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Burnside rings and fusion systems
Sune Precht Reeh
Centre for Symmetry and Deformation
Department of Mathematical Sciences
UC Santa Cruz, May 21, 2014
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The Burnside ring of a finite group
Let G be a finite group. The isomorphism classes of finite
G-sets 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, hence the
transitive G-sets [G/H] for H ≤ 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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For each finite G-set X and any subgroup H ≤ G, we can
count the number of fixed points |X H |. 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.
|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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Fusion systems
A fusion system over a finite p-group S is a category F
where the objects are the subgroups P ≤ S and the
morphisms 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
If S is a subgroup of G, we can turn any G-set X into an
S-set by restricting the G-action to S. The S-set X still
remembers some of the G-sction, for instance the fixed
points |X P | only depend on P ≤ S up to G-conjugation.
In general, 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
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 is a subring of A(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 =
X 1
X Q0 0
#{Q ∼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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The p-local basis
By applying trF
S to each transitive S-set [S/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.
In the case where G has S as a Sylow p-subgroup and
F = FS (G), the sets [G/P ] for P ≤ S also form a basis for
A(F)(p) . However this basis depends on more than just F.
Proposition (R.)
When S is a Sylow p-subgroup of G, [G/S] is invertible in
A(F)(p) . The G-basis and the β-basis are furthermore
related by βP =
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[G/P ]
[G/S]
for all P ≤ S.
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Bisets
The Burnside biset module A(S, T ) is additively
constructed like a Burnside ring, but from sets that have
both a right S-action and a free left T -action that
commute. The composition A(T, R) × A(S, T ) → A(S, R)
is defined by Y ◦ X := Y ×T X.
A(S, S) is the double Burnside ring of S.
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:
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 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) for F × F is F-characteristic and
idempotent. Hence ωF = β∆(S,id) .
As an immediate consequence we even get a formula for
the fixed points of ωF :
Theorem (Boltje-Danz, R.)
|S|
We have |(ωF )∆(P,ϕ) | = |F (P,S)|
for ϕ ∈ F(P, S), and
D
|(ωF ) | = 0 for all other subgroup D ≤ S × S.
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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 from earlier:
(ωF ◦ X)Q =
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#{Q0
X 1
X Q0 .
∼F Q} 0
Q ∼F Q
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Embedding the Burnside ring in the biset ring
In the double Burnside ring A(G, G) of a finite group G,
the transitive bisets [G ×H G] ∼
= [H, id] for H ≤ G satisfy
the same multiplication formulas as the G-sets G/H in
A(G).
Consequently, A(G) is isomorphic to the subring of
A(G, G) generated by [H, id] for H ≤ G.
For a saturated fusion system F we have A(F) and
A(F, F) – the ring of (F × F)-stable virtual bisets. Is
there a similar result for these rings?
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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.
Corollary
The characteristic idempotent ωF ∈ Asemichar (F)(p) is
unique, since A(F)(p) only has idempotents 0 and 1.
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References
[1] Robert Boltje and Susanne Danz, A ghost ring for the left-free
double Burnside ring and an application to fusion systems, Adv.
Math. 229 (2012), no. 3, 1688–1733. MR2871154
[2] Kári Ragnarsson, Classifying spectra of saturated fusion systems,
Algebr. Geom. Topol. 6 (2006), 195–252. MR2199459
(2007f:55013)
[3] Kári Ragnarsson and Radu Stancu, Saturated fusion systems as
idempotents in the double Burnside ring, Geom. Topol. 17 (2013),
no. 2, 839–904. MR3070516
[4] Sune Precht Reeh, The abelian monoid of fusion-stable finite sets
is free, 15 pp., preprint, available at arXiv:1302.4628.
[5] Sune Precht Reeh, Transfer and characteristic idempotents for
saturated fusion systems, 39 pp., preprint, available at
arXiv:1306.4162.
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