Burnside rings and fusion systems
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1 university of copenhagen Burnside rings and fusion systems Sune Precht Reeh Centre for Symmetry and Deformation Department of Mathematical Sciences UC Santa Cruz, May 21, 2014 Slide 1/16
2 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. Slide 2/16
3 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 Y ) H = X H + Y H and (X Y ) H = X H Y H, so the fixed point maps X 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. Slide 3/16
4 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: Hom S (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 F S (G) defined for S Syl p (G) with morphisms for P, Q S. Hom FS (G)(P, Q) := Hom G (P, Q). Slide 4/16
5 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. Slide 5/16
6 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). Slide 6/16
7 A transfer map Theorem (R.) Let F be a saturated fusion system over a finite p-group S. Then there is a transfer map tr F S : A(S) (p) A(F) (p) satisfying tr F S (X) Q = 1 #{Q F Q} Q F Q X 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). Slide 7/16
8 The p-local basis By applying tr F S to each transitive S-set [S/P ], we get an alternative Z (p) -basis for A(F) (p) consisting of β P := tr F 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 = F S (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 β P = [G/P ] [G/S] for all P S. Slide 8/16
9 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]. Slide 9/16
10 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 F S (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 ω F A(S, S) (p), and ω F determines F. Slide 10/16
11 The transfer map for fusion systems, when applied to the product fusion system F F, gives a new construction for ω F : 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 ω F = β (S,id). As an immediate consequence we even get a formula for the fixed points of ω F : Theorem (Boltje-Danz, R.) We have (ω F ) (P,ϕ) = S F(P,S) for ϕ F(P, S), and (ω F ) D = 0 for all other subgroup D S S. Slide 11/16
12 Maps induced by virtual bisets Each (virtual) biset B A(S, S) (p) induces a map A(S) (p) A(S) (p) by X B X = B S X. For F = F S (G), the map induced by the (S, S)-biset G is res G S trg S : 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 ω F A(S, S) (p) coincides with the transfer map from earlier: (ω F X) Q = 1 #{Q F Q} Q F Q X Q. Slide 12/16
13 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? Slide 13/16
14 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). Slide 14/16
15 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 A semichar (F) be the subring of A(F, F) consisting of all semicharacteristic elements. Slide 14/16
16 The ring of semicharacteristic elements Theorem (R.) The ring of semicharacteristic elements A semichar (F) (p) is isomorphic to the Burnside ring A(F) (p) with the basiselement β (P,id) A semichar (F) (p) corresponding to β P A(F) (p). The isomorphism A semichar = (F) (p) A(F)(p) coincides with the map X X/S that quotients out the right S-action of a biset. Corollary The characteristic idempotent ω F A semichar (F) (p) is unique, since A(F) (p) only has idempotents 0 and 1. Slide 15/16
17 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, MR [2] Kári Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006), MR (2007f:55013) [3] Kári Ragnarsson and Radu Stancu, Saturated fusion systems as idempotents in the double Burnside ring, Geom. Topol. 17 (2013), no. 2, MR [4] Sune Precht Reeh, The abelian monoid of fusion-stable finite sets is free, 15 pp., preprint, available at arxiv: [5] Sune Precht Reeh, Transfer and characteristic idempotents for saturated fusion systems, 39 pp., preprint, available at arxiv: Slide 16/16
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