Burnside rings and fusion systems
Sune Precht Reeh
Centre for Symmetry and Deformation Department of Mathematical Sciences
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.
For each finite G-set X and any subgroup H ≤ G, we can count the number of fixed points |XH|. Taking fixed points respects the structure of A(G):
|(X t Y )H| = |XH| + |YH| and |(X × Y )H| = |XH| · |YH|, so the fixed point maps X 7→ |XH| extend to ring
homomorphisms A(G) → Z.
|XH| only depends on H up to G-conjugation, and the collection of numbers |XH| for H ≤ G determines X ∈ A(G) uniquely.
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.
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 |XP| 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 |XQ| = |XP| whenever Q, P are
isomorphic/conjugate in F .
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).
A transfer map
Theorem (R.)
Let F be a saturated fusion system over a finite p-group S.
Then there is a transfer map trFS: A(S)(p)→ A(F )(p) satisfying
trFS(X)Q
= 1
#{Q0 ∼F Q}
X
Q0∼FQ
XQ0
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).
The p-local basis
By applying trFS to each transitive S-set [S/P ], we get an alternative Z(p)-basis for A(F )(p) consisting of
βP := trFS([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 = [G/P ][G/S] for all P ≤ S.
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].
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 .
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.)
We have |(ωF)∆(P,ϕ)| = |F (P,S)||S| for ϕ ∈ F (P, S), and
|(ωF)D| = 0 for all other subgroup D ≤ S × S.
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 ×SX.
For F = FS(G), the map induced by the (S, S)-biset G is resGStrGS: 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
= 1
#{Q0∼F Q}
X
Q0∼FQ
XQ0 .
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?
u n i v e r s i t y o f c o p e n h a g e n
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).
all semicharacteristic elements.
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.
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.
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] K´ari Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006), 195–252. MR2199459
(2007f:55013)
[3] K´ari 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.