Representation rings for fusion systems and dimension functions - - PowerPoint PPT Presentation

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Representation rings for fusion systems and dimension functions

Sune Precht Reeh

joint with Erg¨ un Yal¸ cın

Notes from the flip charts are in green.

Isle of Skye, 22. June 2018 Thanks to support from Maria de Maeztu (MDM-2014-0445). Slide 1/15

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Dimension functions

Related to Yal¸ cın’s talk: Study finite group actions on finite CW-complexes ≃ Sn – with restrictions on isotropy. One way to construct a G-action on an actual sphere: take the unit-sphere S(V ) where V is a real G-representation. For a real G-representation V we define the dimension function for V as Dim(V )(P) := dimR(V P ) for P ≤ G up to conjugation in G. The dimension function for an action of G on a finite homotopy sphere X ≃ Sn is given by XP ∼

p SDim(X)(P)−1 for any p-subgroup P ≤ G and a

prime p.

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Borel-Smith functions

Q: Which functions on the conjugacy classes of subgroups in G arise as dimension functions for real representations/homotopy sphere actions? Necessary: Borel-Smith conditions for a function f on the conjugacy classes of subgroups. (i) If K ⊳ H, H/K ∼ = Z/p, p odd, then 2 | f(K) − f(H). (ii) If K ⊳ H, H/K ∼ = Z/p × Z/p, then f(K) − f(H) =

K<L<H

• f(L) − f(H)
• .

(iii) If K ⊳ H ⊳ L ≤ NG(K), H/K ∼ = Z/2, and if L/K ∼ = Z/4, then 2 | f(K) − f(H),

• r if L/K ∼

= Q8, then 4 | f(K) − f(H). Let Cb(G) denote the set of Borel-Smith functions for G.

Sune Precht Reeh Slide 3/15

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Borel-Smith functions 2

As an example, consider the group C5. Let us denote the irreducible complex characters of C5 by χ1, χ2, . . . , χ5. The irreducible real representations of C5 then have characters χ1, χ2 + χ5, χ3 + χ4. The dimension functions for these are Dim 1 C5 χ1 1 1 χ2 + χ5 2 χ3 + χ4 2 The only Borel-Smith condition that applies to C5 is (i), which states that 2 | f(1) − f(C5). This relation is easily confirmed for the irreducible real representations. In fact every Borel-Smith function is a linear combination of the dimension functions above and hence is the dimension function of some virtual real representation.

Sune Precht Reeh Slide 4/15

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Realizing Borel-Smith functions

Theorem

Let G be a finite group. [tom Dieck] When G is nilpotent, RR(G) → Cb(G) is surjective. [Dotzel-Hamrick] If G is a p-group, and if f ∈ Cb(G) is nonnegative and monotone, then there exists a real representation V such that Dim(V ) = f.

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Realizing Borel-Smith functions 2

Theorem (R.-Yal¸ cın)

Let G be a finite group. If f is a non-negative, monotone Borel-Smith function defined on the prime-power subgroups of G, then there exists a finite G-CW-complex X ≃ Sn such that X only has prime-power isotropy, and Dim(X) = N · f for some N > 0.

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Sketch of proof

f B.S.-function f2 f3 f5 f7 · · · Restriction to p-groups B.S.-function at p V2 V3 V5 V7 · · · Rest of talk Real repr. “at p” X [Hambleton-Yal¸ cın] Finite htpy. sphere with G-action

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Fusion systems

Given a finite group G and a prime p, let S ∈ Sylp(G). The fusion system FS(G) induced by G on S is a category with objects P ≤ S and morphisms HomFS(G)(P, Q) := {cg : P → Q | g ∈ G, g−1Pg ≤ Q}. There is a notion of abstract (saturated) fusion systems F on S, with exotic examples not coming from finite groups and Sylow subgroups. An S-representation V is F-stable if χV (a) = χV (a′) whenever a′ = ϕ(a) for some homomorphism ϕ ∈ F and a, a′ ∈ S. The representation ring RR(F) consists of F-stable virtual representations V ∈ RR(S).

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Fusion systems 2

Consider C5 ⋊ C4 where C4 acts on C5 as the full automorphism group. The fusion system F = FC5(C5 ⋊ C4) at the prime 5 has C5 endowed with the additional conjugation from C4. The trivial representation χ1 is F-stable, but χ2 + χ5 and χ3 + χ4 are not invariant under the C4-action. The indecomposable F-stable representations are χ1 and χ2 + χ3 + χ4 + χ5. Their dimension functions: Dim 1 C5 χ1 1 1 χ2 + · · · + χ5 4 The Borel-Smith functions for C5 are no longer all going to be linear combinations of the above, e.g. the Borel-Smith function (2 0). However, up to multiplying with 2 they are.

Sune Precht Reeh Slide 9/15

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Borel-Smith functions for F

Cb(F) consists of Borel-Smith functions f ∈ Cb(S) such that f is constant on isomorphism classes in F. RR(−) → Cb(−) is a natural transformation of biset functors on p-groups and is pointwise surjective. A general result then gives us that RR(F)(p) → Cb(F)(p) is surjective. If we want a result without p-localization, we need to add an extra condition to the Borel-Smith conditions: (iv) [Bauer] If K ⊳ H, H/K ∼ = Z/p, α ∈ Aut(H/K) is induced by AutF(H), then (order of α) | f(K) − f(H).

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Borel-Smith functions for F 2

For our example fusion system F on C5 induced by C5 ⋊ C4, we have an automorphism of C5/1 of order 4 induced by the C4-action. The condition (iv) then states 4 | f(1) − f(C5). This is enough to ensure that every Borel-Smith function satisfying the additional condition 4 | f(1) − f(C5) will be a linear combination of the dimension functions for F-stable representations, and hence realized by an F-stable virtual real representation.

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Realizing Borel-Smith functions (at p)

Theorem (R.-Yal¸ cın)

Let F be a saturated fusion system on a p-group S (e.g. F = FS(G)), then RR(F) → Cb+(iv)(F) is surjective.

Theorem (R.-Yal¸ cın)

Let F be a saturated fusion system on a p-group S (e.g. F = FS(G)). If f is a nonnegative, monotone Borel-Smith function (possibly satisfying (iv)), then there exists an F-stable real S-representation V such that Dim(V ) = N · f for some N > 0 (depending on F and not f). Open problem: Does N = 1 work for f satisfying (iv)?

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Proof for monotone B.S.-functions

Suppose f is a non-negative, monotone Borel-Smith function for F. Realize f by some real S-representation V that might not be F-stable. χV lives in a finite extension L of Q, so χ′ =

σ∈Gal(L/Q) χσ V is a rational

valued character. There is an m > 0 such that m · χ′ is the character of a rational S-representation W, with Dim(W) = m · |L : Q| · f. That f is F-stable implies that Dim(W) is F-stable which in turn implies that W is F-stable because Dim(−) is injective on rational representations.

• Sune Precht Reeh

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Epilogue

f B.S.-function f2 f3 f5 f7 · · · B.S.-function for Fp V2 V3 V5 V7 · · · Fp-stable real representation X Finite htpy. sphere with G-action

Theorem (R.-Yal¸ cın)

Let G be a finite group. If f is a non-negative, monotone Borel-Smith function defined on the prime-power subgroups of G, then there exists a finite G-CW-complex X ≃ Sn such that X only has prime-power isotropy, and Dim(X) = N · f for some N > 0.

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The End

[1] Stefan Bauer, A linearity theorem for group actions on spheres with applications to homotopy representations, Comment. Math. Helv. 64 (1989), no. 1, 167–172. [2] Tammo tom Dieck, Transformation groups, de Gruyter Studies in Mathematics,

• vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR889050 (89c:57048)

[3] Ronald M. Dotzel and Gary C. Hamrick, p-group actions on homology spheres,

• Invent. Math. 62 (1981), no. 3, 437–442.

[4] Ian Hambleton and Erg¨ un Yal¸ cın, Group actions on spheres with rank one isotropy,

• Trans. Am. Math. Soc. 368 (2016), no. 8, 5951-5977.

[5] Sune Precht Reeh and Erg¨ un Yal¸ cın, Representation rings for fusion systems and dimension functions, Math. Z. 288 (2018), no. 1-2, 509-530.

Sune Precht Reeh Slide 15/15

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Encore

In Yal¸ cın’s talk we heard the p-fusion of Qd(p) mentioned many times. Let us quickly confirm the claim that Qd(p) cannot act on a finite homotopy sphere with rank 1 isotropy. Qd(p) = (Z/p)2 ⋊ SL2(p), S = (Z/p)2 ⋊ 1 1

0 1

• Every nontrivial element of S has order p, and they fit together to form

the center Z along with p2 + p other cyclic subgroups of order p. The cyclic subgroups combine to form p + 1 elementary abelian subgroups of rank 2, and each of these subgroups contain the center along with p other cyclic subgroups.

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Encore 2

Inside S all the nontrivial element of (Z/p)2 are conjugate, so the fusion system of Qd(p) has the additional property that the center Z is conjugate to the p other subgroups of (Z/p)2.

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Encore 3

S (Z/p)2 Z 1 (Cp)2, p copies Cp, p2 + p copies

p p each

Qd(p)

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Encore 4

Now suppose we had a nontrivial dimension function f of some homotopy sphere action with isotropy concentrated in rank 1, this would mean that f(S) = f((Z/p)2) = f((Cp)2) = 0. The quotient S/Z is isomorphic to Z/p × Z/p, so the Borel-Smith condition (ii) then implies that f(Z) = 0. Z is conjugate to the other subgroups of (Z/p)2, so by F-stability of f we have f(Cp) = 0 as well for those Cp lying in (Z/p)2. Borel-Smith condition (ii) applied to (Z/p)2 then implies f(1) = 0, and by monotonicity all of f is trivially 0.

Sune Precht Reeh Slide 19/15