio State Univ. Math. Res. Inst. Publ., vol. 6, de Gruyter, Berlin, 1997, pp. 45–59. [4] P. Fong, B. Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982) 109– 153. [5] I.M. Isaacs, G. Navarro, New refinements of the McKay conjecture for arbitrary finite groups, Ann. of Math. 156 (2002) 333–344. [6] I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Science Publ., 1995. [7] G.O. Michler, J.B. Olsson, Character correspondences in finite general linear, unitary and symmetric groups, Math. Z. 184 (1983) 203–233. [8] J.B. Olsson, On the number of characters in blocks of finite general linear, unitary and symmetric groups, Math. Z. 186 (1984) 41–47. [9] K. Uno, Conjectures on character degrees for the simple Thompson group, Osaka J. Math. 41 (2004) 11–36.
Uno’s invariant conjecture for the general linear and unitary groups in nondefining characteristics Jianbei An Department of Mathematics, University of Auckland, Auckland, New Zealand Received 5 November 2002 Available online 23 December 2004 Communicated by Michel Broué
Abstract Uno’s invariant conjecture, which is a combination of Dade’s invariant conjecture and the Isaacs– Navarro conjecture is proved for general linear and unitary groups in nondefining characteristics. 2004 Elsevier Inc. All rights reserved.
1. Introduction Recently, Isaacs and Navarro [5] proposed a new conjecture which is a significant generalization of the Alperin–McKay conjecture, and Uno [9] raised an alternating sum version of the Isaacs–Navarro conjecture which is a refinement of the Dade conjecture. Dade’s invariant conjecture was proved in [1] for the general linear and unitary groups G in nondefining characteristics. In this paper, we show that Uno’s invariant conjecture also holds for G in nondefining characteristics. It is mentioned by Uno that his projective conjecture implies the Isaacs–Navarro conjecture. The paper is organized as follows. In Section 2, we fix notation, state the Isaacs–Navarro conjecture, the Dade and Uno invariant conjectures in detail. In Section 3 we prove some lemmas and a key proposition, which is due to Paul Fong. In the last section we verify
E-mail address: [email protected]. 0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2004.09.037
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Uno’s invariant conjecture of the general linear and unitary groups in nondefining characteristics.
2. The conjectures Let G be a finite group and r a prime. Given an r-subgroup chain C : P0 < P1 < · · · < Pw
(2.1)
of G, define |C| = w, Ck : P0 < P1 < · · · < Pk , and N(C) = NG (C) = N(P0 ) ∩ N(P1 ) ∩ · · · ∩ N(Pw ).
(2.2)
Then C is said to be radical if it satisfies the following two conditions: (a) P0 = Or (G) and (b) Pk = Or (N(Ck )) for 1 k w, where Or (N(Ck )) is the largest normal r-subgroup of the normalizer N(Ck ). Denote by R = R(G) the set of all radical r-chains of G, Irr(G) the set of all irreducible ordinary characters of G and Blk(G) the set of r-blocks. Let E be an extension of G, F = E/G, C ∈ R(G), ψ ∈ Irr(NG (C)) and let NE (C, ψ) be the stabilizer of (C, ψ) in E. Then NF (C, ψ) = NE (C, ψ)/NG (C) is a subgroup of F . For a subgroup U F , denote by Irr(NG (C), B, d, U ) the set of characters ψ in Irr(NG (C)) such that d(ψ) = d, B(ψ)G = B and NF (C, ψ) = U , where d(ψ) = logr (|G|r ) − logr (ψ(1)r ) is the r-defect of ψ and B(ψ) is the block of NG (C) containing ψ. Let H G, ϕ ∈ Irr(H ) and let α(ϕ) = αr (ϕ) be the integer 0 < α(ϕ) (r − 1) such that the r -part (|H |/ϕ(1))r of |H |/ϕ(1) satisfies |H | α(ϕ) ≡ (mod r). ϕ(1) r Given an integer α 1, let Irr(H, [α]) be the subset of Irr(H ) consisting of characters ϕ such that α(ϕ) ≡ ±α (mod r), and let Irr(H, B, d, U, [α]) = Irr(H, B, d, U ) ∩ Irr(H, [α]) and k(H, B, d, U, [α]) = |Irr(H, B, d, U, [α])|. Note that if r = 2, then Irr(H, [α]) = Irr(H ) and we may always suppose r is odd and 1 α (r − 1)/2. Let B ∈ Blk(G) with a defect group D = D(B) and the Brauer correspondent b ∈ Blk(NG (D)). Then k NG (D), B, d(B), U, [α] k NG (D), B, d(B), [α] = U E
is the number of characters ϕ ∈ Irr(b) such that ϕ has height 0 and α(ϕ) ≡ ±α (mod r), where d(B) is the defect of B.
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Isaacs–Navarro Conjecture [5, Conjecture B]. In the notation above, k G, B, d(B), [α] = k NG (D), B, d(B), [α] . Uno’s Invariant Conjecture [9, Conjecture 3.2]. If Op (G) = 1 and if D(B) > 1, then for any integers d 0 and 1 α (r − 1)/2, (−1)|C|k NG (C), B, d, U, [α] = 0, (2.3) C∈R/G
where R/G is a set of representatives for the G-orbits of R. Set k(NG (C), B, d, U ) = |Irr(NG (C), B, d, U )|. In the notation above the Dade invariant conjecture is stated as follows. Dade’s Invariant Conjecture [3]. If Op (G) = 1 and D(B) > 1, then for any integer d 0, (−1)|C| k NG (C), B, d, U = 0. C∈R/G
Note that if r = 2 or 3, then Uno’s conjecture is equivalent to Dade’s, so in the verification of Uno’s conjecture, we may suppose r 5. The following proposition will be used in Section 4 and its proof is an application of the proofs of [5]. Proposition (2A). Uno’s invariant conjecture holds for all blocks with cyclic defect groups. Proof. Let B ∈ Blk(G) with D = x, and let Ω(D) D be the unique subgroup of order = NG (Ω(D)). Then (2.3) is equivalent to p and G B, d, U, [α] . k G, B, d, U, [α] = k G,
(2.4)
∈ Blk(G), and Lr the Let b ∈ Blk(NG (D)) be the Brauer correspondent of B, bG = B r-regular elements of L = CG (D). By [5, Theorem (2.2)] there is a bijection χ → ψ = ψχ of Irr(B) onto Irr(b) such that
χ(xy) = εχ ψ(xy),
(2.5)
for any y ∈ Lr , where εχ = ±1. As shown in the proof of [5, Theorem (2.1)] (2.5) implies that χ(1)r ≡ ±|G : NG (D)|r ψ(1)r (mod r) and in particular, α(χ) ≡ ±α(ψ)
(mod r).
(2.6)
and B by B. It follows that there exists a bijection Φ : χ → χ˜ of Irr(B) Replace G by G such that α(χ) ≡ ±α(χ˜ ) (mod r). onto Irr(B)
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Let Irr(B, [α]) = Irr(G, [α]) ∩ Irr(B), χ ∈ Irr(B, [α]) and ψ = ψχ ∈ Irr(b, [α]). If τ ∈ −1 −1 NF (B), then τ normalizes D and so x = x τ is also a generator of D. Thus y = y τ ∈ Lr and (cf. [2, Corollary 1.9]) χ τ (xy) = χ x y = εχ ψ x y = εχ ψ τ (xy).
(2.7)
Similarly, χ˜ τ and ψ τ satisfy (2.7) with χ replaced by Φ(χ) = χ˜ . It follows by the defini [α]) are NF (B)-isomorphic. tion of Φ that Φ(χ τ ) = Φ(χ)τ , so that Irr(B, [α]) and Irr(B, This implies (2.4). 2
3. A main proposition and several lemmas We will follow the notation of Section 2. The first lemma is a simple observation. Lemma (3A). (a) Suppose C, C ∈ R(G) such that NG (C) = NG (C ) and NE (C) = NE (C ). Then for any block B ∈ Blk(G), any integers d, α and any subgroup U F k NG (C), B, d, U, [α] = k NG C , B, d, U, [α] .
If moreover, (−1)|C| = −(−1)|C | , then we can omit the two chains C and C from the alternating sum (2.3). (b) Let H be a subgroup of G and ϕ a character of H . Then α IndG H (ϕ) = α(ϕ). Proof. The proof of (a) is trivial. Since IndG H (ϕ)(1) = |G : H |ϕ(1), it follows that α IndG H (ϕ) =
|G|r |H |r = = α(ϕ). |G : H |r ϕ(1)r ϕ(1)r
2
Let X and X be subsets of irreducible characters, τ act on both X and X . A map Φ : X → X is defect preserving if for any ξ ∈ X , d Φ(ξ ) = d(ξ ); Φ is α-preserving if α Φ(ξ ) ≡ α(ξ )
(mod r);
and Φ is compatible with τ if Φ ξ τ = Φ(ξ )τ .
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Next we show that some defect preserving bijections given in [1] are also α-preserving. Let H = H1 × H2 be a direct product of two finite groups H1 and H2 , and let πi be the natural projection from H onto Hi for i = 1, 2. If C ∈ R(H ) is given by (2.1), then in general, Or (Hi ) πi (P1 ) πi (P2 ) · · · πi (Pw ) is not a radical chain of Hi . Let Y = {πi (P ): 0 w} and suppose |Y| = u. Relabel the subgroups in Y such that Y = {Q0 < Q1 < · · · < Qu }. Thus Q0 < Q1 < · · · < Qu is a radical chain of Hi , which is denoted by πi (C). Lemma (3B). Let n be a nonnegative integer, Bi ∈ Blk(Hi ) and R(Bi ) the subfamily of R(Hi ) consisting of chains C such that Blk(NHi (C)) contains a Brauer correspondent of Bi . (a) Suppose Xi ⊆ Irr(Hi ) for i = 1, 2, and ψ is a defect and α-preserving bijection of X1 onto X2 . In addition, let Ki = Hi S(n), and let Xi S(n) be the subset of Irr(Ki ) consisting of all characters covering a character of (Xi )n , where S(n) is the symmetric group on n letters. Then ψ can be extended to a defect and α-preserving bijection Ψ of X1 S(n) onto X2 S(n). Moreover, if τ is an automorphism of Ki for i = 1, 2 such that (Hi )τ = Hi , ψ(η)τ = ψ(ητ ) and y τ = y for each η ∈ X1 and y ∈ S(n), then Ψ is compatible with τ . (b) Let B = B1 × B2 ∈ Blk(H ), and let L(Bi ) be an Hi -invariant subset of R(Bi ), and L(B) a subset of R(B) consisting of all chains C such that πi (C) ∈ L(Bi ) for i = 1, 2. Suppose U Out(H ) is generated by τ1 , . . . , τt and τj = π1 (τj ) × π2 (τj ), where each πi (τj ) is an automorphism of Hi . Let Ui = πi (τ1 ), . . . , πi (τt ) for i = 1, 2. Then
(−1)|C| k NH (C), B, d, U, [α]
C∈L(B)/H
=
d1 +d2 =d, α1 α2 =α
2 i=1
(−1)
|C|
k NHi (C), Bi , di , Ui , [αi ] .
C∈L(Bi )/Hi
Proof. (a) It suffices to check that the defect preserving bijection Ψ given in the proof of [1, (1A)(a)] is also α-preserving. Since Ψ is induced by induction Ind, (3B)(a) follows by Lemma (3A)(b). (b) Similarly, since the bijection ϕ given in the proof (b) of [1, (1A)] satisfies Lemma (3A)(a), the same proof can be applied here with some obvious modifications. 2 Let q be a power of a prime p distinct from r, ε = +1 or −1 and let e be the multiplicative order of εq modulo r, and a = logr (|q e − ε|r ). In addition, let GLε (n, q) = GL(n, q) or U(n, q) according as ε = +1 or −1.
Lemma (3C). Let L = GLε (m, q)×GLε (m , q δ ) be a regular subgroup of G = GLε (n, q), where n = m + m δ or m + 2m δ according as (ε, ε ) = (−1, +1) or (ε, ε ) = (−1, +1). If e | n − m, then the perfect isometry RLG is an α-preserving bijection, up to a sign, between Er (L, (s)) and Er (G, (s)), where s is a semisimple r -element such that CG (s) L.
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Proof. Write n = m + eu for some integer u. First notice that if et is even, then q et /2 ≡ ±1 (mod r). This is trivial when e is odd. If e = 2e is even, then ε = +1 (cf. [4, Proposition (4F)]), et/2 = e t and q e ≡ ±1 (mod r), so the claim is also clear. If χ ∈ Er (L, (s)), then RLG (χ)(1) = |G : L|p χ(1) and so α RLG (χ) =
|G|p |G|r = α(χ). (|G : L|p )r χ(1)r |L|p
But n(n − 1) = m(m − 1) + eu(eu + 2m − 1) and eu(eu + 2m − 1) is even, so |G|p = q n(n−1)/2 ≡ ±q m(m−1)/2 ≡ ±|L|p (mod r). 2 The following proposition is the key step in the verification of Uno’s conjecture for GLε (n, q), its proof is due to Paul Fong. Proposition (3D). Let G = GLε (n, q), and let µ be a partition of n, and χµ the unipotent character of G parametrized by µ. Suppose µ has e-core κ, e-weight w and e-quotient (µ(1) , µ(2), . . . , µ(e) ), so that n = ew + |κ|. If w(i) = |µ(i) | and m = |κ|, then e w α(χµ ) ≡ ±ew q e − ε r α(χκ ) α(ωµ(i) )
(mod r),
(3.1)
i=1
where ωµ(i) ∈ Irr(S(w(i) )) is parametrized by µ(i) and χκ is the unipotent character of GLε (m, q) parametrized by κ. Here α(χκ ) = 1 if κ is a empty set. We may suppose r is odd since (3.1) is trivial when r = 2. Rewrite (3.1) as e w |GLε (m, q)|r |G|r |S(w(i) )|r ≡ ±ew q e − ε r χµ (1)r χκ (1)r ωµ(i) (1)r
(mod r).
(3.2)
i=1
Let ∼ be the following equivalence relation on nonzero integers: a ∼ b (mod r) if ar = br and ar ≡ br (mod r). By convention the r-part ar of a is taken to be positive if a is negative. Note equivalences for ∼ can be multiplied and cancelled. Thus (3.2) will hold if we show w |GLε (m, q)| |G| ∼ ±ew q e − ε χµ (1) χκ (1)
e |S(w(i) )| i=1
ωµ(i) (1)
(mod r).
We analyze the various quantities occurring in (3.3). First, suppose ε = +1, so |G| = q
n(n−1)/2
n k=1
q −1 , k
n k d(µ) k=1 (q − 1) . χµ (1) = q h h∈Hµ (q − 1)
(3.3)
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Here d(µ) = i (i − 1)µi ; µ1 , µ2 , µ3 , . . . are the parts of µ in decreasing order; and Hµ is the multiset of hook lengths of µ. For any positive integers k and n, set [k]q =
qk − 1 , q −1
[n]!q =
n
[k]q .
k=1
If H is a multiset of positive integers and e is a positive integer, let He = {h ∈ H : e | h} and He = {h ∈ H : e h}. Set [H ]!q =
[h]q ,
[He ]!q =
h∈H
[h]q ,
[He ]!q =
h∈He
[h]q .
h∈He
By convention [0]q = [0]!q = [∅]!q = 1. In particular, |G| = q n(n−1)/2−d(µ)(q − 1)n [Hµ ]!q . χµ (1)
(3.4)
Lemma (3E). In the notation above, [Hµ ]!q
(q e − 1)w w ! ∼± e [H ] [Hµ(i) ]!q e κ q (q − 1)we e
(mod r),
i=1
q d(µ) ∼ ±q d(κ)
(mod r).
Proof. Let D be a set of β-numbers for µ. The hooks of µ correspond to pairs (x, y) of integers, where 0 x < y, x ∈ / D, and y ∈ D. The hook (x, y) then has length y − x. Suppose µ∗ is gotten from µ by removing the hook (a, b) of length e. Removing b from D and inserting a into D gives a set D ∗ of β-numbers for µ∗ . We compare hooks of D and D ∗ . A hook (x, y) of D with x = a and y = b remains a hook of D ∗ of the same length. A hook (x, b) of D with x < a corresponds to the hook (x, a) of D ∗ . Moreover, b − x ≡ a − x (mod e) so that [b − x]q ≡ [a − x]q (mod e). A hook (b, y) of D with y > b corresponds to the hook (a, y) of D ∗ . Moreover, y − a ≡ y − b (mod e) so that [y − a]q ≡ [y − b]q (mod r). Lastly, D has e hooks with no counterparts in D ∗ : Namely, (x, b) where x ∈ / D and a x b, and (a, y) where y ∈ D with a y b. These hooks of D have lengths 1, 2, . . . , e, and contribute a factor [1]q [2]q · · · [e − 1]q ≡ ±e(q − 1)−(e−1) (mod r) to [Hµ,e ]q , where Hµ,e = (Hµ )e . Thus [Hµ,e ]!q ∼ ±e(q − 1)−(e−1) [Hµ∗ ,e ]!q (mod r). Now κ arises from µ by removing a sequence of w hooks of length e. Since Hκ,e = Hκ , iteration gives [Hµ,e ]!q ∼ ±ew (q − 1)−w(e−1) [Hκ ]!q
(mod r).
(3.5)
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On the other hand, [Hµ,e ]!q
=
qe − 1 q −1
e q eh − 1 q e − 1 w = [Hµ(i) ]!q e . qe − 1 q −1
w e
i=1 h∈Hµ(i)
(3.6)
i=1
The first equation of (3E) now follows from (3.5) and (3.6). Let µ = (µ1 , µ2 , µ3 , . . .), where µ1 µ2 µ3 · · ·, and let µ = (µ1 , µ2 , µ3 , . . .) be the dual partition of µ, where µ1 µ2 µ3 · · ·. By [6, Chapter I, (1.5), (1.6)] d(µ) =
1 µj µj − 1 . (i − 1)µi = 2 i
(3.7)
j
The node (i, j ) of the Young diagram of µ has hook length hij (µ) = µi + µj − i − j + 1 and content cij (µ) = j − i. Thus (3.7) implies that
hij (µ) = d µ + d(µ) + |µ|,
i,j
cij (µ) = d µ − d(µ).
i,j
In particular, 2d(µ) =
i,j
hij (µ) −
cij (µ) − |µ|.
(3.8)
i,j
Let µ, µ∗ , D, D ∗ be as in the first paragraph of the proof. Recall that each hook in D ∗ corresponds to a hook in D of congruent length modulo e. In addition, D has e hooks with no counterparts in D ∗ , these hooks having lengths 1, 2, . . . , e. Thus i,j
hij (µ) −
i,j
1 hij µ∗ ≡ 1 + 2 + 3 + · · · + e ≡ e(e + 1) (mod e). 2
Moreover, i,j
cij (µ) −
i,j
1 cij µ∗ ≡ e(e + 1) (mod e). 2
Indeed, µ∗ arises from µ by removing the hook (a, b). The nodes in this hook have contents 1, 2, . . . , e (mod e) and sum to 12 e(e + 1) (mod e). Lastly, |µ| − |µ∗ | = e. It follows from (3.8) that 2d(µ) ≡ 2d(µ∗ ) (mod e). Iterating gives 2d(µ) ≡ 2d(κ) (mod e), or equivalently, q d(µ) ∼ ±q d(κ) (mod r). This completes the proof of (3E). 2
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Corollary (3F). The congruence (3.3) reduces to e e |S(w(i) )| [Hµ(i) ]!q e ∼ ± ωµ(i) (1) i=1
(mod r).
i=1
Proof. As shown in the proof of (3C) q n(n−1)/2 ∼ q m(m−1)/2 (mod r). Thus (3F) follows from (3.4) and (3E). 2 By Corollary (3F), to show (3.3) it suffices to show [Hµ(i) ]!q e ∼ ±
|S(w(i) )| ωµ(i) (1)
(mod r).
(3.9)
We change notation for the sake of simplicity. We suppose from now on that r is a prime dividing q − 1 and µ is a partition of w. It suffices to show [Hµ ]!q ∼
|S(w)| ωµ (1)
(mod r).
(3.10)
In other words, q e and µ(i) in (3.9) are now denoted as q and µ in (3.10). Let µ˜ and (µ1 , µ2 , . . . , µr ) be the r-core and r-quotient of µ. The hooks of µ of length divisible by r are in bijection with the hooks of µ1 , µ2 , . . . , µr , a hook of µ of length rm corresponding to a hook in the r-quotient of length m. We iterate this process: Let µ˜ i and (µi1 , µi2 , . . . , µir ) be the r-core and r-quotient of µi ; let µ˜ ij and (µij 1 , µij 2 , . . . , µij r ) be the r-core and r-quotient of µij ; and so on. The configurations {µ˜ i1 i2 ...ik : k 0} and {µi1 i2 ...ik : k 0} are called the r-core tower and the r-quotient tower of µ. Here µ˜ ∅ = µ˜ and µ∅ = µ. Since µ is determined by its r-core and r-quotient, it follows that µ is determined by its r-core tower. It will be convenient to set I = {1, 2, . . . , r} and to represent elements of the k-fold Cartesian product I k by i = (i1 , i2 , . . . , ik ). By convention I 0 = {∅}. Let Hµ be the multiset of hook lengths of µ as before, let Hµ,r = {h ∈ Hµ : r | h}, and let Hµ,r = {h ∈ Hµ : r h}. Set hµ =
h∈Hµ
h,
hµ,r =
h,
hµ,r =
h∈Hµ,r
h.
h∈Hµ,r
The hook-length formula gives ωµ (1) = |S(w)|/ hµ . So (3.10) reduces to [Hµ ]!q ∼ ±hµ
(mod r).
(3.11)
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If r does not divide h, then [h]q = (q h − 1)/(q − 1) = q h−1 + · · · + q + 1 ≡ h (mod r). This implies [Hµ,r ]!q =
[h]q ≡
h∈Hµ,r
h = hµ,r
(mod r).
h∈Hµ,r
On the other hand, qh − 1 q −1
[Hµ,r ]!q =
h∈Hµ,r
=
qr − 1 q −1
= [r]q
n1 rh q −1 qr − 1
n1
i∈I h∈Hµi
[Hµi ]!q r ,
i∈I
where n1 =
i∈I
|µi |. Thus
n [Hµi ]!q r [Hµ ]!q = [Hµ,r ]!q [Hµ,r ]!q ∼ [r]q 1 hµ,r
(mod r).
i∈I
Iteration gives n n n hµi,r [Hµ ]!q ∼ [r]q 1 [r]q r 2 [r]q r 2 3 · · · ×
(mod r),
(3.12)
k0 i∈I k
where nk =
i∈I k
|µi |.
Lemma (3G). [r]q ∼ r (mod r). Proof. Write q − 1 = r a t, where a 1 and gcd(r, t) = 1. Then
q −1= 1+r t r
a
r
r r ai i r t , −1= i i=1
whence [r]q =
r q r − 1 r a(i−1) i−1 = r t . q −1 i i=1
The summand for i = 1 is for 2 i < r are divisible by r 2 since r divides the summandsa(i−1) r r; ; the summand for i = r is divisible by r 2 the binomial coefficient i and r divides r 2 a(r−1) since r divides r . Thus (3G) holds. 2
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We apply (3G) to each of the terms [r]q , [r]q r , [r]q r 2 , . . . in (3.12) to get [Hµ ]!q ∼ r n1 +n2 +n3 +··· ×
hµi,r
(mod r).
(3.13)
k0 i∈I k
But hµ = hµ,r hµ,r n1 =r hµi hµ,r i∈I
= r n1
hµi ,r hµi ,r hµ,r
i∈I
=r
n1 +n2
hµi
i∈I 2
= ··· = r n1 +n2 +···
hµi
,r
hµ,r
i∈I
hµi ,r .
k0 i∈I k
This and (3.13) now imply (3.11). If ε = −1, then e is odd and we replace q by −q in the proof above. This gives a proof of (3.3).
4. The proof of Uno’s invariant conjecture Let V be the underlying space of G = GLε (n, q), H G and let CV (H ) and [V , H ] be the subspaces of V generated by the vectors of V fixed and moved by H , respectively. Let r 5 and let A(R) be the intersections of all the maximal normal abelian subgroups of an r-subgroup R G, and P (R) = Ωa (A(R)). A radical chain C : 1 < P1 < · · · < Pw of G is central radical if Pk = P (Pk ) for 1 k w. Denote by CR = CR(G) the set of all central radical chains of G. Let B ∈ Blk(G), CR(B) = CR ∩ R(B), and let CR∗ (B) be the subfamily of CR(B) consisting of chains C satisfying CV (P1 ) = CV (D), where P1 is the first nontrivial subgroup of C and D = D(B). Let Fq be the set of polynomials serving as elementary divisors for all semisimple elements of G (cf. [1, p. 379]), Fqk = Fqk (r) the subset of F = Fq consisting of all polynomials whose roots have multiplicative order r k . In addition, let Fq (r, a) = Fq0 ∪ Fq1 ∪ · · · ∪ Fqa and Fq (r) = k0 Fqk (r). Then Fq0 = {T − 1} and by [8, (1) and (2)], |Fq (r, a)| =
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1 + (r a − 1)/e and |Fqa+i | = (r a − r a−1 )/e for i 1 (note that the number e here is the e in [7] and [8]). Moreover, by [7, (1.5)], the induced degree δΓ = e or er i according as Γ ∈ Fq (r, a)\{T − 1} or Fqa+i , and each elementary divisor of an r-element is an element of Fq (r). We write
Fq (r, a) = ∆(0, j ): 0 j r a − 1 /e ,
∆(0, 0) = T − 1
and for i 1
Fqa+i (r) = ∆(i, j ): 1 j r a − r a−1 /e . A sequence (wi ) = (w0 , w1 , w2 , . . . , wt ) of nonnegative integers is called an r-weight sequence of u if t
wi r i = u.
(4.1)
i=0
Given such a sequence (wi ), let ΩG ((wi )) be the subset of Irr(B0 (G)) consisting of irreducible characters χy,µ of G = GLε (eu, q) such that a −1)/e 1+(r
a−1 (r a −r )/e
m0,j = w0 ,
j =0
mi,j = wi
j =1
for i 1 and µ = ,j µ ,j , where B0 (G) is the principal block of G, m ,j = m∆( ,j )(y) and each µ ,j is a partition of m ,j , except when = j = 0, in which case µ0,0 is a partition of em0,0 with the empty e-core. Thus ΩG (wi ) (disjoint), Irr B0 (G) = (wi )
where (wi ) runs over all r-weight sequences of u. Since χy,µ (1) = |G : CG (y)|p χµ (1), it follows that α(χy,µ ) =
|G|p α(χµ ) ≡ α(χµ ) |CG (y)|p
(mod r),
where χµ is the unipotent character of CG (y) parametrized by µ. Thus α(χy,µ ) ≡ α(χµ ) =
α(χµ ,j )
(mod r),
(4.2)
,j
where χµ0,0 is the unipotent character of GLε (em0,0 , q) parametrized by µ0,0 and for other j , j , χµ ,j is the unipotent character of GLε (m ,j , q er ) parametrized by µ ,j .
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β
An r-weight sequence (wi ) of (4.1) determines a unique partition λ = (λ0 0 , . . . , λt t ) of u, called an r-weight partition, where λi = r i and βi = wi for all i 0. β β β Let H = GLε (u, q e ) G = GLε (eu, q), λ = (λ0 0 , λ2 2 , . . . , λt t ) a partition of u with λ0 < λ1 < · · · < λt and R = Rλ a radical subgroup of H and G such that CG (R) = CH (R) =
t
β
Ki i ,
NH (R) =
i=0
NG (R) =
t
Ki S(βi ),
i=0 t τi , Ki S(βi ), i=0
where Ki = GLε (λi , q e ) GLε (eλi , q) and τi ∈ GLε (eλi , q) induces a field automorphism of order e on Ki . Thus P (Rλ ) = Rλ and each radical subgroup W of G or H satisfying P (W ) = W and CV (W ) = 0 is conjugate to some Rλ . If K G, denote by CR∗ (K) the subfamily of R(CK (Or (K))) consisting of the chains whose subgroups W satisfies P (W ) = W and CV (W ) = 0. Let CR∗K (λ) be the subfamily of CR∗ (K) consisting of all chains whose first subgroup P1 = Rλ , CR∗K (λ)e the subfamily of CR∗K (λ) consisting of the chains C with |C| even and CR∗K (λ)o = CR∗ (K)\CR∗K (λ)e . If C ∈ CR∗H (λ) is given by (2.1) with Rλ = Or (H ), then g(C) : Rλ < P2 < · · · < Pw is a chain of CR∗ (NH (Rλ )), |g(C)| = |C| − 1, NH (C) = NNH (Rλ ) (g(C)) and g is a bijection of CR∗H (λ) onto CR∗ (NH (Rλ )). We can identify CR∗H (λ) with CR∗ (NH (Rλ )). For K = H or G, let XK0 (λ) = ΩK ((β)) or ∅ according to whether or not λ is an r-weight partition, where β = (β0 , . . . , βt ). In addition, let XK+ (λ) = and XK− (λ) = XK0 (λ)
Irr B0 NK (C)
(4.3)
C∈CR∗K (λ)e /NK (Rλ )
C∈CR∗K (λ)o /NK (Rλ ) Irr(B0 (NK (C))).
Lemma (4A). Let XH0 = ∅ or ΩH ((δi )) according as u is not a power of r or u = r , and let XH+ =
Irr B0 NH (C) ,
C∈CR∗ (H )e /H
XH− = XH0
Irr B0 NH (C) .
C∈CR∗ (H )o /H
Then there exists a defect and α-preserving bijection Ψ of XH+ onto XH− which is also compatible with each τ ∈ Out(H ). Proof. If C0 (H ) ∈ R(H ) with |C0 (H )| = 0, then the decompositions XH0 (λ), Irr B0 (H ) = XH0 λ
CR∗ (H ) = C0 (H ) CR∗H (λ) λ
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are disjoint, where λ runs over the partitions of u with λ = (u). It then suffices to show that there exists a defect and α preserving bijection Ψ of XH+ (λ) onto XH− (λ) which is compatible with each τ ∈ Out(H ). But as shown in the proof of [1, (4C)] there is a bijection of XH+ (λ) onto XH− (λ) which is defect preserving and compatible with each τ ∈ Out(H ), so it suffices to show that Ψ is α-preserving. If u = 1, then XH+ (λ) = XH− (λ) = ∅, XH+ = XH− = XH0 = ΩH ((1, 0)) and Ψ is the identity map. Let Ki = GLε (λi , q e ) and Wi = GLε (λi , q e ) S(βi ), so that Ki < G. By induction, there is a defect and α preserving bijection ψ of XK+i onto XK−i , which is compatible with each τi ∈ Out(Ki ). By (3B)(a), ψ extends to a defect and α preserving bijection Ψi of XK+i S(βi ) and XK−i S(βi ), which is compatible with each τ ∈ Out(GLε (λi βi , q e )), where Wi GLε (λi βi , q e ) H . β But CR∗ (Wi ) = CR∗ (Ki i ), so as shown in the proof of [1, (1E)(a)] XK+i S(βi ) =
Irr B0 NWi (C)
C∈CR∗ (Wi )e /Wi
and XK−i S(βi ) = XK0 i S(βi ) C∈CR∗ (Wi )o /Wi Irr(B0 (NWi (C))). Let πWi be the projection of NH (Rλ ) onto Wi . Then
CR∗H (λ) = C ∈ CR∗ (H ): πWi (C) ∈ CR∗ (Wi ) C0 (H ) and so by Lemma (3B), there is a defect and α-preserving bijection Φ of XH+ (λ) =
t + i=0 XKi S(βi ) onto t i=0
0 XK−i S(βi ) = YH (λ)
Irr B0 NH (C) ,
C∈CR∗H (λ)o /NH (Rλ )
0 where YH (λ) = ti=0 XK0 i S(βi ). Thus it suffices to show that there is a defect and α 0 (λ). preserving bijection of XH0 (λ) onto YH 0 (λ) = ∅, and Φ is a required If some λi is not a power of r, then XK0 i = ∅, XH0 (λ) = YH + − bijection of XH (λ) onto XH (λ). Suppose each λi is a power of r, and we may suppose λi = r i for all i 0. Thus β = (β0 , β1 , . . . , βt ) is an r-weight sequence, XK0 i = ΩKi ((δj i )). It follows by [1, (4A)] 0 (λ) = that there is a defect preserving bijection ψ ∗ between XH0 (λ) = ΩH (β) and YH
t i=0 ΩKi ((δj i )) S(βi ), which is compatible with each τ ∈ Out(H ). As shown in the proof of [1, (4A)] the bijection ψ ∗ is given by induction Ind, so ψ ∗ is also α preserving by Lemma (3A)(b). This proves (4A). 2 Theorem (4B). Let B be an r-block of G = GLε (n, q) with a positive defect. If r and q are relatively prime, then B satisfies Uno’s invariant conjecture.
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Proof. (1) First, we reduce the proof to the subfamily CR of central radical chains. Namely, we show
(−1)|C| k N(C), B, d, U, [α] =
C∈R/G
(−1)|C| k N(C), B, d, U, [α] .
C∈CR/G
Let M be the complementary G-invariant subfamily R\CR of CR in R. It suffices to show that
(−1)|C| k N(C), B, d, U, [α] = 0.
(4.4)
C∈M/G
As shown in the proof of [1, (2D)], there is a bijection ϕ from M to itself satisfying NG (C) = NG (ϕ(C)), ϕ(ϕ(C)) = C, |ϕ(C)| = |C| ± 1 and NE (C) = NE (ϕ(C)) for an extension E of G. Thus (4.4) follows by Lemma (3A)(a). (2) Reduce the proof to the subfamily CR∗ , namely,
(−1)|C| k N(C), B, d, U, [α] =
C∈CR(B)/G
(−1)|C| k N(C), B, d, U, [α] .
C∈CR∗ (B)/G
(4.5) If we replace the initial step of the induction on the dimension in the proof of [1, (3A)] by Proposition (2A), then the same proof can be applied here with some obvious modifications. (3) Reduce the proof to blocks B with D = D(B) satisfying CV (D) = {0}. Let B be a block of G = GLε (V ) labelled by (s, κ), D = D(B), V0 = CV (D) and V+ = [V , D]. Then D = D0 × D+ and we may suppose s ∈ CG (D) = GLε (V0 ) × C+ , where D0 = 1V0 , D+ G+ = GLε (V+ ) and C+ = CG+ (D+ ). Thus s = s0 × s+ with s0 ∈ GLε (V0 ) and s+ ∈ C+ . Let B(0) and B(+) be blocks of G0 = GLε (V0 ) and G+ labelled by (s0 , κ) and (s+ , −), respectively. Then B(0) has defect 0 consisting of a unique irreducible character χ0 with α(χ0 ) = α0 . If L = G0 × G+ G, then RLG is a perfect isometry between Irr(B(0) × B(+)) and Irr(B) which is compatible with the outermorphisms of G (cf. [1, p. 381]). By Lemma (3C), RLG is α-preserving. Since RLG is also defect preserving and each character of Irr(B(0) × B(+)) has the form χ0 × χ+ for some χ+ ∈ Irr(B(+)) and since α(χ0 × χ+ ) = α0 α(χ+ ), it follows that k G, B, d, U, [α] = k G+ , B(+), d, U+ , [α+ ] , where α+ ≡ α/α0 (mod r). Suppose C ∈ CR∗ (B) is given by (2.1) with |C| 1. We may suppose Pt = 1V0 × P+ (t) for all t 1, where P+ (t) G+ . Define ϕ(C) : 1 < P+ (1) < P+ (2) < · · · < P+ (w).
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Then ϕ(C) ∈ CR∗ (B(+)), NG (C) = G0 × NG+ (ϕ(C)) and |C| = |ϕ(C)|. Applying the proof of [1, (3B)], we have k NG (C), B, d, U, [α] = k NG+ ϕ(C) , B(+), d, U+ , [α+ ] .
(4.6)
This implies the step (3). (4) Reduce the proof to the principal block B = B0 . The reduction [1, (3C)] is obtained by several defect preserving bijections and a cancellation. Each bijection is either given by a perfect isometry satisfying Lemma (3C) or by an induction IndH K of some subgroups K H G. By Lemma (3A)(b) and Lemma (3C), these bijections are also α-preserving. The last cancellation give in [1, p. 383] is given by a bijection satisfying Lemma (3A)(a), which is also α-preserving. Applying the modified proof of [1, (3C)], we get the proof of step (4). (5) Final calculation. Suppose B is the principal block of G = GLε (eu, q) for some u 1. The proof is similar the that of Lemma (4A). Note 0 XG (λ), Irr B0 (G) =
CR∗ (G) = C0 (G) CR∗G (λ),
λ
λ
where |C0 (G)| = 0 and λ runs over all partitions of u. So it suffices to show that there exists a defect and α preserving bijection Ψ of XG+ (λ) and XG− (λ) which is compatible with each τ ∈ Out(G). Given integer i 0, let Li = NGLε (eλi ,q) (Ki ) = τi , Ki for some τi ∈ GLε (eλi , q) inducing a field automorphism of order e on Ki . By (4A), there is a defect and α preserving bijection ψ of XK+i onto XK−i , which is compatible with each τ ∈ Out(Ki ). In particular, ψ(χ τi ) = ψ(χ)τi for any χ ∈ XK+i . Since CR∗ (Li ) = CR∗ (Ki ), it follows that ψ extends to a defect and α preserving bijection of XL+i and XL−i , where XL±i is defined similar to that of XK±i with XK0 i replaced by τi , XK0 i . Here τi , XK0 i denotes the irreducible characters of Li covering characters of XK0 i . Similar proof to that
of Lemma (4A) shows that there is a defect and α-preserving bijection Φ of XG+ (λ) = ti=0 XL+i S(βi ) onto t i=0
0 XL−i S(βi ) = YG (λ)
Irr B0 NG (C) ,
C∈CR∗G (λ)o /NG (Rλ )
0 (λ) = t τ , X 0 S(β ). It then suffices to show that there is a defect and α where YG i i=0 i Ki 0 preserving bijection of YG (λ) onto XG0 (λ). 0 (λ) = X 0 (λ) = ∅. Suppose λ is an r-weight If λ is not an r-weight partition, then YG G 0 partition, so that XG (λ) = ΩG ((β)) and we may suppose λi = r i . Thus (δ i ) is an r-weight sequence of r i , XK0 i = ΩKi ((δ i )) consists of irreducible characters ψ∆ of a+i ∗ a+i ∗ or Ki = GLe (r i , q e ) parametrized by (∆, 1) for ∆ ∈ (Fqa+i e ) , where (Fq e ) = Fq e
478
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Fq e (r, a) according as i > 0 or i = 0. If a character χ ∈ τi , ΩKi ((δ i )) is not an exteni sion of the trivial character, then χ has degree e|Ki |p /(q er − ε); if χ is one of the e extensions of the trivial character, then χ(1) = 1. In particular, τi , ΩK (δ i ) = r a − r a−1 /e i
or e + r a − 1 /e
according as i > 0 or i = 0. We my suppose
τ0 , ΩK0 (δ 0 ) = ξ1 , . . . , ξe , ξΓ | Γ ∈ Fq (r, a)\{T − 1}
and τi , ΩKi ((δ i )) = {ξΓ | Γ ∈ Fqa+i } for i > 0, where each ξi ∈ τ0 , ΩK0 ((δ 0 )) is an i extension of the trivial character and ξΓ (1) = e|Ki |p /(q er − ε) for Γ ∈ Fqa+i or Fq (r, a) according as i > 0 or i = 0. It follows that i
e|Ki |r (q er − ε)r α(ξΓ ) = ≡ ± qe − ε r , e(|Ki |p )r
α(ξi ) ≡ ±e q e − ε r
(mod r).
(4.7)
Each character ζ ∈ τi , ΩKi ((δ i )) S(βi ) has the form τ ,K S(β ) ζ = ζ (i, µi ) = Indτii ,Kii S(zii) ξ˜ ωµi , where ξ˜ is an extension of ξ ∈ (τi , ΩKi ((δ i )))βi to (τi , Ki )βi S(zi ) and zi is the type of ξ and ωµi is a irreducible character of the Young subgroup S(zi ). Thus if i > 0 and f = (r a − r a−1 )/e, then µi = (µi,1 , µi,2 , . . . , µi,f ) and each µi,j is a partition of the multiplicity ui,j = mξ∆(i,j) (ξ ) of ξ∆(i,j ) in ξ . If i = 0 and h = (r a − 1)/e, then µ0 = (µ∗0,1 , µ∗0,2 , . . . , µ∗0,e , µ0,1 , . . . , µ0,h ) and each µ∗0,j is a partition of u∗0,j = mξj (ξ ) and each µ0,j is a partition of u0,j = mξ∆(0,j) (ξ ). By Lemma (3A)(b) and (4.7), ±((q e − ε)r )βi α(ωµi ) α(ζ ) = α ξ˜ α(ωµi ) ≡ ±eu0,0 ((q e − ε)r )β0 α(ωµ0 )
if i > 0, if i = 0,
(mod r)
(4.8)
where u0,0 = u∗0,1 + u∗0,2 + · · · + u∗0,e . Let y be a semisimple element of G = GLε (eu, q) such that m∆(i,j ) (y) = ui,j or eu0,0 according as (i, j ) = (0, 0) or (i, j ) = (0, 0). Then y is a uniquely determined r-element, up to conjugacy. In addition, let γi,j = µi,j for all i, j except when i = j = 0, in which case let γ0,0 be the unique
partition of em0,0 with empty e-core and e-quotient (µ∗0,1 , µ∗0,2 , . . . , µ∗0,e ), and let γ = i,j γi,j . Then χy,γ ∈ ΩG ((β)). If χγ is the unipotent character of CG (y) parametrized by γ , then by Proposition (3D) and (4.8), α(χγ ) ≡ ±
t α ζ(i, µi ) (mod r). i=0
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0 If η = ti=0 ζ (i, µi ), then η ∈ YG (λ) = ti=0 τi , ΩKi ((δ i )) S(βi ). Define ϕ(η) = χy,γ . Then by (4.2) α(ϕ(η)) ≡ ±α(η) (mod r). As shown in the proof of [1, (4D)(a)] ϕ is also 0 (λ) onto Ω ((β)), which is also compatible with the a defect preserving bijection of YG G outer automorphisms of G. This completes the proof. 2
Acknowledgments The author would like to thank Paul Fong for Proposition (3D) and its proof, and thank Jorn Olsson for valuable suggestions. The author would also like to thank the Marsden Fund for their support.
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