The mod-2 cohomology. of the finite Coxeter groups. James A. Swenson University of Wisconsin Platteville

Transcription

1 p. 1/1 The mod-2 cohomology of the finite Coxeter groups James A. Swenson swensonj/ University of Wisconsin Platteville

2 p. 2/1 Thank you! Thanks for spending this half-hour here I hope you have fun!

3 p. 3/1 Coxeter Groups Theorem 1 ([Cox35]). The finite Coxeter groups are precisely the finite groups of reflections and rotations in Euclidean space. Every Coxeter group is a direct sum of irreducible Coxeter groups, and these have been classified.

4 p. 3/1 Coxeter Groups Theorem 1 ([Cox35]). The finite Coxeter groups are precisely the finite groups of reflections and rotations in Euclidean space. Every Coxeter group is a direct sum of irreducible Coxeter groups, and these have been classified. In this talk, every Coxeter group is finite.

5 p. 3/1 Coxeter Groups Theorem 1 ([Cox35]). The finite Coxeter groups are precisely the finite groups of reflections and rotations in Euclidean space. Every Coxeter group is a direct sum of irreducible Coxeter groups, and these have been classified. In this talk, every Coxeter group is finite. Some examples: Symmetric groups, dihedral groups, groups of symmetries of regular polyhedra...

6 p. 4/1 Group Cohomology Definition. Let G be a discrete group and BG an Eilenberg-Mac Lane space of type K(G, 1). The mod-2 cohomology of G, denoted by H (G), is the cohomology ring H (BG; F 2 ).

7 p. 5/1 First Examples Example. H (Z 2 ) = H (BZ 2, F 2 ) = H (RP, F 2 ) = F 2 [X].

8 First Examples Example. H (Z 2 ) = H (BZ 2, F 2 ) = H (RP, F 2 ) = F 2 [X]. Example. Let Dih(8) denote the dihedral group with eight elements. We will show (Theorem 16) that H (Dih(8)) = F 2 [X 1,Y 1,Z 2 ]/(X 1 Y 1 ). Here, generators are indexed by degree; e.g., Z 2 H 2 (Dih(8)). p. 5/1

9 p. 6/1 What is H (G) like? Theorem 2 (cf. [Bro82], I.4.2). H (G) = Ext F 2 G(F 2, F 2 ). In particular, H q (G) = { F 2 (q = 0), 0 (q < 0).

10 p. 6/1 What is H (G) like? Theorem 2 (cf. [Bro82], I.4.2). H (G) = Ext F 2 G(F 2, F 2 ). In particular, H q (G) = { F 2 (q = 0), 0 (q < 0). Corollary 3. H is a contravariant functor from the category of groups to the category of graded F 2 -algebras.

11 p. 7/1 In principle, H (G) can be computed Theorem 4 ([Eve61]). H (G) is finitely generated as an F 2 -algebra.

12 p. 7/1 In principle, H (G) can be computed Theorem 4 ([Eve61]). H (G) is finitely generated as an F 2 -algebra. Proposition 5 (e.g. [Bro82], III.10.1). If n > 0, then G H n (G) = 0.

13 p. 7/1 In principle, H (G) can be computed Theorem 4 ([Eve61]). H (G) is finitely generated as an F 2 -algebra. Proposition 5 (e.g. [Bro82], III.10.1). If n > 0, then G H n (G) = 0. Corollary 6. If G is odd, then H (G) = F 2 (concentrated in degree zero).

14 p. 8/1 Computing H (G) Theorem 7 (Künneth). H (G G ) = H (G) H (G ).

15 p. 8/1 Computing H (G) Theorem 7 (Künneth). H (G G ) = H (G) H (G ). Example (e.g. [Hat02], 3.17). H ((Z 2 ) d ) = F 2 [X 1,...,X d ]. (These generators are all of degree one).

16 p. 8/1 Computing H (G) Theorem 7 (Künneth). H (G G ) = H (G) H (G ). Example (e.g. [Hat02], 3.17). H ((Z 2 ) d ) = F 2 [X 1,...,X d ]. (These generators are all of degree one). Corollary 8. If G is odd, then H (G G ) = H (G).

17 p. 9/1 A computational method Proposition 9 (e.g. [Nak61], 1.3). If f Inn(G), then H (f) is the identity.

18 p. 9/1 A computational method Proposition 9 (e.g. [Nak61], 1.3). If f Inn(G), then H (f) is the identity. Definition ([Qui71]). The Quillen category of G is denoted C(G). Its objects are the elementary abelian 2-subgroups of G. The morphisms are generated by inclusions and inner automorphisms of G.

19 p. 9/1 A computational method Proposition 9 (e.g. [Nak61], 1.3). If f Inn(G), then H (f) is the identity. Definition ([Qui71]). The Quillen map q G is the map which makes the following diagram commute for every A: H (G) q G lim A C(G) H (A) H (A)

20 p. 10/1 The Quillen map Theorem 10 ([Qui71]). The kernel and cokernel of the Quillen map are nilpotent.

21 p. 10/1 The Quillen map Theorem 10 ([Qui71]). The kernel and cokernel of the Quillen map are nilpotent. Theorem 11 ([FL]). If G is a finite Coxeter group, then the Quillen map is an isomorphism: H (G) = lim A C(G) H (A).

22 p. 11/1 Naming the cohomology classes Corollary 12. If G is a finite Coxeter group, then a class f H (G) is equivalent to a coherent family i.e., one for which {f A H (A) : A C(G)} (f A ) A B = f B and (c g ) (f A ) = f (g A), where g G and c g : g A A sends x g 1 xg.

23 p. 12/1 f A is stable (c g ) (f A ) = f (g A) (f A ) A B = f B. Proposition 13 ([CE99], 9.3). For each A, f A is stable i.e., the following diagram commutes: H (A) c g H ( g A) H (A g A)

24 p. 12/1 f A is stable (c g ) (f A ) = f (g A) (f A ) A B = f B. Corollary 15. For each A, f A H (A) N G(A)/A.

25 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy.

26 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy. Compute the action of N G (A)/A on H 1 (A).

27 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy. Compute the action of N G (A)/A on H 1 (A). Find the invariant ring H (A) N G(A).

28 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy. Compute the action of N G (A)/A on H 1 (A). Find the invariant ring H (A) N G(A). See which invariants are stable.

29 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy. Compute the action of N G (A)/A on H 1 (A). Find the invariant ring H (A) N G(A). See which invariants are stable. Find a lot of coherent families of stable elements.

30 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy. Compute the action of N G (A)/A on H 1 (A). Find the invariant ring H (A) N G(A). See which invariants are stable. Find a lot of coherent families of stable elements. Prove that an arbitrary coherent family can be generated by the chosen families.

31 p. 13/1 Computations using Quillen s Theorem Find the maximal A C(G) up to conjugacy. Compute the action of N G (A)/A on H 1 (A). Find the invariant ring H (A) N G(A). See which invariants are stable. Find a lot of coherent families of stable elements. Prove that an arbitrary coherent family can be generated by the chosen families. Find the relation ideal.

32 p. 14/1 Example: Dih(2m) Theorem 16. If G = Dih(2m), then F 2 [X 1 ] m odd, H (G) = F 2 [X 1,Y 1 ] m 2 (mod 4), F 2 [X 1,Y 1,Z 2 ]/(X 1 Y 1 ) 4 m. Proof: Unless 4 m, all maximal elementary 2-subgroups are conjugate. If A is a representative, then H (G) = H (A).

33 p. 15/1 H (Dih(2m)), 4 m Proof (cont.): Let 4 m. Let {A 1,A 2 } represent the two conjugacy classes of maximal elementary abelian 2-subgroups. We have N G (A i ) = Dih(8).

34 p. 15/1 H (Dih(2m)), 4 m Proof (cont.): If f Ai is stable, then f Ai H (A i ) N G(A i )/A i = F 2 [u,v] Σ 2 = F 2 [e 1,e 2 ] where e k denotes the k th elementary symmetric function. Every element of F 2 [e 1,e 2 ] is stable. Denoting each coherent family by the pair (f A1,f A2 ), fix {X 1 = (e 1, 0),Y 1 = (0,e 1 ),Z 2 = (e 2,e 2 )}.

35 p. 16/1 Example: H (E 6 ) Consider the group E 6, whose Coxeter graph is:

36 p. 16/1 Example: H (E 6 ) Theorem 17. If G = E 6, then H (G) is isomorphic to F 2 [X 1,X 2,X 3,X 3,X 4,X 5 ] (R 2,R 3,X 3 R 1,R 4,X 5 R 1,X 1 R 5,X 3 R 5) where the generators are indexed by their degrees, and the relations have degrees [6, 8, 9, 10, 11, 11, 13] respectively.

37 p. 17/1 Future questions H (E 8 ) and its relationship to other H (G)

38 p. 17/1 Future questions H (E 8 ) and its relationship to other H (G) Compare other detecting families of subgroups

39 p. 17/1 Future questions H (E 8 ) and its relationship to other H (G) Compare other detecting families of subgroups Stable splittings of BG

40 p. 17/1 Future questions H (E 8 ) and its relationship to other H (G) Compare other detecting families of subgroups Stable splittings of BG Invariant theory and commutative algebra

41 p. 17/1 Future questions H (E 8 ) and its relationship to other H (G) Compare other detecting families of subgroups Stable splittings of BG Invariant theory and commutative algebra Group cohomology and commutative algebra

42 p. 17/1 Future questions H (E 8 ) and its relationship to other H (G) Compare other detecting families of subgroups Stable splittings of BG Invariant theory and commutative algebra Group cohomology and commutative algebra...

43 p. 18/1 Works cited [Bro82] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, [CE99] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, With an appendix by David A. Buchsbaum, Reprint of the 1956 original. MR MR (2000h:18022) [Cox35] H. S. M. Coxeter, The complete enumeration of finite groups r 2 i = (r ir j ) k ij, J. London Math. Soc. 10 (1935), [Eve61] Leonard Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), MR MR (25 #1191) [FL] Mark Feshbach and Jean Lannes, Personal communication. [Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, [Nak61] Minoru Nakaoka, Homology of the infinite symmetric group, Ann. of Math. (2) 73 (1961), [Qui71] Daniel Quillen, The spectrum of an equivariant cohomology ring I, Ann. of Math. (2) 94 (1971), MR MR (45 #7743)