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Burnside rings of fusion systems
Sune Precht Reeh
Centre for Symmetry and Deformation
Department of Mathematical Sciences
PhD defence, August 11, 2014
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Outline
An example
1
The Burnside ring of a finite group
2
Fusion stable elements
3
The characteristic idempotent
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An example: Rotating a cube
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The Burnside ring of a finite group
Let G be a finite group. All finite sets with an action of G
(up to isomorphism) form an abelian monoid with disjoint
union as addition.
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The Burnside ring of a finite group
Let G be a finite group. All finite sets with an action of G
(up to isomorphism) form an abelian monoid with disjoint
union as addition.
The Grothendieck group of this monoid is the Burnside
ring of G, denoted A(G). The multiplication in A(G) is
given by Cartesian products. We call the elements of A(G)
virtual G-sets.
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The Burnside ring of a finite group
Let G be a finite group. All finite sets with an action of G
(up to isomorphism) form an abelian monoid with disjoint
union as addition.
The Grothendieck group of this monoid is the Burnside
ring of G, denoted A(G). The multiplication in A(G) is
given by Cartesian products. We call the elements of A(G)
virtual G-sets.
Every finite G is the disjoint union of its orbits, so the
transitive G-sets [G/H] for subgroups H of G form an
additive basis for A(G).
The basis element [G/H] only depends on H up to
conjugation in G.
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The Burnside ring of a finite group
For each finite G-set X and any subgroup H ≤ G, we can
count the number of points in X that are fixed by H, i.e.
the number |X H |.
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The Burnside ring of a finite group
For each finite G-set X and any subgroup H ≤ G, we can
count the number of points in X that are fixed by H, i.e.
the number |X H |.
|X H | only depends on H up to G-conjugation, and the
collection of numbers |X H | for H ≤ G determines
X ∈ A(G) uniquely.
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The Burnside ring of a finite group
For each finite G-set X and any subgroup H ≤ G, we can
count the number of points in X that are fixed by H, i.e.
the number |X H |.
|X H | only depends on H up to G-conjugation, and the
collection of numbers |X H | for H ≤ G determines
X ∈ A(G) uniquely.
Taking fixed points respects the structure of A(G):
|(X t Y )H | = |X H | + |Y H | and |(X × Y )H | = |X H | · |Y H |,
so the fixed point maps X 7→ |X H | extend to ring
homomorphisms A(G) → Z.
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Fusion systems
A fusion system over a finite p-group S consists of the
subgroups P ≤ S and collections of maps/homomorphism
F(Q, P ) between the subgroups. The maps must satisfy:
• HomS (P, Q) ⊆ F(P, Q) ⊆ Inj(P, Q) for all P, Q ≤ S.
• Every ϕ ∈ F(P, Q) factors in F as an isomorphism
P → ϕP followed by an inclusion ϕP ,→ Q.
A saturated fusion system satisfies a few additional axioms
that play the role of Sylow’s theorems.
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Fusion systems
A fusion system over a finite p-group S consists of the
subgroups P ≤ S and collections of maps/homomorphism
F(Q, P ) between the subgroups. The maps must satisfy:
• HomS (P, Q) ⊆ F(P, Q) ⊆ Inj(P, Q) for all P, Q ≤ S.
• Every ϕ ∈ F(P, Q) factors in F as an isomorphism
P → ϕP followed by an inclusion ϕP ,→ Q.
A saturated fusion system satisfies a few additional axioms
that play the role of Sylow’s theorems.
The canonical example of a saturated fusion system is
FS (G) defined for S ∈ Sylp (G) with morphisms
HomFS (G) (P, Q) := HomG (P, Q).
for P, Q ≤ S.
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The Burnside ring of a fusion system
Let F be a saturated fusion system over S. A finite S-set
X (or an element of A(S)) is said to be F-stable if
|X Q | = |X P | whenever Q, P are isomorphic/conjugate in
F.
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The Burnside ring of a fusion system
Let F be a saturated fusion system over S. A finite S-set
X (or an element of A(S)) is said to be F-stable if
|X Q | = |X P | whenever Q, P are isomorphic/conjugate in
F.
Theorem (R.)
The isomorphism classes of F-stable finite S-sets form a
free abelian monoid. The irreducible stable sets are in
1-to-1 correspondence with the subgroups of S up to
F-conjugation.
The Grothendieck group of this monoid is the Burnside
ring of F, denoted A(F), and it is also the subring of
A(S) formed by all the F-stable elements.
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A transfer map
Theorem (R.)
Let F be a saturated fusion system over a finite p-group S.
Then there is a transfer map trF
S : A(S)(p) → A(F)(p)
satisfying
F
trS (X)Q =
for all X ∈ A(S)(p) and Q ≤ S.
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A transfer map
Theorem (R.)
Let F be a saturated fusion system over a finite p-group S.
Then there is a transfer map trF
S : A(S)(p) → A(F)(p)
satisfying
F
trS (X)Q =
#{Q0
X 1
X Q0 ∼F Q} 0
for all X ∈ A(S)(p) and Q ≤ S.
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Q ∼F Q
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A transfer map
Theorem (R.)
Let F be a saturated fusion system over a finite p-group S.
Then there is a transfer map trF
S : A(S)(p) → A(F)(p)
satisfying
F
trS (X)Q =
#{Q0
X 1
X Q0 ∼F Q} 0
Q ∼F Q
for all X ∈ A(S)(p) and Q ≤ S.
Furthermore, π is a homomorphism of A(F)(p) -modules
and restricts to the identity on A(F)(p) .
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A transfer map
Theorem (R.)
Let F be a saturated fusion system over a finite p-group S.
Then there is a transfer map trF
S : A(S)(p) → A(F)(p)
satisfying
F
trS (X)Q =
#{Q0
X 1
X Q0 ∼F Q} 0
Q ∼F Q
for all X ∈ A(S)(p) and Q ≤ S.
Furthermore, π is a homomorphism of A(F)(p) -modules
and restricts to the identity on A(F)(p) .
We get an alternative Z(p) -basis for A(F)(p) consisting of
βP := trF
S ([S/P ]) which only depends on P ≤ S up to
F-conjugation.
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Bisets
The double Burnside ring A(S, T ) is additively constructed
like a Burnside ring, but from sets that have both a right
S-action and a left S-action that commute. The
composition/multiplication A(S, S) × A(S, S) → A(S, S) is
defined by Y ◦ X := Y ×S X.
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Bisets
The double Burnside ring A(S, T ) is additively constructed
like a Burnside ring, but from sets that have both a right
S-action and a left S-action that commute. The
composition/multiplication A(S, S) × A(S, S) → A(S, S) is
defined by Y ◦ X := Y ×S X.
Warning: The obvious bijection A(S, S) ∼
= A(S × S) does
not preserve the multiplication.
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Bisets
The double Burnside ring A(S, T ) is additively constructed
like a Burnside ring, but from sets that have both a right
S-action and a left S-action that commute. The
composition/multiplication A(S, S) × A(S, S) → A(S, S) is
defined by Y ◦ X := Y ×S X.
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Bisets
The double Burnside ring A(S, T ) is additively constructed
like a Burnside ring, but from sets that have both a right
S-action and a left S-action that commute. The
composition/multiplication A(S, S) × A(S, S) → A(S, S) is
defined by Y ◦ X := Y ×S X.
We are particularly interested in (virtual) (S, S)-bisets
where all stabilizers have the form of twisted diagonals
∆(P, ϕ) = {(ϕ(x), x) | x ∈ P } ≤ S × S
for some P ≤ S and ϕ ∈ F(P, S).
For each twisted diagonal, we denote the transitive biset
[S × S/∆(P, ϕ)] by [P, ϕ].
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The characteristic idempotent
If G induces a fusion system on S, we can ask what
properties G has as an (S, S)-biset in relation to FS (G).
Linckelmann-Webb wrote down the essential properties as
the following definition:
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The characteristic idempotent
If G induces a fusion system on S, we can ask what
properties G has as an (S, S)-biset in relation to FS (G).
Linckelmann-Webb wrote down the essential properties as
the following definition:
An element Ω ∈ A(S, S)(p) is said to be F-characteristic if
• Ω is F-stable with respect to both S-actions,
• every stabilizer of Ω has the form
∆(P, ϕ) = {(ϕ(x), x) | x ∈ P } for some ϕ ∈ F(P, S),
• |Ω|/|S| is invertible in Z(p) .
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The characteristic idempotent
If G induces a fusion system on S, we can ask what
properties G has as an (S, S)-biset in relation to FS (G).
Linckelmann-Webb wrote down the essential properties as
the following definition:
An element Ω ∈ A(S, S)(p) is said to be F-characteristic if
• Ω is F-stable with respect to both S-actions,
• every stabilizer of Ω has the form
∆(P, ϕ) = {(ϕ(x), x) | x ∈ P } for some ϕ ∈ F(P, S),
• |Ω|/|S| is invertible in Z(p) .
Theorem (Broto-Levi-Oliver)
Every saturated fusion system F has a (non-unique)
characteristic biset.
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The characteristic idempotent
If G induces a fusion system on S, we can ask what
properties G has as an (S, S)-biset in relation to FS (G).
Linckelmann-Webb wrote down the essential properties as
the following definition:
An element Ω ∈ A(S, S)(p) is said to be F-characteristic if
• Ω is F-stable with respect to both S-actions,
• every stabilizer of Ω has the form
∆(P, ϕ) = {(ϕ(x), x) | x ∈ P } for some ϕ ∈ F(P, S),
• |Ω|/|S| is invertible in Z(p) .
Theorem (Ragnarsson-Stancu)
Every saturated fusion system F has a unique
F-characteristic idempotent ωF ∈ A(S, S)(p) , and ωF
determines F.
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The characteristic idempotent
The transfer map for fusion systems, when applied to the
product fusion system F × F, gives a new construction for
ωF :
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The characteristic idempotent
The transfer map for fusion systems, when applied to the
product fusion system F × F, gives a new construction for
ωF :
Theorem (R.)
Let F be a saturated fusion system. The element
β∆(S,id) ∈ A(S, S)(p) associated to F × F is
F-characteristic and idempotent. Hence ωF = β∆(S,id) .
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Maps induced by virtual bisets
Each (virtual) biset B ∈ A(S, S)(p) induces a map
A(S)(p) → A(S)(p) by X 7→ B ◦ X = B ×S X.
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Maps induced by virtual bisets
Each (virtual) biset B ∈ A(S, S)(p) induces a map
A(S)(p) → A(S)(p) by X 7→ B ◦ X = B ×S X.
For F = FS (G), the map induced by the (S, S)-biset G is
G
resG
S trS : A(S) → A(F), which sends [S/P ] to [G/P ].
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Maps induced by virtual bisets
Each (virtual) biset B ∈ A(S, S)(p) induces a map
A(S)(p) → A(S)(p) by X 7→ B ◦ X = B ×S X.
For F = FS (G), the map induced by the (S, S)-biset G is
G
resG
S trS : A(S) → A(F), which sends [S/P ] to [G/P ].
Theorem (R.)
The map A(S)(p) → A(F)(p) induced by the characteristic
idempotent ωF ∈ A(S, S)(p) coincides with the transfer
map trF
S from earlier:
(ωF ◦ X)Q =
X 1
X Q0 .
#{Q0 ∼F Q} 0
Q ∼F Q
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(Semi)characteristic elements
Not only is the characteristic idempotent ωF related to
trF
S . There turns out to be a close relation between all the
F-characteristic elements and the Burnside ring A(F)(p) :
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Recall:
For a saturated fusion system F an element Ω ∈ A(S, S)(p)
is F-characteristic if
• Ω is F-stable with respect to both S-actions,
• every stabilizer of Ω has the form
∆(P, ϕ) := {(ϕ(x), x) | x ∈ P } for some ϕ ∈ F(P, S),
• |Ω|/|S| is invertible in Z(p) .
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Define:
For a saturated fusion system F an element Ω ∈ A(S, S)(p)
is F-semicharacteristic if
• Ω is F-stable with respect to both S-actions,
• every stabilizer of Ω has the form
∆(P, ϕ) := {(ϕ(x), x) | x ∈ P } for some ϕ ∈ F(P, S),
• |Ω|/|S| is invertible in Z(p) .
Let Asemichar (F) be the subring of A(F, F) consisting of
all semicharacteristic elements.
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The ring of semicharacteristic elements
Theorem (R.)
The ring of semicharacteristic elements Asemichar (F)(p) is
isomorphic to the Burnside ring A(F)(p) with the
basiselement β∆(P,id) ∈ Asemichar (F)(p) corresponding to
βP ∈ A(F)(p) .
∼
=
The isomorphism Asemichar (F)(p) −
→ A(F)(p) coincides
with the map X 7→ X/S that quotients out the right
S-action of a biset.
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The ring of semicharacteristic elements
Theorem (R.)
The ring of semicharacteristic elements Asemichar (F)(p) is
isomorphic to the Burnside ring A(F)(p) with the
basiselement β∆(P,id) ∈ Asemichar (F)(p) corresponding to
βP ∈ A(F)(p) .
∼
=
The isomorphism Asemichar (F)(p) −
→ A(F)(p) coincides
with the map X 7→ X/S that quotients out the right
S-action of a biset.
Corollary
The characteristic idempotent ωF ∈ Asemichar (F)(p) is
unique, since A(F)(p) only has idempotents 0 and 1.
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THE END(?)
Thank you for your attention!
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