THE GLUING PROBLEM FOR SOME FUSION SYSTEMS

SEJONG PARK

Abstract. We answer the gluing problem of blocks of finite groups [3, 4.2] for tame blocks and the principal p-block of PSL_{3}(p) for p odd. In particular, we show that the gluing problem for the principal p-block of PSL_{3}(p) does not have a unique solution when p ≡ 1 mod 3.

1. Introduction

Let F be a (saturated) fusion system on a finite p-group P and let k be an algebraically closed field. Let [S(F^{c})] be the poset of F-conjugacy classes [σ] of chains

σ = (R_{0} < R_{1} < ⋯ < R_{n}), n ≥ 0

of F-centric subgroups R_{i} of P, with partial order induced by taking subchains. Also, let

Aut_{F}(σ) = {α ∈ Aut_{F}(R_{n}) | α(R_{i}) = R_{i} for all i}.

For any positive integer i, there is a covariant functor

A_{F}^{i}: [S(F^{c})] → Ab

sending [σ] ∈ [S(F^{c})] to H^{i}(Aut_{F}(σ), k^{×}), where Ab denotes the category of abelian groups.

In [2], using the contractiblity of [S(F^{c})] [5, 1.1], Linckelmann shows that for every fusion system F there exists a cohomology exact sequence as follows:

Theorem 1.1 ([2, 1.1]). Let F be a fusion system on a finite p-group P and let k be an algebraically closed field. Then there exists an exact sequence of abelian groups

0 → H^{1}([S(F^{c})], A_{F}^{1}) → H^{2}(F^{c}, k^{×}) → H^{0}([S(F^{c})], A_{F}^{2}) → H^{2}([S(F^{c})], A_{F}^{1}) → H^{3}(F^{c}, k^{×}).

In particular, the group H^{2}(F^{c}, k^{×}) is finite, of order coprime to char(k) if char(k) is positive.

The importance of this exact sequence comes from the gluing problem of blocks of finite groups [3, 4.2], which has bearing on reformulations of Alperin’s weight conjecture [3, 4.5, 4.7][4, 4.3]. If F is the fusion system of a block of a finite group, then it determines an element of H^{0}([S(F^{c})], A_{F}^{2}) by the work of Külshammer and Puig [1, 1.8, 1.12]. The gluing problem asks whether it is the image of the map

H^{2}(F^{c}, k^{×}) → H^{0}([S(F^{c})], A_{F}^{2}).

In particular, if H^{2}([S(F^{c})], A_{F}^{1}) = 0, then the gluing problem will have solutions, the number of which is equal to the order of H^{1}([S(F^{c})], A_{F}^{1}).

In this paper, we compute H^{i}([S(F^{c})], A_{F}^{1}) (i = 1, 2) for fusion systems F of tame blocks and the principal p-block of PSL_{3}(p) for p odd; in particular, we show that the gluing problem for the principal p-block of PSL_{3}(p) does not have a unique solution when 3 | (p − 1).

Theorem 1.2. Let P be a dihedral, semidihedral, or (generalized) quaternion 2-group, and let F be a fusion system on P. Let k be an algebraically closed field of characteristic 2. Then we have

H^{2}(F^{c}, k^{×}) = H^{0}([S(F^{c})], A_{F}^{2}) = 0.

In particular, the gluing problem for tame blocks has the zero class as a unique solution.

Date: August 17, 2009.

Theorem 1.3. Let p be an odd prime number and let P be an extraspecial group of order p^{3} and exponent p. Then for any fusion system F on P we have

H^{2}([S(F^{c})], A_{F}^{1}) = 0.

If F = F_{P} (PSL_{3}(p)), then we have

H^{1}([S(F^{c})], A_{F}^{1}) = \begin{cases}0, & if 3 ∤ (p − 1) \\ ℤ/3, & if 3 | (p − 1). \end{cases}

In particular, the gluing problem for the principal p-block of PSL_{3}(p) has a unique solution if 3 ∤ (p − 1), and three solutions if 3 | (p − 1).

We prove these two theorems in §2, §3, respectively.

2. Tame fusion systems

Let P be either a dihedral group D_{2^{n}}(n ≥ 2), a semidihedral group SD_{2^{n}}(n ≥ 4), or a (generalized) quaternion group Q_{2^{n}}(n ≥ 3) of order 2^{n}. It is well known that the subgroups R of P are cyclic, dihedral, semidihedral, or quaternion, and their automorphism groups are 2-groups except when R ≅ D_{4} or Q_{8}, in which cases we have Aut(D_{4}) ≅ S_{3}, Aut(Q_{8}) ≅ S_{4}. From this, one can easily deduce the following proposition.

Proposition 2.1. Let F be any fusion system on P and let R be an F-centric subgroup of P. (1) If R ≇ D_{4}, Q_{8}, then Out_{F}(R) is a 2-group.

(2) If R ≅ D_{4}, Q_{8} and R < P, then Out_{F}(R) ≅ C_{2} or S_{3}.

(3) If R ≅ D_{4}, Q_{8} and R = P, then Out_{F}(R) = 1 or C_{3}.

Corollary 2.2. Let F be any fusion system on P and let σ be a chain of F-centric subgroups of P. Let k be an algebraically closed field of characteristic 2. We have

H^{1}(Aut_{F}(σ), k^{×}) ≅ \begin{cases}ℤ/3, & if σ = (P), P ≅ D_{4} or Q_{8}, Out_{F}(P) ≅ C_{3} \\ 0, & otherwise \end{cases}

and

H^{2}(Aut_{F}(σ), k^{×}) = 0.

Proof. Since k is an algebraically closed field of characteristic 2, we have

H^{1}(A, k^{×}) = Hom(A, k^{×}) ≅ Hom(A/([A, A]O^{p′}(A)), k^{×})

for any finite group A. Thus we have

H^{1}(C_{2}, k^{×}) = H^{1}(S_{3}, k^{×}) = 0, H^{1}(C_{3}, k^{×}) ≅ ℤ/3.

Similarly, we have H^{2}(A, k^{×}) ≅ H^{2}(A/O_{p}(A), k^{×}) for any finite group A. Also, it is well known that

H^{2}(C_{3}, k^{×}) = H^{2}(S_{3}, k^{×}) = 0.

Now the result follows from Proposition 2.1. □

Proof of Theorem 1.2. Let C = [S(F^{c})] and A^{i} = A_{F}^{i}. By Corollary 2.2, we have A^{2} = 0, and hence H^{0}(C, A^{2}) = 0. By Theorem 1.1, it remains to show H^{1}(C, A^{1}) = 0.

Case 1: Out_{F}(P) = 1. Then A^{1} = 0 and so H^{1}(C, A^{1}) = 0.

Case 2: P ≅ D_{4}, Out_{F}(P) ≅ C_{3}. Then F^{c}, and hence C, has one object. Thus H^{1}(C, A^{1}) = 0.

Case 3: P ≅ Q_{8}, Out_{F}(P) ≅ C_{3}. Then P has a unique (up to F-conjugacy) F-centric proper subgroup R ≅ C_{4}. Thus the poset C and the functor A^{1} are as follows:

\xymatrix{ & [P] \ar[dl] &  & ℤ/3 \ar[dl] \\ [R < P] &  & 0 \\  & [R] \ar[ul] &  & 0 \ar[ul]}

Thus we have H^{1}(C, A^{1}) = 0. □

3. The extraspecial group of order p^{3} and exponent p, p odd

Let p be an odd prime and let P be the extraspecial group of order p^{3} and exponent p. Ruiz and Viruel [6] classified all fusion systems F on P. First let us recall some basic facts from [6]. Explicitly, one can view P as a Sylow p-subgroup of SL_{3}(p) as follows:

P = \{(\begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) | x, y, z ∈ 𝔽_{p}\} .

In particular, we have

Z(P) = [P, P] = Φ(P) = \{(\begin{matrix}1 & 0 & z \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) | z ∈ 𝔽_{p}\} .

The sequence of groups

1 → Inn(P) ⟶^{ι} Aut(P) ⟶^{π} Aut(P/Z(P)) → 1,

where ι is the inclusion and π sends each α ∈ Aut(P) to the induced automorphism uZ(P) ↦ α(u)Z(P) of P/Z(P), is split exact. More precisely, Aut(P/Z(P)) ≅ GL_{2}(p) and through this

isomorphism and the splitting map, each (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(p) acts on P as follows:

(\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & a & \frac{1}{2}ac \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & b & \frac{1}{2}bd \\ 0 & 1 & d \\ 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & 0 & ad − bc \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Also we have Inn(P) ≅ C_{p} × C_{p} and through this isomorphism, each (a, b) ∈ C_{p} × C_{p} acts as

(\begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & x & z + ay − bx \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) .

There are exactly p + 1 proper centric subgroups of P:

V_{i} = \{(\begin{matrix}1 & x & z \\ 0 & 1 & ix \\ 0 & 0 & 1 \end{matrix}) | x, z ∈ 𝔽_{p}\} (0 ≤ i < p), V_{p} = \{(\begin{matrix}1 & 0 & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) | y, z ∈ 𝔽_{p}\} .

All V_{i} (0 ≤ i ≤ p) are elementary abelian normal subgroups of P of order p^{2}. Let A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈

GL_{2}(p) act on the V_{i} as follows:

(\begin{matrix}1 & x & z \\ 0 & 1 & ix \\ 0 & 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & ax + bz & cx + dz \\ 0 & 1 & i(ax + bz) \\ 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) ↦ (\begin{matrix}1 & 0 & cy + dz \\ 0 & 1 & ay + bz \\ 0 & 0 & 1 \end{matrix}) ,

and we denote the action of A on P and V_{i} by A_{P} and A_{V_{i}}, respectively.

Now let F be an arbitrary fusion system on P, and let k be an algebraically closed field of characteristic p. The chains in [S(F^{c})] have length at most 1, and hence

(1)  H^{2}([S(F^{c})], A_{F}^{1}) = 0.

We use the following lemma for computing H^{1}([S(F^{c})], A_{F}^{1}). It enables us to work with a cochain complex smaller than the one induced from the standard projective resolution of the constant covariant functor \underline{ℤ}: [S(F^{c})] → Ab.

Lemma 3.1 ([4, 3.2]). Let F be a fusion system on a finite p-group P and let A: [S(F^{c})] → Ab be a covariant functor. Let C(A) be the cochain complex of abelian groups whose component in degree n ≥ 0 is equal to

C(A)^{n} = ⨁_{[σ]} A([σ])

where the direct sum is taken over the set of F-conjugacy classes [σ] of chains σ of F-centric subgroups of P of length n, and whose coboundary maps δ^{n}: C(A)^{n−1} → C(A)^{n} are given by

δ^{n}(α)([σ]) = ∑_{i=0}^{n}(−1)^{i}A([σ(i)] → [σ])(α([σ(i)]))

where α ∈ C(A)^{n−1}, σ = (R_{0} < ⋯ < R_{n}), and σ(i) = (R_{0} < ⋯ < R_{i−1} < R_{i+1} < ⋯ < R_{n}). The we have

H^{n}([S(F^{c}], A) ≅ H^{n}(C(A))

for any integer n ≥ 0.

3.1. F = F_{P} (PSL_{3}(p)), 3 ∤ (p − 1). From [6], we have that

(1) Out_{F}(P) ≅ ⟨(\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix})_{P} ⟩ × ⟨(\begin{matrix}1 & 0 \\ 0 & ζ \end{matrix})_{P} ⟩ (𝔽_{p}^{×} = ⟨ζ⟩);

(2) the F-conjugacy classes among the V_{i} are {V_{0}}, {V_{p}}, {V_{i} | 1 ≤ i ≤ p − 1};

(3) V_{i} is F-radical if and only if i = 0, p;

(4) Aut_{F}(V_{0}) ≅ Aut_{F}(V_{p}) ≅ GL_{2}(p).

Then we have

Aut_{F}(V_{1} < P) ≅ (C_{p} × C_{p}) ⋊ ⟨(\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{P} ⟩ ,

and the restriction map

Aut_{F}(V_{1} < P) → Aut_{F}(V_{1})
α ↦ α|_{V_{1}}

is surjective by Alperin’s fusion theorem. The above map has kernel ≅ C_{p} contained in Inn(P) and

it sends (\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{P} to (\begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix})_{V_{1}}. It follows that

Aut_{F}(V_{1}) ≅ C_{p} ⋊ ⟨(\begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix})_{V_{1}}⟩ .

Also we have Aut_{F}(V_{0} < P) = Aut_{F}(V_{p} < P) = Aut_{F}(P), and the restriction maps to V_{0} and V_{p} are given as follows:

Aut_{F}(P) → Aut_{F}(V_{0})  Aut_{F}(P) → Aut_{F}(V_{p})
(\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix})_{P} ↦ (\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{V_{0}}  (\begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix})_{P} ↦ (\begin{matrix}1 & 0 \\ 0 & ζ \end{matrix})_{V_{p}}

(\begin{matrix}1 & 0 \\ 0 & ζ \end{matrix})_{P} ↦ (\begin{matrix}1 & 0 \\ 0 & ζ \end{matrix})_{V_{0}}  (\begin{matrix}1 & 0 \\ 0 & ζ \end{matrix})_{P} ↦ (\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{V_{p}}

Thus the poset [S(F^{c})] and the functor A_{F}^{1} are as follows:

[figure: [P]; ℤ/(p − 1) ⊕ ℤ/(p − 1); π; [V_{0} < P]; ℤ/(p − 1) ⊕ ℤ/(p − 1); [V_{0}]; ℤ/(p − 1); i; [V_{p} < P]; ℤ/(p − 1) ⊕ ℤ/(p − 1); [V_{p}]; ℤ/(p − 1); j; [V_{1} < P]; ℤ/(p − 1); [V_{1}]; ℤ/(p − 1)]

where π(1, 0) = π(0, 1) = 1, i(1) = (2, 1), j(1) = (1, 2), and all other maps on the right hand side are the identity maps. Thus the cochain complex C(A_{F}^{1}) of Lemma 3.1, after splicing off the identity map on ℤ/(p − 1) induced from the inclusion [V_{1}] → [V_{1} < P], is

4(ℤ/(p − 1)) ⟶^{δ^{1}} 4(ℤ/(p − 1)) → 0 → ⋯

where the image of δ^{1} is generated by (1, 0, 1, 0), (0, 1, 0, 1), (2, 1, 0, 0), and (0, 0, 1, 2). Thus δ^{1} is a bijection, and hence

(2)  H^{1}([S(F^{c})], A_{F}^{1}) = 0.

3.2. F = F_{P} (PSL_{3}(p)), 3 | (p − 1). From [6], we have that

(1) Out_{F}(P) = ⟨(\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})⟩ × ⟨(\begin{matrix}1 & 0 \\ 0 & ζ^{3} \end{matrix})⟩ (𝔽_{p}^{×} = ⟨ζ⟩);

(2) the F-conjugacy classes among the V_{i} are \{V_{0}\}, \{V_{p}\}, \{V_{ζ^{3i}} | 0 ≤ i < \frac{p−1}{3}\}, \{V_{ζ^{3i+1}} | 0 ≤ i < \frac{p−1}{3}\}, \{V_{ζ^{3i+2}} | 0 ≤ i < \frac{p−1}{3}\};

(3) V_{i} is F-radical if and only if i = 0, p;

(4) Aut_{F}(V_{0}) ≅ Aut_{F}(V_{p}) ≅ SL_{2}(p) ⋊ C_{(p−1)/3}.

Then we have

Aut_{F}(V_{i} < P) ≅ (C_{p} × C_{p}) ⋊ ⟨(\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})⟩ (0 < i < p),

and the restriction map

Aut_{F}(V_{i} < P) → Aut_{F}(V_{i})
α ↦ α|_{V_{i}}

is surjective by Alperin’s fusion theorem for 0 < i < p. The above map has kernel ≅ C_{p} contained

in Inn(P) and it sends (\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{P} to (\begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix})_{V_{i}}. It follows that

Aut_{F}(V_{i}) ≅ C_{p} ⋊ ⟨(\begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix})_{V_{i}}⟩ (0 < i < p).

Also we have Aut_{F}(V_{0} < P) = Aut_{F}(V_{p} < P) = Aut_{F}(P), and the restriction maps to V_{0} and V_{p} are given as follows:

Aut_{F}(P) → Aut_{F}(V_{0})  Aut_{F}(P) → Aut_{F}(V_{p})
(\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{P} ↦ (\begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix})_{V_{0}}  (\begin{matrix}ζ & 0 \\ 0 & ζ \end{matrix})_{P} ↦ (\begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix})_{V_{p}}

(\begin{matrix}1 & 0 \\ 0 & ζ^{3} \end{matrix})_{P} ↦ (\begin{matrix}1 & 0 \\ 0 & ζ^{3} \end{matrix})_{V_{0}}  (\begin{matrix}1 & 0 \\ 0 & ζ^{3} \end{matrix})_{P} ↦ (\begin{matrix}ζ^{3} & 0 \\ 0 & ζ^{3} \end{matrix})_{V_{p}}

Thus the poset [S(F^{c})] and the functor A_{F}^{1} are as follows:

[figure: [P]; ℤ/(p − 1) ⊕ ℤ/(\frac{p−1}{3}); π; π; π; [V_{0} < P]; ℤ/(p − 1) ⊕ ℤ/(\frac{p−1}{3}); [V_{0}]; ℤ/(\frac{p−1}{3}); i; [V_{p} < P]; ℤ/(p − 1) ⊕ ℤ/(\frac{p−1}{3}); [V_{p}]; ℤ/(\frac{p−1}{3}); j; [V_{1} < P]; ℤ/(p − 1); [V_{1}]; ℤ/(p − 1); [V_{ζ} < P]; ℤ/(p − 1); [V_{ζ}]; ℤ/(p − 1); [V_{ζ^{2}} < P]; ℤ/(p − 1); [V_{ζ^{2}}]; ℤ/(p − 1)]

where π(1, 0) = 1, π(0, 1) = 0, i(1) = (3, 1), j(1) = (3, 2), and all other maps on the right hand side are the identity maps. Thus the cochain complex C(A_{F}^{1}) of Lemma 3.1, after splicing off the three identity maps on ℤ/(p − 1) induced from the inclusions [V_{ζ^{i}}] → [V_{ζ^{i}} < P] (i = 0, 1, 2) is

ℤ/(p − 1) ⊕ 3(ℤ/(\frac{p − 1}{3})) ⟶^{δ^{1}} 2(ℤ/(p − 1) ⊕ ℤ/(\frac{p − 1}{3})) → 0 → ⋯

where the image of δ^{1} is generated by (1, 0, 1, 0), (0, 1, 0, 1), (3, 1, 0, 0), and (0, 0, 3, 2). Thus δ^{1} is injective, and comparing the order of the groups, we get

(3)  H^{1}([S(F^{c})], A_{F}^{1}) ≅ ℤ/3.

Now Theorem 1.3 follows from (1), (2), and (3).

References

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