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

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a 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

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Dimension functions Related to Yal¸ cın’s talk: Study finite group actions on finite CW-complexes ≃ S n – 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 ) := dim R ( V P ) for P ≤ G up to conjugation in G . The dimension function for an action of G on a finite homotopy sphere X ≃ S n is given by X P ∼ p S Dim ( X )( P ) − 1 for any p -subgroup P ≤ G and a prime p . Sune Precht Reeh Slide 2/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a 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 ) = � f ( L ) − f ( H ) . K<L<H (iii) If K ⊳ H ⊳ L ≤ N G ( K ) , H/K ∼ = Z / 2, and if L/K ∼ = Z / 4, then 2 | f ( K ) − f ( H ), or if L/K ∼ = Q 8 , then 4 | f ( K ) − f ( H ). Let C b ( G ) denote the set of Borel-Smith functions for G . Sune Precht Reeh Slide 3/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Borel-Smith functions 2 As an example, consider the group C 5 . Let us denote the irreducible complex characters of C 5 by χ 1 , χ 2 , . . . , χ 5 . The irreducible real representations of C 5 then have characters χ 1 , χ 2 + χ 5 , χ 3 + χ 4 . The dimension functions for these are Dim 1 C 5 χ 1 1 1 χ 2 + χ 5 2 0 χ 3 + χ 4 2 0 The only Borel-Smith condition that applies to C 5 is (i) , which states that 2 | f (1) − f ( C 5 ). 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

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Realizing Borel-Smith functions Theorem Let G be a finite group. [tom Dieck] When G is nilpotent, R R ( G ) → C b ( G ) is surjective. [Dotzel-Hamrick] If G is a p -group, and if f ∈ C b ( G ) is nonnegative and monotone, then there exists a real representation V such that Dim ( V ) = f . Sune Precht Reeh Slide 5/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a 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 ≃ S n such that X only has prime-power isotropy, and Dim ( X ) = N · f for some N > 0 . Sune Precht Reeh Slide 6/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Sketch of proof f B.S.-function Restriction to p -groups · · · B.S.-function at p f 2 f 3 f 5 f 7 Rest of talk Real repr. “at p ” V 2 V 3 V 5 V 7 · · · [Hambleton-Yal¸ cın] Finite htpy. sphere with G -action X Sune Precht Reeh Slide 7/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Fusion systems Given a finite group G and a prime p , let S ∈ Syl p ( G ). The fusion system F S ( G ) induced by G on S is a category with objects P ≤ S and morphisms Hom F S ( G ) ( P, Q ) := { c g : P → Q | g ∈ G, g − 1 Pg ≤ 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 χ V ( a ) = χ V ( a ′ ) whenever a ′ = ϕ ( a ) for some homomorphism ϕ ∈ F and a, a ′ ∈ S . The representation ring R R ( F ) consists of F -stable virtual representations V ∈ R R ( S ). Sune Precht Reeh Slide 8/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Fusion systems 2 Consider C 5 ⋊ C 4 where C 4 acts on C 5 as the full automorphism group. The fusion system F = F C 5 ( C 5 ⋊ C 4 ) at the prime 5 has C 5 endowed with the additional conjugation from C 4 . The trivial representation χ 1 is F -stable, but χ 2 + χ 5 and χ 3 + χ 4 are not invariant under the C 4 -action. The indecomposable F -stable representations are χ 1 and χ 2 + χ 3 + χ 4 + χ 5 . Their dimension functions: Dim 1 C 5 χ 1 1 1 χ 2 + · · · + χ 5 4 0 The Borel-Smith functions for C 5 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

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Borel-Smith functions for F C b ( F ) consists of Borel-Smith functions f ∈ C b ( S ) such that f is constant on isomorphism classes in F . R R ( − ) → C b ( − ) is a natural transformation of biset functors on p -groups and is pointwise surjective. A general result then gives us that R R ( F ) ( p ) → C b ( 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 Aut F ( H ), then (order of α ) | f ( K ) − f ( H ). Sune Precht Reeh Slide 10/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Borel-Smith functions for F 2 For our example fusion system F on C 5 induced by C 5 ⋊ C 4 , we have an automorphism of C 5 / 1 of order 4 induced by the C 4 -action. The condition (iv) then states 4 | f (1) − f ( C 5 ). This is enough to ensure that every Borel-Smith function satisfying the additional condition 4 | f (1) − f ( C 5 ) will be a linear combination of the dimension functions for F -stable representations, and hence realized by an F -stable virtual real representation. Sune Precht Reeh Slide 11/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a 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 = F S ( G ) ), then R R ( F ) → C b + (iv) ( F ) is surjective. Theorem (R.-Yal¸ cın) Let F be a saturated fusion system on a p -group S (e.g. F = F S ( 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) ? Sune Precht Reeh Slide 12/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a 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. χ V lives in a finite extension L of Q , so χ ′ = � σ ∈ Gal ( L / Q ) χ σ 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 Slide 13/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a Epilogue f B.S.-function B.S.-function for F p f 2 f 3 f 5 f 7 · · · F p -stable real representation · · · V 2 V 3 V 5 V 7 Finite htpy. sphere with G -action X 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 ≃ S n such that X only has prime-power isotropy, and Dim ( X ) = N · f for some N > 0 . Sune Precht Reeh Slide 14/15

u n i v e r s i t a t a u t o n o m a d e b a r c e l o n a 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