1 Introduction
In 𝑝-local representation theory of finite groups, when dealing with an object 𝑃 of a fusion system, we often consider attributes of 𝑃 that are not invariants of 𝑃.
That is, we often consider conditions on 𝑃, constructions determined by 𝑃, numbers associated with 𝑃 that are not isomorphism invariants with respect to the fusion system.
The two conjectures below imply that certain attributes of 𝑃 are, despite appearances, invariants after all.
To better explain what we mean, we must first set the scene.
Throughout, we let 𝒪 be a complete local Noetherian commutative unital ring with an algebraically closed residue field 𝔽 of prime characteristic 𝑝.
The hypothesis on 𝒪 implies that either
O
=
F
or else 𝒪 is a complete discrete valuation ring.
We understand any 𝒪-module, and in particular any algebra over 𝒪, to be finitely generated and free over 𝒪 or over 𝔽.
We understand a basis for an 𝒪-module to be an 𝒪-basis or an 𝔽-basis.
A basis Ω for an algebra over 𝒪 is said to be unital provided every element of 𝒪 is a unit.
Throughout, we let 𝐺 be a finite group.
We deem 𝐺-algebras to be over 𝒪.
We presume familiarity with the theory of 𝐺-algebras, Brauer pairs, almost-source algebras, fusion systems, as discussed by Linckelmann [5, Chapters 5, 6, 8].
We shall freely make use of much notation and terminology from there. However, adopting two definitions by Craven [1, Definitions 4.1, 4.11], we understand the term fusion system in the broad sense, as distinct from saturated fusion system.
For a finite
G
×
G
-set Γ, the image of
c
∈
Γ
, under the action of
(
f
,
g
)
∈
G
×
G
, can be written as
(
f
,
g
)
c
or as
f
c
g
-
1
.
When the latter notation prevails, we call Γ a
(
G
,
G
)
-biset.
Regard 𝐺 as a
(
G
,
G
)
-biset by left and right multiplication.
Extending 𝒪-linearly,
O
G
becomes a permutation
O
(
G
×
G
)
-module.
The scenario of main concern in this paper is as follows.
Let 𝑏 be a block of
O
G
.
Let 𝐷 be a defect group of the block algebra
O
G
b
.
Let 𝐴 be an almost-source 𝐷-algebra of
O
G
b
.
Let ℱ be the fusion system on 𝐷 associated with 𝐴.
By Green’s Indecomposability Criterion, 𝐴 is a permutation
O
(
D
×
D
)
-submodule of
O
G
.
Let Ω be a
D
×
D
-stable basis for 𝐴.
By the Krull–Schmidt Theorem, the
(
D
,
D
)
-biset Ω is well-defined up to isomorphism.
A result of Linckelmann, Theorem 7.2 below, describes how ℱ is determined by Ω.
One clause of the description involves a condition that is not ℱ-isomorphism invariant.
Theorem 7.5 asserts that the condition can be omitted when 𝑏 is the principal block of
O
G
.
Most of this paper, though, is concerned with some apparently stronger properties of 𝐴 and Ω.
The notion of an ℱ-semicharacteristic biset is defined by Gelvin–Reeh [3].
We shall recall the definition in Section 6.
The next result is presented in [5, Proposition 8.7.11] for source algebras, but the proof carries over immediately to almost-source algebras.
Theorem 1.1
Theorem 1.1 (Linckelmann)
Let 𝑏 be a block of
O
G
.
Let 𝐷 be a defect group of the block algebra
O
G
b
.
Let 𝐴 be an almost-source 𝐷-algebra of
O
G
b
.
Write ℱ for the fusion system on 𝐷 associated with 𝐴.
If 𝐴 has a unital
D
×
D
-stable basis Ω, then the
(
D
,
D
)
-biset Ω is ℱ-semicharacteristic.
If, furthermore, 𝐴 is a source algebra, then Ω is ℱ-characteristic.
The weaker of the two conjectures below asserts that some almost-source algebra of
O
G
b
do have a stable basis.
The stronger asserts that the source algebras of
O
G
b
have stable bases.
The next theorem gives two necessary and sufficient criteria for the stability hypothesis.
We understand an idempotent decomposition in a ring to be a sum of finitely many mutually orthogonal idempotents.
When the summands are primitive idempotents, we call the sum a primitive idempotent decomposition.
Given a pointed group
U
μ
on any 𝐺-algebra 𝐴, we define the multiplicity of
U
μ
, denoted
m
A
(
U
μ
)
, to be the number of elements of 𝜇 appearing in a primitive idempotent decomposition of the unity element of the 𝑈-fixed subalgebra
A
U
.
Now suppose 𝐴 is an interior 𝐺-algebra.
Let
U
μ
and
V
ν
be pointed groups on 𝐴.
Suppose
U
≅
V
, and let
ϕ
:
V
→
U
be a group isomorphism.
We call 𝜙 an isofusion
V
ν
→
U
μ
provided, in the sense of Puig [6, Section 2], 𝜙 is an “𝐴-fusion”
V
ν
→
U
μ
.
In Section 4, we shall be reviewing that notion, and we shall also be noting some further characterizations of isofusions.
Theorem 1.2
Let 𝑏 be a block of
O
G
with defect group 𝐷 and almost-source 𝐷-algebra 𝐴.
Write ℱ for the fusion system on 𝐷 associated with 𝐴.
Then the following three conditions are equivalent.
For every ℱ-isomorphism
ϕ
:
Q
→
P
, there is a bijective correspondence between the local points 𝛾 of 𝑃 on 𝐴 and the local points 𝛿 of 𝑄 on 𝐴 whereby
γ
↔
δ
if and only if 𝜙 is an isofusion
Q
δ
→
P
γ
.
Furthermore, when
γ
↔
δ
, we have
m
A
(
P
γ
)
=
m
A
(
Q
δ
)
.
For every ℱ-isomorphism
ϕ
:
Q
→
P
, there is a bijective correspondence between the points 𝛼 of 𝑃 on 𝐴 and the points 𝛽 of 𝑄 on 𝐴 whereby
α
↔
β
if and only if 𝜙 is an isofusion
Q
β
→
P
α
.
Furthermore, when
α
↔
β
, we have
m
A
(
P
α
)
=
m
A
(
Q
β
)
There exists a unital
D
×
D
-stable basis for 𝐴.
We shall prove that theorem in Section 8.
When the equivalent conditions (a), (b), (c) hold, we call 𝐴 a uniform almost-source 𝐷-algebra of
O
G
b
.
Condition (b) can be mnemonically summarized as follows: multiplicities of pointed groups are ℱ-invariant, condition (a) likewise.
Let us explain another way of interpreting condition (a) as an assertion of ℱ-invariance.
Recall that, for each ℱ-centric subgroup 𝑃 of 𝐷, there is a unique local point
λ
P
A
of 𝑃 on 𝐴.
Given ℱ-centric subgroups 𝑃 and 𝑄 of 𝐷 and an ℱ-isomorphism
ϕ
:
Q
→
P
, then 𝜙 is an isofusion
Q
λ
Q
A
→
P
λ
P
A
.
Condition (a), confined to the centric objects of ℱ, says that the function
P
↦
m
A
(
P
λ
P
A
)
is constant on ℱ-isomorphism classes.
Conjecture 1.3
Let 𝑏 be a block of
O
G
.
Then the block algebra
O
G
b
has a uniform almost-source algebra.
Part of our evidence for the conjecture is the following result, proved in Section 8.
Theorem 1.4
Suppose 𝐺 is 𝑝-solvable.
Given a block 𝑏 of
O
G
, then the block algebra
O
G
b
has a uniform almost-source algebra.
By the uniqueness of source algebras up to conjugation, if some source algebra of
O
G
b
is uniform, then every source algebra of
O
G
b
is uniform.
Conjecture 1.5
For any block 𝑏 of
O
G
, the source algebras of
O
G
b
are uniform.
In Section 8, we shall show that Conjecture 1.5 holds in the following special case.
Recall a fusion system ℱ on a finite 𝑝-group 𝐷 satisfies
F
=
N
F
(
D
)
if and only if every ℱ-isomorphism is a restriction of an ℱ-automorphism of 𝐷.
Proposition 1.6
Let 𝑏 be a block of
O
G
with defect group 𝐷 and source 𝐷-algebra 𝐴.
Write ℱ for the fusion system on 𝐷 associated with 𝐴.
If
F
=
N
F
(
D
)
, then 𝐴 is uniform.
In particular, 𝐴 is uniform whenever 𝐷 is abelian or
D
⊴
G
.
Let us outline the organization of the paper.
The next three sections may be of use in applications quite different from our main concerns here.
Section 2 supplies a necessary and sufficient criterion for an interior 𝐺-algebra to have a
G
×
G
-stable basis.
In Section 3, we show that, for any 𝐺-algebra, distinct points of a 𝑝-subgroup cannot share a defect pointed subgroup.
In Section 4, we present a new approach to fusions in interior 𝐺-algebras.
The remaining four sections are focused on proofs of the new results stated in this introduction.
Sections 5 and 6 establish the crux of Theorem 1.2 but, for simplicity, in a more abstract setting.
A
(
D
,
D
)
-biset Ω is said to be ℱ-divisible provided
D
×
1
and
1
×
D
act freely on Ω and, given any isomorphism 𝜙 between subgroups of 𝐷, then 𝜙 is an ℱ-isomorphism if and only if there exists
w
∈
Ω
satisfying
ϕ
(
y
)
w
y
-
1
=
w
for all 𝑦 in the domain of 𝜙.
In Section 7, we shall prove the following theorem.
Theorem 1.7
Let 𝑏, 𝐷, 𝐴, ℱ be as in Theorem 1.1.
Let Ω be a
D
×
D
-stable basis for 𝐴.
If ℱ is 𝑝-constrained or 𝐺 is 𝑝-solvable or 𝑏 is the principal block of
O
G
, then Ω is ℱ-divisible.
In the final section, we shall complete the proofs of Theorems 1.2, 1.4 and Proposition 1.6.
2 Unital stable bases
Our hypothesis on 𝒪 implies that every algebra over 𝒪 has a unital basis.
More generally, for any finite 𝑝-group 𝐷, any permutation 𝐷-algebra has a unital 𝐷-stable basis; indeed, a further generalization, with a weaker condition on the coefficient ring, can be found in [5, Proposition 5.8.13] by Linckelmann.
The purpose of this section is to give a criterion for an interior 𝐺-algebra to have a unital
G
×
G
-stable basis.
As a reminder of a convention announced in Section 1, let us repeat that any 𝒪-module 𝑋 is deemed to have a finite basis (over 𝒪 or over 𝔽); we mean a finite subset Ω such that Ω is an 𝒪-basis when 𝑋 is 𝒪-free, an 𝔽-basis when 𝑋 is annihilated by
J
(
O
)
.
We understand the rank of 𝑋 to be the well-defined natural number
|
Ω
|
.
We define
X
¯
=
X
/
J
(
O
)
X
as an 𝔽-module.
We write
w
¯
and
Ω
¯
for the images in
X
¯
of an element 𝑤 of 𝑋 and a subset Ω of 𝑋.
Lemma 2.1
Given an 𝒪-free 𝒪-module 𝑋 and
Ω
⊆
X
such that the reduction map
X
→
X
¯
restricts to a bijection
Ω
→
Ω
¯
, then Ω is an 𝒪-basis for 𝑋 if and only if
Ω
¯
is an 𝔽-basis for
X
¯
.
Proof
This easy exercise demands, at most, just a hint: when
Ω
¯
is an 𝔽-basis, express each element of Ω as an 𝒪-linear combination of elements of an 𝒪-basis for 𝑋; then consider the determinant of the square matrix formed by the coefficients.
∎
Lemma 2.2
Let Ω be a basis for an 𝒪-module 𝑋.
For each
w
∈
Ω
, let
v
w
∈
X
.
Taking 𝜆 to run over the elements of 𝒪, then, for all except at most
|
Ω
|
values of the reduction
λ
¯
∈
F
, the set
{
w
+
λ
v
w
:
w
∈
Ω
}
is a basis for 𝑋.
Proof
By the previous lemma, we may assume that 𝑋 is annihilated by
J
(
O
)
.
Hence, in fact, we may assume that
O
=
F
.
Let
w
(
λ
)
=
w
+
λ
v
w
.
Let
M
(
λ
)
be the matrix, with rows and columns indexed by Ω, such that the
(
w
′
,
w
)
-entry
M
w
′
,
w
(
λ
)
of
M
(
λ
)
is given by
w
(
λ
)
=
∑
w
′
M
w
′
,
w
(
λ
)
w
′
.
The set
{
w
(
λ
)
:
w
∈
Ω
}
is an 𝔽-basis for 𝑀 if and only if
M
(
λ
)
is invertible.
The function
λ
↦
det
(
M
(
λ
)
)
is a polynomial function over 𝔽 with degree at most Ω and with non-zero constant term.
Therefore,
det
(
M
(
λ
)
)
=
0
for at most
|
Ω
|
values of 𝜆.
∎
Lemma 2.3
Let 𝐴 be an algebra over 𝒪, let 𝑛 be the rank of 𝐴, and let
a
∈
A
and
u
∈
A
×
.
Taking 𝜆 to run over the elements of 𝒪, then, for all except at most 𝑛 values of the reduction
λ
¯
∈
F
, we have
a
+
λ
u
∈
A
×
.
Proof
Since units of
A
¯
lift only to units in 𝐴, we may assume that
O
=
F
.
Let
ρ
:
A
→
End
O
(
A
)
be the regular representation.
We have
a
+
λ
u
∈
A
×
if and only if
ρ
(
a
+
λ
u
)
∈
End
O
(
A
)
×
, equivalently,
ρ
(
-
u
-
1
a
)
-
λ
.
id
A
∈
End
O
(
A
)
×
; in other words, 𝜆 is not an eigenvalue of
ρ
(
-
u
-
1
a
)
.
∎
When discussing interior 𝐺-algebras, we shall freely use notation and terminology from Linckelmann [5, Chapter 5], but let us reiterate a few conventions.
Let 𝐴 be an interior 𝐺-algebra.
For
g
∈
G
and
a
∈
A
, we write
g
a
=
σ
A
(
g
)
a
, similarly for
a
g
, where
σ
A
:
G
→
A
×
is the structural homomorphism of 𝐴.
As usual, we regard 𝐴 as an
O
G
-module via
σ
A
.
That is, 𝑔 sends 𝑎 to
a
g
=
g
a
g
-
1
.
We also regard 𝐴 as an
O
(
G
×
G
)
-module, with
(
f
,
g
)
∈
G
×
G
sending 𝑎 to
f
a
g
-
1
.
Theorem 2.4
Let 𝐴 be an interior 𝐺-algebra.
Then 𝐴 has a unital
G
×
G
-stable basis if and only if 𝐴 has a
G
×
G
-stable basis Ω such that, for all
w
∈
Ω
, the stabilizer
N
G
×
G
(
w
)
fixes a unit of 𝐴.
Proof
One direction is obvious.
Conversely, suppose there exists Ω as specified.
For
a
∈
A
, we write
N
(
a
)
=
N
G
×
G
(
a
)
.
Letting 𝑤 run over representatives of the
G
×
G
-orbits of Ω, we choose elements
u
(
w
)
∈
A
×
∩
A
N
(
w
)
.
Now, letting 𝑤 run over all the elements of Ω, we let
w
↦
u
(
w
)
be the unique function such that
u
(
f
w
g
-
1
)
=
f
u
(
w
)
g
-
1
for all 𝑤 and all
f
,
g
∈
G
.
We have
N
(
u
(
f
w
g
-
1
)
)
=
N
(
f
,
g
)
(
u
(
w
)
)
.
So
N
(
w
)
=
N
(
u
(
w
)
)
for all
w
∈
Ω
.
Let
n
=
|
Ω
|
.
Lemma 2.3 implies that, letting 𝜆 run over the elements of 𝒪, then, for each
w
∈
Ω
, the element
w
λ
-
w
+
λ
u
(
w
)
is a unit for all except at most 𝑛 values of
λ
¯
.
Lemma 2.2 implies that 𝐴 has basis
Ω
λ
=
{
w
λ
:
w
∈
Ω
}
for all except at most 𝑛 values of
λ
¯
.
Therefore,
Ω
λ
is a unital basis for all except at most
n
n
+
1
values of 𝜆.
∎
Corollary 2.5
Let 𝑒 be an idempotent of
Z
(
O
G
)
.
Let 𝑆 be a Sylow 𝑝-subgroup of 𝐺.
Then the algebra
O
G
e
has a unital
S
×
S
-stable basis.
Proof
Let
A
=
O
G
e
.
We can regard 𝐴 as a direct summand of the permutation
O
(
S
×
S
)
-module
O
G
.
So 𝐴 has an
S
×
S
-stable basis Ω; moreover, any such Ω is isomorphic to an
S
×
S
-subset of the basis 𝐺 of
O
G
.
Given
w
∈
Ω
, then
N
S
×
S
(
w
)
=
N
S
×
S
(
g
)
for some
g
∈
G
.
On the other hand, viewing 𝐴 as an algebra, the unit
g
e
∈
A
also has stabilizer
N
S
×
S
(
g
e
)
=
N
S
×
S
(
g
)
.
The required conclusion now follows by applying the latest theorem to 𝐴 as an interior 𝑆-algebra.
∎
3 A uniqueness theorem
The little theorem in this section, and the subsequent remark, will be needed in Section 5.
Let us first establish some notation concerning relative multiplicities.
Given a 𝐺-algebra 𝐴 and pointed groups
T
τ
and
U
μ
on 𝐴 with
T
≤
U
, we define the relative multiplicity of
T
τ
in
U
μ
, denoted
m
A
(
T
τ
,
U
μ
)
, to be the number of elements of 𝜏 that appear in a primitive idempotent decomposition of 𝑖 in the algebra
A
T
, where 𝑖 is any element of 𝜇.
Observe that, given
U
≤
V
≤
G
, then, by considering a primitive idempotent decomposition in
A
V
and refining to a primitive idempotent decomposition in
A
U
, we obtain
m
A
(
U
μ
)
=
∑
ν
m
A
(
U
μ
,
V
ν
)
m
A
(
V
ν
)
,
where 𝜈 runs over the points of 𝑉 on 𝐴.
Similarly, given pointed groups
U
μ
and
W
ω
on 𝐴 and
U
≤
V
≤
W
, then
m
A
(
U
μ
,
W
ω
)
=
∑
ν
m
A
(
U
μ
,
V
ν
)
m
A
(
V
ν
,
W
ω
)
with 𝜈 running as before.
Now let 𝐷 be a finite 𝑝-group, and suppose 𝐴 a 𝐷-algebra.
A theorem of Puig, given by Linckelmann in [5, Theorem 5.12.20], implies that, given a point 𝛼 of 𝐷 on 𝐴 and a defect pointed subgroup
P
γ
of
D
α
, then any element of 𝛼 can be expressed in the form
tr
P
D
(
i
)
, where
i
∈
γ
and the idempotents
i
g
of 𝐴 are mutually orthogonal as
g
P
runs over the left cosets of 𝑃 in 𝐷.
In [5, Section 5.12], it is explained how the theorem can be seen as a generalization of Green’s Indecomposability Criterion.
Theorem 3.1
Given a 𝐷-algebra 𝐴, a local pointed group
P
γ
and points 𝛼 and 𝛽 of 𝐷 on 𝐴 such that
P
γ
is a defect pointed subgroup of
D
α
and
D
β
, then
α
=
β
.
Proof
Let
a
∈
α
and
b
∈
β
.
By Puig’s generalization of Green’s Indecomposability Theorem, there exist
i
,
j
∈
γ
such that
a
=
tr
P
D
(
i
)
and
b
=
tr
P
D
(
j
)
, with
i
g
.
i
=
0
=
j
g
.
j
for all
g
∈
D
-
P
.
Let
r
∈
(
A
P
)
×
such that
i
=
j
r
.
Define
u
=
tr
P
D
(
i
r
j
)
and
v
=
tr
P
D
(
j
r
-
1
i
)
.
By direct calculation,
u
v
=
i
and
v
u
=
j
.
So the idempotents 𝑖 and 𝑗 of
A
P
are associate, hence conjugate.
∎
Proof of the following remark is a straightforward application of Mackey decomposition.
Remark 3.2
Given a 𝐷-algebra 𝐴, a point 𝛼 of 𝐷 on 𝐴 and a defect pointed subgroup
P
γ
of
D
α
, then
m
A
(
P
γ
,
D
α
)
=
|
N
D
(
P
γ
)
:
P
|
.
4 Fusions in interior 𝐺-algebras
We shall discuss Puig’s notion of fusion between pointed groups.
Some of the criteria we shall give for the existence of morphisms do not appear to be widely known.
Throughout this section, we let 𝐴 be an interior 𝐺-algebra.
Given a group isomorphism 𝜃, we write
cod
(
θ
)
for the codomain,
dom
(
θ
)
for the domain, and we define
Δ
(
θ
)
=
{
(
θ
(
v
)
,
v
)
:
v
∈
dom
(
θ
)
}
as a subgroup of
cod
(
θ
)
×
dom
(
θ
)
.
Note that, given
U
,
V
≤
G
and a group isomorphism
ϕ
:
V
→
U
, then the
Δ
(
ϕ
)
-fixed 𝒪-submodule
A
ϕ
=
A
Δ
(
ϕ
)
is the 𝒪-submodule of elements
a
∈
A
such that
ϕ
(
v
)
a
=
a
v
for all
v
∈
V
.
Condition (a) in the next lemma was considered by Puig [6, 2.5, 2.12].
He used it in his definition of an “𝐴-fusion”.
Lemma 4.1
Let
U
μ
and
V
ν
be pointed groups on 𝐴.
Let
ϕ
:
V
→
U
be a group isomorphism.
Choose
i
∈
μ
and
j
∈
ν
.
Then the following two conditions are equivalent:
there exists
r
∈
A
×
such that
ϕ
(
v
)
i
=
(
v
j
)
r
;
there exist
s
∈
i
A
ϕ
j
and
s
′
∈
j
A
ϕ
-
1
i
such that
i
=
s
s
′
and
j
=
s
′
s
.
Moreover, the equivalent conditions (a) and (b) are independent of the choices of 𝑖 and 𝑗.
Proof
Assuming (a) and putting
s
=
i
r
j
and
s
′
=
j
r
-
1
i
, we deduce (b).
Conversely, assume (b).
Since associate idempotents of a semiperfect ring are conjugate, there exists
q
∈
A
×
such that
i
=
q
j
q
-
1
.
Putting
r
=
s
+
(
1
-
i
)
q
(
1
-
j
)
and
r
′
=
s
′
+
(
1
-
j
)
q
-
1
(
1
-
i
)
, then
r
r
′
=
1
=
r
′
r
.
So
r
∈
A
×
.
A straightforward manipulation yields the equality in (a).
Confirmation of the rider is routine.
∎
When conditions (a) and (b) hold, we call 𝜙 an isofusion
V
ν
→
U
μ
and we call
(
s
,
s
′
)
a 𝜙-witness
j
↦
i
.
Witnesses can be combined in the following way.
Let
U
μ
,
V
ν
,
W
ω
be pointed groups on 𝐴.
Let
i
∈
μ
,
j
∈
ν
,
k
∈
ω
.
Let
(
s
,
s
′
)
:
j
↦
i
,
(
t
,
t
′
)
:
k
↦
j
be witnesses for group isomorphisms
ϕ
:
V
→
U
,
ψ
:
W
→
V
, respectively.
Then
(
s
t
,
t
′
s
′
)
:
k
↦
i
is a witness for
ϕ
ψ
.
Also,
(
s
′
,
s
)
is a witness for
ϕ
-
1
.
The next two lemmas are implicit in [6, 2.5, 2.6, 2.7].
They can be proved easily and routinely using condition (a) in Lemma 4.1.
Alternatively, they can be proved very quickly using the above comments about witnesses.
Lemma 4.2
Lemma 4.2 (Puig)
Let
U
,
V
≤
G
, and let
ϕ
:
V
→
U
be a group isomorphism.
For each point 𝜈 of 𝑉 on 𝐴, there is at most one point 𝜇 of 𝑈 on 𝐴 such that 𝜙 is an isofusion
V
ν
→
U
μ
.
For each point 𝜇 of 𝑈 on 𝐴, there is at most one point 𝜈 of 𝑉 on 𝐴 such that 𝜙 is an isofusion
V
ν
→
U
μ
.
Given points 𝜇 of 𝑈 and 𝜈 of 𝑉 on 𝐴 such that 𝜙 is an isofusion
V
ν
→
U
μ
, then
ϕ
-
1
is an isofusion
U
μ
→
V
ν
.
Lemma 4.3
Lemma 4.3 (Puig)
There is a groupoid whose objects are the pointed groups on 𝐴 and whose isomorphisms are the isofusions, the composition being the usual composition of group isomorphisms.
In other words, given a pointed group
U
μ
on 𝐴, then the identity automorphism
id
U
is an isofusion
U
μ
→
U
μ
; moreover, the isofusions between the pointed groups on 𝐴 are closed under composition and inversion.
We shall be making frequent use of the latest lemma, without further mention.
Consideration of arbitrary points of a 𝑝-group, not just the local points, will be crucial in the proof of Theorem 5.2 below.
That should not be surprising since all the points of 𝑉 on 𝐴 appear in the formula for
m
A
(
U
μ
,
W
ω
)
in Section 3.
We mention, without proof, that when 𝐴 is the principal 2-block algebra of the symmetric group
S
5
, there exist local pointed groups
U
μ
≤
W
ω
and
U
<
V
<
W
such that all the points 𝜈 of 𝑉 on 𝐴 satisfying
U
μ
<
V
ν
<
W
ω
are non-local.
We shall be needing a lemma describing how isofusions restrict to pointed subgroups.
Lemma 4.4
Let
U
μ
and
V
ν
be pointed groups on 𝐴.
Let
ϕ
:
V
ν
→
U
μ
be an isofusion.
Then the pointed subgroups
S
σ
of
U
μ
and the pointed subgroups
T
τ
of
V
ν
are in a bijective correspondence such that
S
σ
↔
T
τ
if and only if 𝜙 restricts to an isofusion
T
τ
→
S
σ
.
Furthermore, when
S
σ
↔
T
τ
, we have
m
A
(
S
σ
,
U
μ
)
=
m
A
(
T
τ
,
V
ν
)
,
and 𝜙 restricts to an isomorphism
N
V
(
T
τ
)
→
N
U
(
S
σ
)
.
Proof
Fix a pointed subgroup
T
τ
≤
V
ν
.
Define
S
=
ϕ
(
T
)
.
We shall show that there exists a point 𝜎 of 𝑆 on 𝐴 such that
m
A
(
S
σ
,
U
μ
)
≥
m
A
(
T
τ
,
V
ν
)
and 𝜙 restricts to an isofusion
T
τ
→
S
σ
and to a monomorphism
N
V
(
T
τ
)
→
N
U
(
S
σ
)
.
That will suffice because the inequality will imply that
S
σ
≤
U
μ
, whereupon replacement of 𝜙 by
ϕ
-
1
and an appeal to Lemma 4.2 will yield the non-strictness of the inequality, the bijectivity of the injection
T
τ
↦
S
σ
and the bijectivity of the monomorphism
N
V
(
T
τ
)
→
N
U
(
S
σ
)
.
Let
i
∈
μ
and
j
∈
ν
.
Let
(
c
,
d
)
be a 𝜙-witness
j
↦
i
.
Choose
k
∈
τ
such that
k
≤
j
.
It is easily checked that
c
k
d
is a primitive idempotent of
A
S
and
(
c
k
,
k
d
)
is a 𝜙-witness
k
↦
c
k
d
.
Letting 𝜎 be the point of 𝑆 on 𝐴 owning
c
k
d
(the point to which
c
k
d
belongs), then
S
σ
≤
U
μ
.
By the rider of Lemma 4.1, 𝜎 is independent of the choice of 𝑘.
So, given any set
{
k
1
,
…
,
k
m
}
of mutually orthogonal elements of
τ
∩
j
A
T
j
, then
{
c
k
1
d
,
…
,
c
k
m
d
}
is a set of mutually orthogonal elements of
σ
∩
i
A
S
i
.
Taking 𝑚 to be as large as possible, we deduce the required inequality of multiplicities.
It remains only to show that, given
g
∈
N
V
(
T
τ
)
, then
ϕ
(
g
)
∈
N
U
(
S
σ
)
.
We have
(
c
k
d
)
ϕ
(
g
)
=
c
.
k
g
.
d
and
k
g
∈
τ
, so, arguing as before,
(
c
k
d
)
ϕ
(
g
)
∈
σ
, as required.
∎
The rest of this section is concerned with a characterization of isofusions that pertains only to local points.
We shall be making use of Brauer maps, as defined in [5, Definitions 5.4.2, 5.4.10] for
O
G
-modules and, in particular, for 𝐺-algebras.
For a 𝑝-subgroup
P
≤
G
and
O
G
-module 𝑀, we write the 𝑃-relative Brauer map on 𝑀 as
br
P
:
M
P
→
M
(
P
)
.
When an element of the Brauer quotient
M
(
P
)
is written in the form
x
¯
, it is to be understood that
x
¯
=
br
P
(
x
)
and
x
∈
M
P
.
We shall sometimes employ the overbar just as a reminder that
x
¯
is an element of a Brauer quotient, and we shall not always need to consider the choice of lift 𝑥.
Let 𝑃 and 𝑄 be 𝑝-subgroups of 𝐺, and let 𝜙 be a group isomorphism
Q
→
P
.
Viewing 𝐴 as an
O
(
G
×
G
)
-module, we write the
Δ
(
ϕ
)
-relative Brauer map on 𝐴 as
br
ϕ
:
A
ϕ
→
A
(
ϕ
)
.
Let 𝑅 be another 𝑝-subgroup of 𝐺 and let
ψ
:
R
→
Q
be an isomorphism.
The multiplication operation
A
×
A
→
A
restricts to an 𝒪-bilinear map
A
ϕ
×
A
ψ
→
A
ϕ
ψ
which induces an 𝔽-bilinear map
A
(
ϕ
)
×
A
(
ψ
)
→
A
(
ϕ
ψ
)
, written
(
u
¯
,
v
¯
)
↦
u
¯
*
v
¯
, where we define
u
¯
*
v
¯
=
u
v
¯
.
To see that ∗ is well-defined, observe that, by an application of the Frobenius relations,
u
v
∈
ker
(
br
ϕ
ψ
)
whenever
u
∈
ker
(
br
ϕ
)
or
v
∈
ker
(
br
ψ
)
.
We call ∗ a localized multiplication.
Some similar bilinear maps appear in [7, 3.2] by Puig–Zhou.
The localized multiplications, taken together, inherit an associativity property: given yet another 𝑝-subgroup 𝑆 of 𝐺 and a group isomorphism
χ
:
S
→
R
, then the expression
u
¯
*
v
¯
*
w
¯
is unambiguous for any
u
¯
∈
A
(
ϕ
)
,
v
¯
∈
A
(
ψ
)
,
w
¯
∈
A
(
χ
)
; in fact,
u
¯
*
v
¯
*
w
¯
=
u
v
w
¯
.
Also note that, putting
ϕ
=
id
P
, the localized multiplication
A
(
P
)
×
A
(
P
)
→
A
(
P
)
is the usual multiplication on the Brauer quotient
A
(
P
)
.
Lemma 4.5
Let
P
γ
and
Q
δ
be local pointed groups on 𝐴.
Let
i
∈
γ
and
j
∈
δ
.
Let
ϕ
:
Q
→
P
be a group isomorphism.
Then 𝜙 is an isofusion
Q
δ
→
P
γ
if and only if there exist
s
¯
∈
i
¯
*
A
(
ϕ
)
*
j
¯
and
t
¯
∈
j
¯
*
A
(
ϕ
-
1
)
*
i
¯
such that
i
¯
=
s
¯
*
t
¯
and
j
¯
=
t
¯
*
s
¯
.
Proof
One direction is clear.
Conversely, suppose such
s
¯
and
t
¯
exist.
Let
b
=
i
s
j
and
y
=
j
t
i
, which belong to
i
A
ϕ
j
and
j
A
ϕ
-
1
i
, respectively.
Then
b
y
¯
=
i
¯
, and hence
i
-
b
y
∈
i
A
P
i
∩
ker
(
br
P
)
⊆
J
(
i
A
P
i
)
.
But
i
A
P
i
is a local ring, so
b
y
has an inverse 𝑎 in
i
A
P
i
.
Putting
x
=
a
b
, then
x
∈
i
A
ϕ
j
and
i
=
x
y
.
We have
(
y
x
)
2
=
y
i
x
=
y
x
, which belongs to the local ring
j
A
Q
j
.
So
j
=
y
x
.
We have shown that
(
x
,
y
)
is a 𝜙-witness
j
↦
i
.
∎
5 Uniformity
For any interior 𝐺-algebra 𝐴, we shall introduce a category
F
uni
(
A
)
whose objects are the 𝑝-subgroups of 𝐺 and whose morphisms are some group monomorphisms.
In subsequent sections, strong further hypotheses will be imposed.
Our motive for the present generality is not any anticipated breadth of application.
The removal of irrelevant conditions is simply for the sake of clarity.
Theorem 5.1
Let 𝐴 be an interior 𝐺-algebra.
Let 𝑃 and 𝑄 be 𝑝-subgroups of 𝐺, and let
ϕ
:
Q
→
P
be a group isomorphism.
Then the following three conditions are equivalent.
The localized multiplications
*
:
A
(
ϕ
)
×
A
(
ϕ
-
1
)
→
A
(
P
)
and
*
:
A
(
ϕ
-
1
)
×
A
(
ϕ
)
→
A
(
Q
)
are surjective.
There exist
u
¯
∈
A
(
ϕ
)
and
v
¯
∈
A
(
ϕ
-
1
)
such that
1
A
(
P
)
=
u
¯
*
v
¯
and
1
A
(
Q
)
=
v
¯
*
u
¯
.
There is a bijective correspondence between the local points 𝛾 of 𝑃 on 𝐴 and the local points 𝛿 of 𝑄 on 𝐴 whereby
γ
↔
δ
provided 𝜙 is an isofusion
Q
δ
→
P
γ
.
Furthermore,
m
A
(
P
γ
)
=
m
A
(
Q
δ
)
when
γ
↔
δ
.
Proof
We first deal with a degenerate case.
Suppose
A
(
P
)
=
0
or
A
(
Q
)
=
0
.
Given
u
∈
A
ϕ
, we then have
br
ϕ
(
u
)
=
br
P
(
1
)
*
br
ϕ
(
u
)
*
br
Q
(
1
)
.
So
A
(
ϕ
)
=
0
and
A
(
ϕ
-
1
)
=
0
.
It is now easy to deduce that, if at least one of (a), (b), (c) holds, then both
A
(
P
)
and
A
(
Q
)
vanish; hence all three of (a), (b), (c) hold.
So we may suppose that
A
(
P
)
and
A
(
Q
)
are non-zero.
Assuming (a), we shall deduce (b).
By the surjectivity of one of the local multiplication operations,
1
A
(
P
)
=
u
¯
*
v
¯
for some
u
¯
∈
A
(
ϕ
)
,
v
¯
∈
A
(
ϕ
-
1
)
.
We shall show that
1
A
(
Q
)
=
v
¯
*
u
¯
for any such
u
¯
and
v
¯
.
Given
a
¯
∈
A
(
P
)
, then
a
¯
=
a
¯
*
u
¯
*
v
¯
and
a
¯
*
u
¯
∈
A
(
ϕ
)
.
So the 𝔽-linear map
-
*
v
¯
:
A
(
ϕ
)
→
A
(
P
)
is surjective.
Similarly,
u
¯
*
-
:
A
(
Q
)
→
A
(
ϕ
)
is surjective.
Therefore,
dim
F
(
A
(
P
)
)
≤
dim
F
(
A
(
ϕ
)
)
≤
dim
F
(
A
(
Q
)
)
.
Replacing 𝜙 with
ϕ
-
1
yields two further inequalities; hence
dim
F
(
A
(
P
)
)
=
dim
F
(
A
(
ϕ
)
)
=
dim
F
(
A
(
Q
)
)
=
dim
F
(
A
(
ϕ
-
1
)
)
.
Since
1
A
(
P
)
=
u
¯
*
v
¯
*
u
¯
*
v
¯
, the composite 𝔽-map
(
u
¯
*
-
)
∘
(
v
¯
*
-
)
∘
(
u
¯
*
-
)
∘
(
v
¯
*
-
)
:
A
(
ϕ
)
←
A
(
Q
)
←
A
(
ϕ
)
←
A
(
Q
)
←
A
(
ϕ
)
is the identity on
A
(
ϕ
)
.
By considering dimensions, the composite
v
¯
*
u
¯
*
-
=
(
v
¯
*
-
)
∘
(
u
¯
*
-
)
:
A
(
Q
)
←
A
(
ϕ
)
←
A
(
Q
)
must be an 𝔽-isomorphism.
But
v
¯
*
u
¯
is easily seen to be an idempotent of
A
(
Q
)
.
Therefore,
1
A
(
Q
)
=
v
¯
*
u
¯
.
We have deduced (b).
Now assuming (b), we shall deduce (c).
Consider a primitive idempotent decomposition
1
=
∑
j
∈
J
j
in
A
Q
.
For each
j
∈
J
, write
j
¯
=
br
Q
(
j
)
, and
let
J
0
=
{
j
∈
J
:
j
¯
≠
0
}
.
For each
j
∈
J
0
, the element
u
¯
*
j
¯
*
v
¯
∈
A
(
P
)
is a primitive idempotent, which we lift to a primitive idempotent
i
j
∈
A
P
.
We have
1
=
∑
j
∈
J
0
i
j
¯
and
1
=
∑
j
∈
J
0
j
¯
as primitive idempotent decompositions in
A
(
P
)
and
A
(
Q
)
, respectively.
We claim that, for every local point 𝛿 of 𝑄 on 𝐴, there exists a local point 𝛾 of 𝑃 on 𝐴 such that
ϕ
:
Q
δ
→
P
γ
.
Plainly, 𝛿 intersects with
J
0
.
Choose
j
∈
δ
∩
J
0
, write
i
=
i
j
, and let 𝛾 be the local point of 𝑃 owning 𝑖.
Defining
s
=
i
u
j
and
t
=
j
v
i
, then
s
¯
and
t
¯
satisfy the hypothesis of Lemma 4.5.
The claim is now established.
Replacing 𝜙 with
ϕ
-
1
and applying Lemma 4.2, we deduce that the condition
ϕ
:
Q
δ
→
P
γ
characterizes a bijective correspondence between the local points 𝛾 of 𝑃 on 𝐴 and the local points 𝛿 of 𝑄 on 𝐴.
Suppose
γ
↔
δ
.
We complete the deduction of (c) by observing that
m
A
(
P
γ
)
=
|
{
j
∈
J
0
:
i
j
¯
∈
br
P
(
γ
)
}
|
=
|
{
j
∈
J
0
:
j
¯
∈
br
Q
(
δ
)
}
|
=
m
A
(
Q
δ
)
.
Assuming (c), we shall deduce (b).
Let
1
=
∑
i
∈
I
i
and
j
=
∑
j
∈
J
j
be primitive idempotent decompositions in
A
P
and
A
Q
, respectively.
Define
J
0
as before and
I
0
similarly.
By the assumption, there is a bijection
J
0
∋
j
↦
i
j
∈
I
0
such that, for each
j
∈
J
0
, there exists a 𝜙-witness
(
u
j
,
v
j
)
:
j
↦
i
j
.
Let
u
=
∑
j
∈
J
0
u
j
and
v
=
∑
j
∈
J
0
v
j
.
Then
u
v
=
∑
j
∈
J
0
i
j
and
v
u
=
∑
j
∈
J
0
j
.
So
u
¯
*
v
¯
=
∑
j
∈
J
0
i
j
¯
=
∑
i
∈
I
0
i
¯
=
1
A
(
P
)
and
v
¯
*
u
¯
=
1
A
(
Q
)
.
We have deduced (b).
Finally, assuming (b), then any
t
¯
∈
A
(
P
)
satisfies
t
¯
=
t
¯
*
1
A
(
P
)
=
t
u
¯
*
v
¯
, so the localized multiplication
A
(
ϕ
)
×
A
(
ϕ
-
1
)
→
A
(
P
)
is surjective, likewise the localized multiplication
A
(
ϕ
-
1
)
×
A
(
ϕ
)
→
A
(
Q
)
.
We have deduced (a).
∎
When the three equivalent conditions in Theorem 5.1 hold, we call 𝜙locally 𝐴-uniform.
Theorem 5.2
Let 𝐴 and
ϕ
:
Q
→
P
be as in the previous theorem.
Then the following three conditions are equivalent.
Every isomorphism restricted from 𝜙 is locally 𝐴-uniform.
There is a bijective correspondence between the points 𝛼 of 𝑃 on 𝐴 and the points 𝛽 of 𝑄 on 𝐴 whereby
α
↔
β
provided 𝜙 is an isofusion
Q
β
→
P
α
.
Furthermore,
m
A
(
P
α
)
=
m
A
(
Q
β
)
when
α
↔
β
.
Proof
We shall show that each condition implies the next and the last implies the first.
Assume (a).
Let 𝛽 be a point of 𝑄 on 𝐴.
Consider a local pointed subgroup
T
τ
≤
Q
β
.
Let
S
=
ϕ
(
T
)
.
By the assumption, there exists a unique local point 𝜎 of 𝑆 on 𝐴 such that 𝜙 restricts to an isofusion
T
τ
→
S
σ
.
Let 𝒱 be the set of points 𝜈 of 𝑄 on 𝐴 such that
T
τ
≤
Q
ν
.
Let 𝒰 be the set of points 𝜇 of 𝑃 on 𝐴 such that
S
σ
≤
P
μ
.
We claim that there is a bijection
U
↔
V
such that
μ
↔
ν
if and only if 𝜙 is an isofusion
Q
ν
→
P
μ
; moreover, when those equivalent conditions hold,
m
A
(
P
μ
)
=
m
A
(
Q
ν
)
.
Since
β
∈
V
, the claim implies the existence of a point 𝛼 of 𝑃 on 𝐴 such that 𝜙 is an isofusion
Q
β
→
P
α
and
m
A
(
P
α
)
=
m
A
(
Q
β
)
.
Again replacing 𝜙 with
ϕ
-
1
and applying Lemma 4.2, part (d) will follow. (Note that, in the application of the claim,
T
τ
does not appear in the conclusion.
The role of
T
τ
is to supply an opportunity for an inductive proof of the claim.)
To demonstrate the claim, we argue by induction on
|
Q
:
T
|
.
By the assumption,
m
A
(
S
σ
)
=
m
A
(
T
τ
)
.
That is,
∑
μ
∈
U
m
A
(
S
σ
,
P
μ
)
m
A
(
P
μ
)
=
∑
ν
∈
V
m
A
(
T
τ
,
Q
ν
)
m
A
(
Q
ν
)
.
Let
V
′
be the set of
ν
∈
V
such that
T
τ
is not a defect pointed subgroup of
V
ν
.
Define
U
′
⊆
U
similarly for
S
σ
.
Given
ν
∈
V
′
, then
Q
ν
′
has a defect pointed subgroup
T
τ
′
′
strictly containing
T
τ
.
Let
S
′
=
ϕ
(
T
′
)
.
By the assumption, there exists a unique local point
σ
′
of
S
′
on 𝐴 such that 𝜙 restricts to an isofusion
T
τ
′
′
→
S
σ
′
′
.
By the inductive hypothesis, there exists a point
μ
′
of 𝑃 on 𝐴 such that 𝜙 is an isofusion
Q
ν
′
→
P
μ
′
and
m
A
(
P
μ
′
)
=
m
A
(
Q
ν
′
)
.
By Lemma 4.4,
S
σ
<
S
σ
′
′
≤
P
μ
′
.
So
μ
′
∈
U
′
.
The same lemma also implies that
m
A
(
S
σ
,
P
μ
′
)
=
m
A
(
T
τ
,
Q
ν
′
)
.
Replacing 𝜙 with
ϕ
-
1
, there is a bijective correspondence
U
′
∋
μ
′
↔
ν
′
∈
V
′
characterized by the condition that 𝜙 is an isofusion
Q
ν
′
→
P
μ
′
.
Therefore,
∑
μ
′
∈
U
′
m
A
(
S
σ
,
P
μ
′
)
m
A
(
P
μ
′
)
=
∑
ν
′
∈
V
′
m
A
(
T
τ
,
Q
ν
′
)
m
A
(
Q
ν
′
)
.
Comparing with the similar equality established earlier, we deduce that
U
=
U
′
if and only if
V
=
V
′
.
In that case, the claim is now clear.
So we may assume that
U
′
⊂
U
and
V
′
⊂
V
.
By Theorem 3.1, the differences are singleton, say,
U
-
U
′
=
{
μ
}
and
V
-
V
′
=
{
ν
}
.
So
m
A
(
S
σ
,
P
μ
)
m
A
(
P
μ
)
=
m
A
(
T
τ
,
Q
ν
)
m
A
(
Q
ν
)
.
By Lemma 4.4,
|
N
P
(
S
σ
)
|
=
|
N
Q
(
T
τ
)
|
.
Since
S
σ
and
T
τ
are defect pointed subgroups of
P
μ
and
Q
ν
, respectively, Remark 3.2 yields
m
A
(
S
σ
,
P
μ
)
=
|
N
P
(
S
σ
)
:
S
|
=
|
N
Q
(
T
τ
)
:
T
|
=
m
A
(
T
τ
,
Q
ν
)
.
Therefore,
m
A
(
P
μ
)
=
m
A
(
Q
ν
)
.
The claim is demonstrated.
The deduction of (b) is complete.
Assume (b).
Again, let
1
=
∑
j
∈
J
j
be a primitive idempotent decomposition in
A
Q
.
By the assumption, there is a primitive idempotent decomposition
1
=
∑
j
∈
J
i
j
in
A
P
such that, writing 𝛼 and 𝛽 for the points 𝑃 and 𝑄 owning
i
j
and 𝑗, respectively, then 𝜙 is an isofusion
Q
β
→
P
α
.
For each
j
∈
J
, let
(
s
j
,
t
j
)
be a 𝜙-witness
j
↦
i
j
.
Let
s
=
∑
j
∈
J
s
j
and
t
=
∑
j
∈
J
t
j
.
A short calculation yields
s
t
=
1
=
t
s
.
In particular,
s
∈
A
×
∩
A
ϕ
.
We have deduced (c).
Finally, assume (c).
Let
u
∈
A
×
∩
A
ϕ
.
Define
v
=
u
-
1
.
Then
v
∈
A
×
∩
A
ϕ
-
1
, and the elements
u
¯
∈
A
(
ϕ
)
,
v
¯
∈
A
(
ϕ
-
1
)
satisfy
1
A
(
P
)
=
u
¯
*
v
¯
and
1
A
(
Q
)
=
v
¯
*
u
¯
.
We have deduced (a).
∎
When the three equivalent conditions in Theorem 5.2 hold, we say that 𝜙 is 𝐴-uniform.
We form a category
F
uni
(
A
)
such that the objects are the 𝑝-subgroups of 𝐺 and the morphisms are the composites of the inclusions and the 𝐴-uniform isomorphisms.
For any Sylow 𝑝-subgroup 𝑆 of 𝐺, we let
F
S
uni
(
A
)
denote the full subcategory of
F
uni
(
A
)
such that the objects of
F
S
uni
(
A
)
are the subgroups of 𝑆.
Plainly,
F
S
uni
(
A
)
is a fusion system on 𝑆, and
F
S
uni
(
A
)
is well-defined up to isomorphism, in fact, well-defined up to 𝐺-conjugation, independently of the choice of 𝑆.
We call
F
S
uni
(
A
)
the uniform fusion system of 𝐴 on 𝑆.
6 Bifree bipermutation algebras
Let 𝐷 be a finite 𝑝-group.
We shall discuss the uniform fusion system
F
uni
(
A
)
for various kinds of interior 𝐷-algebra 𝐴.
We shall be imposing hypotheses as we need them.
Again, the generality is in the service of clarity.
We shall find, in Section 8, that whenever
F
uni
(
A
)
comes to be of concern in our contexts of application, it will coincide with the usual fusion system.
For the time being, though, the logic of our discussion requires us to treat
F
uni
(
A
)
as a potential innovation.
That having been said, Question 6.2 does raise a possibility that the generality may be of interest in itself.
We write
I
(
D
)
to denote the set of group isomorphisms 𝜙 such that
cod
(
ϕ
)
and
dom
(
ϕ
)
are subgroups of 𝐷.
Observe that the condition
Δ
(
ϕ
)
=
N
characterizes a bijective correspondence
ϕ
↔
N
between the isomorphisms 𝜙 in
I
(
D
)
and those subgroups
N
≤
D
×
D
that intersect trivially with
D
×
1
and
1
×
D
.
Let Ω be a
(
D
,
D
)
-biset.
We call Ωbifree provided
D
×
1
and
1
×
D
act freely on Ω.
Thus, Ω is bifree if and only if, for all
w
∈
Ω
, there exists
ϕ
∈
I
(
D
)
such that
N
D
×
D
(
w
)
=
Δ
(
ϕ
)
.
We define a bipermutation 𝐷-algebra to be an interior 𝐷-algebra 𝐴 such that 𝐴 has a
D
×
D
-stable basis.
Suppose 𝐴 is a bipermutation 𝐷-algebra, and let Ω be a
D
×
D
-stable basis.
Generalizing a comment made in Section 1, we note that, by Green’s Indecomposability Theorem and the Krull–Schmidt Theorem, Ω is well-defined as a
(
D
,
D
)
-biset up to isomorphism.
When the left action of 𝐷 and the right action of 𝐷 on Ω are free, we say that 𝐴 is bifree.
Theorem 6.1
Let 𝐴 be a bifree bipermutation 𝐷-algebra.
Then the following two conditions are equivalent.
Given
ϕ
∈
I
(
D
)
such that
A
(
ϕ
)
≠
0
, then 𝜙 is 𝐴-uniform.
𝐴 has a unital
D
×
D
-stable basis.
Proof
Let Ω be a
D
×
D
-stable basis for 𝐴.
Given
ϕ
∈
I
(
D
)
, then
A
(
ϕ
)
≠
0
if and only if
Ω
Δ
(
ϕ
)
≠
∅
.
So (b) implies (a).
Conversely, suppose (a) holds.
By Theorem 2.4, (b) will follow when we have shown that, given
w
∈
Ω
, then
N
D
×
D
(
w
)
fixes a unit of 𝐴.
By the bifreeness,
N
D
×
D
(
w
)
=
Δ
(
ϕ
)
for some
ϕ
∈
I
(
D
)
.
We have
A
(
ϕ
)
≠
0
, so, by the assumption, 𝜙 is 𝐴-uniform and
Δ
(
ϕ
)
fixes a unit of 𝐴.
We have deduced (b).
∎
When the two equivalent conditions in the latest theorem hold, we say that the bifree bipermutation 𝐷-algebra 𝐴 is uniform.
The term uniform bifree bipermutation 𝐺-algebra is rather long, but a decision on how to improve it might best await resolution of the following question.
Question 6.2
Is every bifree bipermutation 𝐷-algebra uniform?
We see no reason to expect the affirmative answer, but we have been unable to find a counter-example.
Consider a finite 𝐷-set Γ.
It is an easy exercise to show that, if 𝐷 is non-cyclic, then, for some Γ, the bipermutation 𝐷-algebra
End
O
(
O
Γ
)
does not have a unital
D
×
D
-stable basis.
However, it is also easy to show that, given 𝐷 and Γ such that
End
O
(
O
Γ
)
does not have a unital
D
×
D
-stable basis, then
End
O
(
O
Γ
)
is not bifree.
Proposition 6.3
Let 𝑒 be an idempotent of
Z
(
O
G
)
.
Let 𝑆 be a Sylow 𝑝-subgroup of 𝐺.
Let
A
=
O
G
e
as an interior 𝑆-algebra by restriction.
Then 𝐴 is a uniform bifree bipermutation 𝑆-algebra and
F
uni
(
A
)
coincides with the fusion system
F
S
(
G
)
of 𝐺 on 𝑆.
Proof
Corollary 2.5 says that 𝐴 is a uniform bifree bipermutation 𝑆-algebra.
The proof of the corollary shows the required equality of fusion systems.
∎
In particular, every group fusion system can be realized as the uniform fusion system of a uniform bifree bipermutation algebra.
At the time of writing, the authors know nothing about whether some or all exotic fusion systems can be realized in that way.
As in [3, Definition 3.4] by Gelvin–Reeh, a bifree
(
D
,
D
)
-biset Ω is said to be ℱ-semicharacteristic provided the following condition holds: given an isomorphism 𝜙 in
I
(
D
)
, then 𝜙 is an ℱ-isomorphism if and only if
Ω
Δ
(
ϕ
)
≠
∅
; furthermore, in that case,
|
Ω
Δ
(
cod
(
ϕ
)
)
|
=
|
Ω
Δ
(
ϕ
)
|
=
|
Ω
Δ
(
dom
(
ϕ
)
)
|
.
It is easy to see that, given an ℱ-semicharacteristic
(
D
,
D
)
-biset Ω, then the integer
|
Ω
|
/
|
D
|
is coprime to 𝑝 if and only if Ω is ℱ-characteristic as defined by Linckelmann [5, Definition 8.7.9] or Ragnarsson–Stancu [8].
Theorem 6.4
Let 𝐴 be a uniform bifree bipermutation 𝐷-algebra.
Then every
D
×
D
-stable basis for 𝐴 is an
F
uni
(
A
)
-semicharacteristic
(
D
,
D
)
-biset.
Proof
Let Ω be any
D
×
D
-stable basis for 𝐴, and let
ϕ
:
Q
→
P
be an
F
uni
(
A
)
-isomorphism.
By the hypothesis on 𝐴, there exists
u
∈
A
×
∩
A
ϕ
.
The bijection
u
Ω
∋
u
w
↔
w
∈
Ω
restricts to a bijection
(
u
Ω
)
Δ
(
ϕ
)
↔
Ω
Q
.
So
|
(
u
Ω
)
Δ
(
ϕ
)
|
=
|
Ω
Q
|
.
But
u
Ω
≅
Ω
as
(
P
,
Q
)
-bisets.
So
|
Ω
Δ
(
ϕ
)
|
=
|
Ω
Q
|
, and similarly for
|
Ω
P
|
.
∎
Theorem 6.4, together with Proposition 6.3, recovers the main result of Gelvin [2], which asserts that, in the notation of the proposition, any
S
×
S
-stable basis of
O
G
b
is
F
S
(
G
)
-semicharacteristic.
In Proposition 8.1, we shall confirm that, for uniform almost-source algebras, the uniform fusion system discussed in the present section coincides with the fusion system discussed in Section 1.
Incidentally, that will realize Theorem 1.1 as a special case of Theorem 6.4.
7 Divisibility of some stable bases
We shall be discussing the fusion system of an almost-source algebra and Linckelmann’s characterization of the fusion system in terms of a stable basis of the almost-source algebra.
The characterization mentions a condition that is not an isomorphism invariant, namely, the condition that a given object is fully centralized.
We shall show that, in some cases, that condition can be omitted.
For the moment, let 𝐷 be any finite 𝑝-group, let ℱ be a fusion system on 𝐷, and let Ω be a
(
D
,
D
)
-biset.
Recall, from Section 1, that Ω is ℱ-divisible
provided Ω is bifree and, given any isomorphism 𝜙 in
I
(
D
)
, then 𝜙 is an ℱ-isomorphism if and only if
Δ
(
ϕ
)
fixes an element of Ω.
Now let 𝐷 be a defect group of a block 𝑏 of
O
G
.
Let 𝐴 be an almost-source 𝐷-algebra of
O
G
b
.
Write ℓ for the unity element of 𝐴.
For each
P
≤
D
, we make the usual identification
(
O
G
)
(
P
)
=
F
C
G
(
P
)
.
Let us recall some facts that can be found, for instance, in [5, Sections 6.4, 8.7] by Linckelmann.
We have an inclusion of idempotents
br
P
(
ℓ
)
≤
b
P
for a unique block
b
P
of
F
C
G
(
P
)
, and we have an inclusion of Brauer pairs
(
P
,
b
P
)
≤
(
D
,
b
D
)
.
For
P
,
Q
≤
D
, the ℱ-isomorphisms
Q
→
P
are the conjugation isomorphisms
Q
∋
y
↦
y
g
, with
g
∈
G
satisfying
(
P
,
b
P
)
=
(
Q
,
b
Q
)
g
.
The next remark is well known.
Remark 7.1
Let 𝑏 be a block of
O
G
with defect group 𝐷 and almost-source algebra 𝐴.
Let Ω be a
D
×
D
-stable basis for 𝐴.
Then Ω is isomorphic to its opposite biset and
|
Ω
Δ
(
ϕ
)
|
=
|
Ω
Δ
(
ϕ
-
1
)
|
for any
ϕ
∈
I
(
D
)
.
Proof
This follows from Gelvin [2, Proposition 6, Lemma 7].
∎
The next result is due to Linckelmann [5, Theorem 8.7.1].
Theorem 7.2
Theorem 7.2 (Linckelmann)
Let 𝑏 be a block of
O
G
with defect group 𝐷 and almost-source algebra 𝐴.
Let ℱ be the fusion system on 𝐷 associated with 𝐴.
Let Ω be a
D
×
D
-stable basis for 𝐴.
Given an isomorphism 𝜙 in
I
(
D
)
, if
Δ
(
ϕ
)
fixes a point of Ω, then 𝜙 is an ℱ-isomorphism; furthermore, the converse holds when
dom
(
ϕ
)
or
cod
(
ϕ
)
is fully ℱ-centralized.
Can we omit the clause on the domain or codomain being fully ℱ-centralized?
More precisely, given 𝑏, must Ω be divisible for some almost-source algebra 𝐴 of
O
G
b
?
Conjecture 1.3 implies the affirmative.
Given 𝑏, must Ω be divisible for some source algebra and hence for every almost-source algebra of
O
G
b
?
To that, Conjecture 1.5 implies the affirmative.
In the next section, we shall be discussing some cases where Ω has the stronger property of being ℱ-semicharacteristic.
In the rest of the present section, we shall be proving the ℱ-divisibility of Ω in some cases where we have been unable to prove the stronger condition.
Following Craven [1, Definition 3.69], we call ℱconstrained provided
C
D
(
O
p
(
F
)
)
=
Z
(
O
p
(
F
)
)
.
By [1, Proposition 4.46], if ℱ is constrained, then
O
p
(
F
)
is the minimum among the ℱ-radical ℱ-centric subgroups of 𝐷.
Proposition 7.3
In the notation of Theorem 7.2, if ℱ is 𝑝-constrained, then Ω is ℱ-divisible.
Proof
We must show that, given
P
,
Q
≤
D
and an ℱ-isomorphism
ϕ
:
Q
→
P
, then
Δ
(
ϕ
)
fixes an element of Ω.
Let
R
=
O
p
(
F
)
.
Then 𝜙 extends to an ℱ-isomorphism
ψ
:
Q
R
→
P
R
.
Since 𝑅 is ℱ-centric,
P
R
and
Q
R
are ℱ-centric and hence fully ℱ-centralized.
So
Ω
Δ
(
ψ
)
≠
∅
.
In particular,
Ω
Δ
(
ϕ
)
≠
∅
.
∎
By [1, Corollary 5.95], the group fusion system of a 𝑝-solvable group is solvable.
Part of [1, Lemma 5.90] says that all saturated subsystems of a solvable fusion system are solvable.
So, if 𝐺 is 𝑝-solvable, then ℱ is solvable.
A theorem of Aschbacher [1, Theorem 5.91]
asserts that every solvable fusion system is constrained.
Therefore, Proposition 7.3 has the following corollary.
Corollary 7.4
In the notation of Theorem 7.2, if 𝐺 is 𝑝-solvable, then Ω is ℱ-divisible.
Alternatively, the corollary can be deduced from Proposition 7.3 by using [5, Theorem 8.1.8] to reduce to the case where
O
p
′
(
G
)
=
1
, then applying a theorem of Hall–Higman [1, Lemma 5.93].
To present one more case where we can guarantee divisibility, we need some preliminaries.
Consider a 𝑝-subgroup 𝑃 of 𝐺.
Recall the condition
br
P
(
γ
)
V
≠
0
characterizes a bijective correspondence
γ
↔
[
V
]
between the local points 𝛾 of 𝑃 on
O
G
and the isomorphism classes
[
V
]
of simple
F
C
G
(
P
)
-modules 𝑉.
Let
σ
P
G
denote the local point of 𝑃 on
O
G
corresponding to the trivial
F
C
G
(
P
)
-modules.
The next result, well-known, can be viewed as a version of Brauer’s Third Main Theorem.
A proof of it is implicit in [5, Theorem 6.3.14] by Linckelmann.
For convenience, we extract the argument.
Theorem 7.5
Let 𝑄 and 𝑃 be 𝑝-subgroups of 𝐺.
Then
Q
≤
P
if and only if
Q
σ
Q
G
≤
P
σ
P
G
.
Proof
We may assume that
O
=
F
and
Q
≤
P
.
Let
η
G
:
F
G
→
F
be the augmentation map.
Given an idempotent 𝑒 of
F
G
, then
η
G
(
e
)
=
1
if and only if 𝑒 acts as the identity on a trivial
F
G
-module.
That is equivalent to the condition that
k
≤
e
for some
k
∈
σ
1
G
.
So, letting
i
∈
σ
P
G
and
j
∈
σ
Q
G
, we have
η
C
G
(
P
)
(
i
)
=
1
=
η
C
G
(
Q
)
(
j
)
.
For all
x
∈
(
F
G
)
P
, we have
η
C
G
(
P
)
(
br
P
(
x
)
)
=
η
G
(
x
)
and similarly with 𝑄 in place of 𝑃.
Hence
η
C
G
(
Q
)
(
br
Q
(
i
)
)
=
η
G
(
i
)
=
η
C
G
(
P
)
(
br
P
(
i
)
)
.
Therefore,
η
C
G
(
Q
)
(
br
Q
(
i
j
)
)
=
η
C
G
(
Q
)
(
br
Q
(
i
)
)
.
η
C
G
(
Q
)
(
br
Q
(
j
)
)
=
1
.
So
i
j
∉
J
(
(
F
G
)
Q
)
.
It follows that
i
≥
j
′
for some conjugate
j
′
of 𝑗 in
(
F
G
)
Q
.
∎
Note that, from Theorem 7.5, we immediately recover the version of Brauer’s Third Main Theorem that is explicitly stated in [5, Theorem 6.3.14]: if 𝑏 is the principal block of
O
G
, then
b
P
is the principal block of
F
C
G
(
P
)
for each
P
≤
D
.
In particular, if 𝑏 is the principal block, then 𝐷 is a Sylow 𝑝-subgroup of 𝐺 and
F
=
F
D
(
G
)
, the fusion system of 𝐺 on 𝐷.
Theorem 7.6
In the notation of Theorem 7.2, if 𝑏 is the principal block of
O
G
, then Ω is ℱ-divisible.
Proof
Given
P
,
Q
≤
D
and
g
∈
G
such that
P
=
Q
g
, then
P
σ
P
G
=
(
Q
σ
Q
G
)
g
.
So any ℱ-isomorphism
ϕ
:
Q
→
P
is an isofusion
Q
σ
Q
G
→
P
σ
P
G
.
By Theorem 7.5, we have
P
σ
P
G
≤
D
σ
D
G
.
But
A
∩
σ
D
G
≠
∅
.
So we can define
σ
P
A
=
A
∩
σ
P
G
as a local point of 𝑃 on 𝐴, likewise
σ
Q
A
.
As 𝜙 is an isofusion
Q
σ
Q
A
→
P
σ
P
A
, Lemma 4.5 implies that
A
(
ϕ
)
and
A
(
ϕ
-
1
)
are non-zero.
∎
The proof of Theorem 1.7 is complete.
8 Uniformity of some almost-source algebras
Again, we let 𝐷 be a defect group of a block 𝑏 of
O
G
.
We shall prove Theorems 1.2, 1.4 and Proposition 1.6.
First, though, we are now ready to note that Conjecture 1.3 implies another characterization of the fusion system of an almost-source algebra.
Proposition 8.1
Let ℱ be the fusion system of a uniform almost-source 𝐷-algebra 𝐴 of
O
G
b
.
Then
F
=
F
uni
(
A
)
.
Proof
We need only show that ℱ and
F
uni
(
A
)
have the same isomorphisms.
By Theorem 7.2, every
F
uni
(
A
)
-isomorphism is an ℱ-isomorphism.
For the converse, consider an ℱ-isomorphism 𝜙.
Since ℱ is a saturated fusion system, Alperin’s Fusion Theorem guarantees that
ϕ
=
ϕ
1
…
ϕ
n
, where
ϕ
1
,
…
,
ϕ
n
, respectively, are isomorphisms restricted from ℱ-automorphisms
ψ
1
,
…
,
ψ
n
of ℱ-centric subgroups of 𝐷.
Letting
r
1
∈
A
×
∩
A
ψ
1
,
…
,
r
n
∈
A
×
∩
A
ψ
n
, then
r
1
…
r
n
∈
A
×
∩
A
ϕ
.
So 𝜙 is an
F
uni
(
A
)
-isomorphism.
∎
Via the proposition, Theorem 1.1 is now recovered as a special case of Theorem 6.4.
Let us prove Theorem 1.2.
Since the ℱ-isomorphisms are closed under restriction to group isomorphisms, (a) and (b) are equivalent, and they can be expressed as follows.
Every ℱ-isomorphism is uniform.
On the other hand, by Theorem 6.1, condition (c) can be reformulated as follows.
Given
ϕ
∈
I
(
D
)
such that
A
(
ϕ
)
≠
0
, then 𝜙 is uniform.
Theorem 7.2 immediately shows that (y) implies (z).
Conversely, assume condition (z), and let 𝜙 be an ℱ-isomorphism.
Invoking Alperin’s Fusion Theorem, write
ϕ
=
ϕ
1
…
ϕ
m
, where each
ϕ
t
is an ℱ-isomorphism restricted from an ℱ-automorphism
ψ
t
of an ℱ-centric subgroup
P
t
≤
D
.
Each
P
t
is fully ℱ-centralized.
So, by Theorem 7.2 again,
A
(
ψ
t
)
≠
0
.
By the assumption,
ψ
t
is uniform.
Hence, each
ϕ
t
is uniform and 𝜙 is uniform.
We have deduced (y).
Theorem 1.2 is now proved.
To prove Theorem 1.4, some preparation is needed.
Recall 𝑏 is said to be of principal type provided
br
P
(
b
)
is a block of
F
C
G
(
P
)
for all
P
≤
D
.
That is equivalent to the condition that the restriction
Res
D
(
O
G
b
)
G
is an almost-source 𝐷-algebra of
O
G
b
.
Corollary 2.5 immediately yields the next result.
Proposition 8.2
If 𝑏 is of principal type, the interior 𝐷-algebra
Res
D
(
O
G
b
)
G
is a uniform almost-source 𝐷-algebra of
O
G
b
.
The following theorem is part of [4, Proposition 3.1] by Harris–Linckelmann.
Theorem 8.3
Theorem 8.3 (Harris–Linckelmann)
Suppose 𝐺 is 𝑝-solvable.
Then there exists a subgroup
H
≤
G
and a block 𝑐 of
O
H
such that
𝐷 is a defect group of 𝑐 and a Sylow 𝑝-subgroup of 𝐻,
O
G
b
=
⨁
f
H
,
g
H
⊆
G
f
O
H
c
g
-
1
as 𝒪-modules,
O
G
b
≅
Ind
G
(
O
H
c
)
H
as interior 𝐺-algebras and
b
=
tr
F
G
(
c
)
.
We also need the following sufficient criterion for an idempotent of
(
O
G
b
)
D
to be an almost-source idempotent.
Lemma 8.4
Let 𝑏 be a block of
O
G
with defect group 𝐷.
Let 𝑐 be an idempotent of
(
O
G
b
)
D
such that
br
D
(
c
)
≠
0
.
Define
A
=
c
O
G
c
as an interior 𝐷-subalgebra of
O
G
b
.
Suppose that, for all
P
≤
D
, the Brauer quotient
A
(
P
)
≅
br
P
(
c
)
F
C
G
(
P
)
br
P
(
c
)
has a unique block.
Then 𝐴 is an almost source algebra of
O
G
b
.
Proof
Since 𝐴 is a permutation 𝐷-algebra and
br
D
(
c
)
≠
0
, each
br
P
(
c
)
≠
0
.
Let
b
P
be a block of
F
C
G
(
P
)
such that
br
P
(
c
)
b
P
≠
0
.
Then
br
P
(
c
)
b
P
is a block of
br
P
(
c
)
F
C
G
(
P
)
br
P
(
c
)
.
By the uniqueness hypothesis,
br
P
(
c
)
≤
b
P
.
∎
We now prove Theorem 1.4.
Let 𝐻 and 𝑐 be as in Theorem 8.3.
Put
A
=
c
O
G
c
as an interior 𝐷-algebra.
By part (3) of Theorem 8.3,
A
=
O
H
c
.
By the other two parts of that theorem, together with Proposition 8.1, 𝐴 is a uniform almost-source algebra of
O
H
c
.
Given
P
≤
D
, then
A
(
P
)
is isomorphic to the block algebra
F
C
G
(
P
)
br
P
(
c
)
.
Hence, via Lemma 8.4, 𝐴 is an almost-source 𝐷-algebra of
O
G
b
.
The proof of Theorem 1.4 is complete.
The rest of this section is a discussion of the special case where 𝐴 is a source algebra.
The next result realizes the multiplicities in Theorem 1.2 as relative multiplicities.
Corollary 8.5
Suppose that 𝑏 is a block of
O
G
with defect group 𝐷 and source 𝐷-algebra 𝐴.
Write ℱ for the fusion system on 𝐷 associated with 𝐴.
Let
λ
D
be the unique local point of 𝐷 on
O
G
such that
λ
D
∩
A
≠
∅
.
Then the following three conditions are equivalent.
For every ℱ-isomorphism
ϕ
:
Q
→
P
, there is a bijective correspondence between the local points 𝛾 of 𝑃 on
O
G
satisfying
P
γ
≤
D
λ
D
, the local points 𝛿 of 𝑄 on
O
G
satisfying
Q
δ
≤
D
λ
D
.
The correspondence is such that
γ
↔
δ
if and only if 𝜙 is an isofusion
Q
δ
→
P
γ
.
Furthermore, when
γ
↔
δ
, we have
m
O
G
(
P
γ
,
D
λ
D
)
=
m
O
G
(
Q
δ
,
D
λ
D
)
.
Upon removing the term local from (a), the condition still holds.
There exists a unital
D
×
D
-stable basis for 𝐴.
Proof
For
P
≤
D
, the condition
β
′
=
A
∩
β
characterizes a bijective correspondence between the points
β
′
of 𝑃 on 𝐴 and the points 𝛽 of 𝑃 on
O
G
satisfying
A
∩
β
≠
∅
.
Moreover,
β
′
is local if and only if 𝛽 is local.
Noting that
ℓ
∈
λ
D
, we have
m
A
(
P
β
′
)
=
m
O
G
(
P
β
,
D
λ
D
)
.
It
is now clear that the assertion to be proved is a special case of Theorem 1.2.
∎
By Theorem 1.1, when the equivalent conditions in the corollary hold, any unital
D
×
D
-stable basis for 𝐴 is ℱ-characteristic.
We mention that some applications of characteristic bisets appear in [8] by Ragnarsson–Stancu.
To comment on the sense in which condition (a) of the corollary expresses an invariance property, let us note that, for any ℱ-centric
P
≤
D
, there is a unique local point
λ
P
of 𝑃 on
O
G
such that
P
λ
P
≤
D
λ
D
.
Defining
m
A
(
P
)
=
m
O
G
(
P
λ
P
,
D
λ
D
)
,
uniformity implies that
m
A
(
P
)
is ℱ-invariant.
We mean to say, if 𝐴 is uniform, then
m
A
(
P
)
depends only on the ℱ-isomorphism class of 𝑃.
To fulfill the obligations taken on in Section 1, it remains only to prove Proposition 1.6.
Still supposing that 𝐴 is a source algebra of
O
G
b
, now supposing also that
F
=
N
F
(
D
)
, let 𝜙 be an ℱ-isomorphism.
Let 𝜓 be an extension of 𝜙 to an ℱ-automorphism of 𝐷.
Let
r
∈
A
×
such that
(
r
,
r
-
1
)
is a 𝜓-witness
ℓ
↦
ℓ
.
Then
r
∈
A
Δ
(
ψ
)
⊆
A
Δ
(
ϕ
)
.
So, by Theorem 2.4, 𝐴 has a unital
D
×
D
-stable basis.
The proof of Proposition 1.6 is complete.
und
Matthew Gelvin
