Representation rings for fusion systems and dimension ...mat.uab.cat/~reeh/slides/2018-06-22 Isle of Skye.pdf · 6/22/2018 · Representation rings for fusion systems and dimension

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Representation rings for fusion systemsand dimension functions

Sune Precht Reehjoint with Ergun Yalcın

Notes from the flip charts are in green.

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

Slide 1/15


Dimension functions

Related to Yalcın’s talk: Study finite group actions on finiteCW-complexes ' Sn – with restrictions on isotropy.

One way to construct a G-action on an actual sphere: take theunit-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 sphereX ' Sn is given by XP ∼

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

prime p.

Sune Precht Reeh

Slide 2/15


Borel-Smith functions

Q: Which functions on the conjugacy classes of subgroups in G arise asdimension functions for real representations/homotopy sphere actions?

Necessary: Borel-Smith conditions for a function f on the conjugacyclasses of subgroups.

(i) If K CH,H/K ∼= Z/p, p odd, then 2 | f(K)− f(H).

(ii) If K CH,H/K ∼= Z/p× Z/p, thenf(K)− f(H) =

∑K<L<H

(f(L)− f(H)

).

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


Borel-Smith functions 2As an example, consider the group C5. Let us denote the irreduciblecomplex characters of C5 by χ1, χ2, . . . , χ5. The irreducible realrepresentations of C5 then have characters χ1, χ2 + χ5, χ3 + χ4. Thedimension functions for these are

Dim 1 C5

χ1 1 1χ2 + χ5 2 0χ3 + χ4 2 0

The only Borel-Smith condition that applies to C5 is (i), which statesthat 2 | f(1)− f(C5). This relation is easily confirmed for the irreduciblereal representations. In fact every Borel-Smith function is a linearcombination of the dimension functions above and hence is the dimensionfunction of some virtual real representation.Sune Precht Reeh

Slide 4/15


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 nonnegativeand monotone, then there exists a real representation V such thatDim(V ) = f .

Sune Precht Reeh

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Realizing Borel-Smith functions 2Theorem (R.-Yalcın)

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

Sune Precht Reeh

Slide 6/15


Sketch of proof

fB.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-Yalcın]

Finite htpy. sphere with G-action

Sune Precht Reeh

Slide 7/15


Fusion systems

Given a finite group G and a prime p, let S ∈ Sylp(G). The fusion systemFS(G) induced by G on S is a category with objects P ≤ S andmorphisms

HomFS(G)(P,Q) := {cg : P → Q | g ∈ G, g−1Pg ≤ Q}.

There is a notion of abstract (saturated) fusion systems F on S, withexotic 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 representationsV ∈ RR(S).

Sune Precht Reeh

Slide 8/15


Fusion systems 2Consider C5 o C4 where C4 acts on C5 as the full automorphism group.The fusion system F = FC5(C5 o C4) at the prime 5 has C5 endowedwith the additional conjugation from C4.

The trivial representation χ1 is F-stable, but χ2 + χ5 and χ3 + χ4 arenot invariant under the C4-action. The indecomposable F-stablerepresentations are χ1 and χ2 + χ3 + χ4 + χ5. Their dimension functions:

Dim 1 C5

χ1 1 1χ2 + · · ·+ χ5 4 0

The Borel-Smith functions for C5 are no longer all going to be linearcombinations 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


Borel-Smith functions for FCb(F) consists of Borel-Smith functions f ∈ Cb(S) such that f isconstant on isomorphism classes in F .

RR(−)→ Cb(−) is a natural transformation of biset functors on p-groupsand is pointwise surjective. A general result then gives us thatRR(F)(p) → Cb(F)(p) is surjective.

If we want a result without p-localization, we need to add an extracondition to the Borel-Smith conditions:

(iv) [Bauer] If K CH, H/K ∼= Z/p, α ∈ Aut(H/K) is induced byAutF (H), then (order of α) | f(K)− f(H).

Sune Precht Reeh

Slide 10/15


Borel-Smith functions for F 2For our example fusion system F on C5 induced by C5 o C4, we have anautomorphism of C5/1 of order 4 induced by the C4-action. Thecondition (iv) then states 4 | f(1)− f(C5).

This is enough to ensure that every Borel-Smith function satisfying theadditional condition 4 | f(1)− f(C5) will be a linear combination of thedimension functions for F-stable representations, and hence realized byan F-stable virtual real representation.

Sune Precht Reeh

Slide 11/15


Realizing Borel-Smith functions (at p)

Theorem (R.-Yalcın)

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.


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

Open problem: Does N = 1 work for f satisfying (iv)?

Sune Precht Reeh

Slide 12/15


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) χ

σV is a rational

valued character.

There is an m > 0 such that m · χ′ is the character of a rationalS-representation W , with Dim(W ) = m · |L : Q| · f .

That f is F-stable implies that Dim(W ) is F-stable which in turnimplies that W is F-stable because Dim(−) is injective on rationalrepresentations. �

Sune Precht Reeh

Slide 13/15


Epilogue fB.S.-function

f2 f3 f5 f7 · · ·B.S.-function for Fp

V2 V3 V5 V7 · · ·Fp-stable real representation

XFinite htpy. sphere with G-action


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

Sune Precht Reeh

Slide 14/15


The End

[1] Stefan Bauer, A linearity theorem for group actions on spheres with applications tohomotopy 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 Ergun Yalcın, Group actions on spheres with rank one isotropy,Trans. Am. Math. Soc. 368 (2016), no. 8, 5951-5977.

[5] Sune Precht Reeh and Ergun Yalcın, Representation rings for fusion systems anddimension functions, Math. Z. 288 (2018), no. 1-2, 509-530.

Sune Precht Reeh

Slide 15/15

http://www.ams.org/mathscinet-getitem?mr=889050

http://www.ams.org/mathscinet-getitem?mr=889050


Encore

In Yalcın’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 finitehomotopy sphere with rank 1 isotropy.

Qd(p) = (Z/p)2 o SL2(p), S = (Z/p)2 o 〈(

1 10 1

)〉

Every nontrivial element of S has order p, and they fit together to formthe center Z along with p2 + p other cyclic subgroups of order p. Thecyclic subgroups combine to form p+ 1 elementary abelian subgroups ofrank 2, and each of these subgroups contain the center along with p othercyclic subgroups.

Sune Precht Reeh

Slide 16/15


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

Sune Precht Reeh

Slide 17/15


Encore 3

S

(Z/p)2

Z

1

(Cp)2, p copies

Cp, p2 + p copies

p p each

Qd(p)

Sune Precht Reeh

Slide 18/15


Encore 4Now suppose we had a nontrivial dimension function f of somehomotopy sphere action with isotropy concentrated in rank 1, this wouldmean 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-Smithcondition (ii) then implies that f(Z) = 0.

Z is conjugate to the other subgroups of (Z/p)2, so by F-stability of fwe 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, andby monotonicity all of f is trivially 0.

Sune Precht Reeh

Slide 19/15

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Representation rings for fusion systems and dimension ...mat.uab.cat/~reeh/slides/2018-06-22 Isle of Skye.pdf · 6/22/2018 · Representation rings for fusion systems and dimension