Representation Stability and FI Modules

Transcription

1 Representation Stability and FI Modules Exposition on work by Church Ellenberg Farb Jenny Wilson University of Chicago Topology Student Workshop 2012 Jenny Wilson (University of Chicago) Representation Stability TSW / 17

2 Background: Classical Homological Stability {Y n } n is a sequence of groups or topological spaces, with inclusions φ n : Y n Y n+1 Definition (Homological Stability) The sequence {Y n } is homologically stable (over a ring R) if for each k 1, the map is an isomorphism for n >> k. (φ n ) : H k (Y n ; R) H k (Y n+1 ; R) Jenny Wilson (University of Chicago) Representation Stability TSW / 17

3 Examples of Homologically Stable Sequences (Nakaoka 1961) Symmetric groups S n (Arnold 1968, Cohen 1972) Braid groups B n (McDuff 1975, Segal 1979) Configuration spaces of open manifolds (Charney 1979, Maazen 1979, van der Kallen 1980) Linear groups, arithmetic groups (such as SL n (Z)) (Harer 1985) Mapping class groups of surfaces with boundary (Hatcher 1995) Automorphisms of free groups Aut(F n ) (Hatcher Vogtmann 2004) Outer automorphisms of free groups Out(F n ) Jenny Wilson (University of Chicago) Representation Stability TSW / 17

4 Generalizing Homological Stability What can we say when H k (Y n ; R) does not stabilize? More generally, let {V n } n be a sequence of R-modules. Suppose V n has an action by a group G n. Our objective: A notion of stability for {V n } n that takes into account the G n symmetries. In this talk: G n = S n, the symmetric group R = Q, and V n are finite dimesional vector spaces Jenny Wilson (University of Chicago) Representation Stability TSW / 17

5 An Example: The Permutation Representation Example (The Permutation Representation) Consider the permutation representation V n = Q n = e 1,..., e n. For each n, V n decomposes into two irreducibles: { } { Q n } = a(e 1 + e e n ) a 1 e a n e n ai = 0 Jenny Wilson (University of Chicago) Representation Stability TSW / 17

6 An Example: The Permutation Representation Some properties of the permutation representation The decomposition into irreducibles looks the same for every n. { } { Q n } = a(e 1 + e e n ) a 1 e a n e n ai = 0 The dimension of V n grows polynomially in n dim(v n ) = n The characters χ n of V n have a nice global description χ n (σ) = #1 cycles of σ for all σ S n, for all n. Jenny Wilson (University of Chicago) Representation Stability TSW / 17

7 Some Representation Theory Some facts about S n representations over Q Every S n representation decomposes uniquely as a sum of irreducibles. Irreducibles are indexed by partitions λ of n, depicted by Young diagrams. Jenny Wilson (University of Chicago) Representation Stability TSW / 17

8 V = V S 8 rep 8 Obstacle ( ) How canv we compare = V irreducibles for different values S 9 repof n? 9 ( ) Solution V = V S 10 rep 10 Two irreducibles are the same if only the top rows of their Young diagrams differ. Example (The Permutation Representation V n = Q n ) Q 1 = V Q 2 = V Q 3 = V Q 4 = V Q 5 = V V V V V Jenny Wilson (University of Chicago) Representation Stability TSW / 17

9 The Definition of an FI module Definition (Church Ellenberg Farb) (The Category FI) Denote by FI the category of Finite sets with Injective maps Jenny Wilson (University of Chicago) Representation Stability TSW / 17

10 The Definition of an FI module Definition (Church Ellenberg Farb) (FI Modules) A (rational) FI module is a functor V : FI Q-Vect Jenny Wilson (University of Chicago) Representation Stability TSW / 17

11 Finite Generation of FI Modules Definition (Generation) If V is an FI module, and S n V n, then the FI module generated by S is the smallest sub FI module containing the elements of S. Definition (Finite Generation) An FI module is finitely generated if it has a finite generating set. Example (The Permutation Representation V n = Q n ) The permutation representation V n = Q n = e 1,..., e n is generated by e 1 V 1. Jenny Wilson (University of Chicago) Representation Stability TSW / 17

12 Consequences of Finite Generation Theorem (Church Ellenberg Farb) Let V be a finitely-generated FI module. Then for n >> 1 The decomposition into irreducible S n representations stabilizes. dim(v n ) is polynomial in n The characters χ n of V n are given by a (unique) polynomial in the variables X r X r (σ) = #r cycles of σ for all σ S n, for all n. Any sub FI module of V also has these properties. We call the sequence {V n } n uniformly representation stable. Jenny Wilson (University of Chicago) Representation Stability TSW / 17

13 Some Representation Stable Cohomology Sequences (Church Farb) {H k (P n ; Q)} n The pure braid group (Jimenez-Rolland) {H k (PMod(Σ n g,r ); Q)} n The pure MCG of an n-puncture surface Σ n g,r (Church) {H k (PConf n (M); Q)} n Ordered configuration space of a manifold M (Putman) {H k (PMod n (M); F)} n The pure MCG of an n-puncture manifold (Putman) Eg, {H k (SL n (Z, l); F)} n Certain congruence subgroups (Wilson) {H k (PΣ n ; Q)} n automorphism group PΣ n The pure symmetric of the free group Jenny Wilson (University of Chicago) Representation Stability TSW / 17

14 Open Question Problem Compute the characters, and the stable decompositions into irreducibles, in the above examples. Jenny Wilson (University of Chicago) Representation Stability TSW / 17

15 My Research My current project To develop a unified FI module theory for the three families of classical Weyl groups. Jenny Wilson (University of Chicago) Representation Stability TSW / 17

16 References Further Reading T Church, B Farb. Representation theory and homological stability, preprint, T Church, J Ellenberg, B Farb. FI modules: A new approach to stability for S n representations, preprint, Jenny Wilson (University of Chicago) Representation Stability TSW / 17

17 The End Acknowledgements Many thanks to Benson Farb, Tom Church, Jordan Ellenberg, and Rita Jimenez-Rolland for their help and guidance. Jenny Wilson (University of Chicago) Representation Stability TSW / 17