Equivariant Euler characteristic

Transcription

1 1 1 Københavns Universitet Louvain-la-Neuve, September 09, 2015 Euler characteristics of centralizer subcategories /home/moller/projects/euler/orbit/presentation/louvainlaneuve.tex

2 Local Euler characteristics at p of an A-group Let be a finite A-group and p a prime (such that p ). S p (Sp+ ) is the poset of (nontrivial) p-subgroup of (A) is the poset of A-p-subgroups H C S p N (H), H C S p C S p+ N (H)/H (A), is a p-local A-subgroup (A), poset of nontrivial A-p-subgroups of N (H)/H χ(c p+ S (A)) is a p-local Euler characteristic of N (H)/H C (A) is the subgroup of fixed by A p the number of p-singular elements in p the p-part of the group order A H, H A = H N (H)/H A The p-local Euler characteristics are globally constrained

3 lobal constraints on local Euler characteristics Theorem: 1 There is an inclusion-exclusion principle for the number C (A) p of p-singular A-centralized elements of : H C p S (A) χ(c p+ S (A)) C H (A) = C (A) p N (H)/H 2 For any A-normalized p-subgroup K of K H C p S (A) 3 C (A) p χ(c p+ S (A)) χ(c p+ S (A)) = 1 N (H)/H

4 lobal constraints for trivial action When A acts trivially on the global constraints are 1 χ(s p+ N (H)/H ) H = p (inclusion-exclusion) 2 H S p χ(s p+ N (H)/H ) = 1 K H S p 3 p χ(s p+ ) (Brown s theorem) Corollary: Brown s theorem of 1975 ( p χ(s p+ )) and Frobenius theorem of 1907 ( p p ) are equivalent. Corollary: Σ(q) p = Σ(q) 2 p for an untwisted finite group of Lie type Σ(q) in defining characteristic p, Σ = A, B,...,.

5 Reality check: = L 3 (F 2 ) = A 2 (F 2 ), p = = 168 = 8 3 7, 2 = 8, 2 = 64 = conjugacy classes of 2-subgroups 4 = 2 Π(A2) conjugacy classes of 2-radical (H = O 2 N (H)) 2-subgroups (Borel Tits) χ(s 2+ N (H)/H ) : N (H) H = 64 [H] [S p ] χ(s 2+ N (H)/H ) : N (H) = 1 [H] [S p ] 3 8 = 2 χ(s 2+ ) = 8 H : N (H) χ(s 2+ N (H)/H) χ(s 2+ N (H)/H ) : N (H) H χ(s 2+ N (H)/H ) : N (H)

6 Motivation for studying Euler characteristics of centralizer subcategories Why are Euler characteristics of centralizer subcategories relevant? Because of their connection to equivariant Euler characteristics! Many familiar posets are equivariant posets and their equivariant Euler characteristics carry interesting information. Here are some examples:

7 s of finite A-categories finite poset (finite category) C A group acting on C χ(c), χ(c) = χ(c) 1 χ r (C, A) = 1 χ(c C (X)) Q, r = 1, 2,... A χ r (C, A) = 1 A X C r (A) X C r (A) χ(c C (X)) Q, r = 1, 2,... C r (A) the set of commuting r-tuples X = (x 1,..., x r ) of A-elements C C (X) the subcategory of C fixed (centralized) by all autofunctors of the r-tuple X = (x 1,..., x r ) χ r (C, A) = χ r 1 (C C (x), C A (x)) (recursion) [x] [A]

8 Equivariant posets Many familiar posets are A-posets Poset A Brown poset S p+ Partition poset Π n Σ n Boolean poset B n Σ n Subspace poset L n (F q ) L n (F q ) and have equivariant (reduced) Euler characteristics χ r (C, A) for r = 1, 2,.... χ 1 (C, A) = χ 1 (C, A) + 1 χ 2 (C, A) = χ 2 (C, A) + k(a) χ 3 (C, A) = χ 3 (C, A) + [x] [A] k(c A (x))

9 s of Brown posets Nontrivial p-subgroups of ordered by inclusion finite group S p+ p prime number dividing the order of z p () number of irreducible C-reps of p-defect 0 conjugation action 0 χ(s p+ ) = 0 P Sp+ : P (Quillen Conjecture) 1 χ 1 (S p+ 2 χ 2 (S p+ 3 χ 3 (S p+, ) = 0 (Webb Theorem), ) = z p() (Alperin Weight Conjecture), ) =?

10 Class equation interpretation of Webb s Theorem (χ 1 ) [x] [] [x] [] [x] [] : C (x) = [x] [] [x] = χ(c p+ S (x)) : C (x) = χ(c p+ S (x)) : C (x) = 0 p x = χ(c p+ S (x)) = 0 For the simple group = L 3 (F 2 ) of order 168: x : C (x) χ(c S 2+ (x)) χ(c S 3+ χ(c S 7+ (x)) (x))

11 Alperin s Weight Conjecture (χ 2 ) The (Knörr Robinson formulation of the) Alperin Weight Conjecture is true for A S p +abelian χ 2 (S p+, ) = z p() with cyclic p-sylow subgroup solvable χ(c p+ S (A))ϕ 2 (A) = z p () with a nontrivial normal p-subgroup ( = z p () = 0) L n (F q ) where p is the characteristic of F q The Mathieu groups M 11, M 12, M 22, M 23, M 24 and the Janko groups J 1, J 2, J 3 at all primes p dividing the group order (computer verifications)

12 Alperin s Weight Conjecture for L 3 (F 2 ), p = 3 = L 3 (F 2 ), p = 3, z p () = 3, = 168 contains six classes of abelian subgroups of order prime to 3 A : N (A) ϕ 2 (A) χ(c p+ S (A)) χϕ 2 : N (A) The sum of the numbers of the bottom row is A S p +abelian Why? χ(c p+ S (A))ϕ 2 (A) = 504 = = z p ()

13 χ 3 (S 2+, ) for alternating and symmetric groups (χ 3) n χ 3 (S 2+ A n, A n ) χ 3 (S 2+ Σ n, Σ n ) Let p be the characteristic of F q. Then χ 2 (S p+ L n(f q), L n(f q )) = χ 2 (L n(f q ), L n (F q )) as the Brown poset S p+ L n(f q) and the building L n(f q ) are L n (F q )-homotopy equivalent z p (L n (F q )) = q 1 where p is the characteristic of F q χ r (L n(f q ), ) are the equivariant Euler characteristics of an n-dimensional F q -representation of

14 s of partition posets Partitions of {1,..., n} ordered by refinement Π n Σ n obvious action Remove smallest and largest element: Π n = Π n {{{1},..., {n}}, {{1,..., n}}} 0 χ(π n) = ( 1) n 1 (n 1)! (Stanley) 1 χ 1 (Π n, Σ n ) = 0 (me!) 2 χ 2 (Π n, Σ n ) = µ(n) µ(n/2) 3 χ 3 (Π n, Σ n ) = 4, 4, 5, 6, 16, 8, 2,... (not in OEIS) χ 3 (Π n, Σ n ) = 0, 4, 26, 33, 108, 162, 358,... (not in OEIS) µ(n) µ(n/2) = 2, 1, 1, 1, 2, 1, 0, 0,... (A092673) /home/vqd218/homedir/math/magma/partition.prg

15 s of Boolean lattices Subsets of {1,..., n} ordered by inclusion B n Σ n obvious action Remove smallest and largest element: B n = B n {, {1,..., n}} 0 χ(b n) = ( 1) n (Stanley) 1 χ 1 (B n, Σ n ) = 0 2 χ 2 (B n, Σ n ) = p even(n) p odd (n) 3 χ 3 (B n, Σ n ) = 3, 1, 0, 10, 8, 12, 1, 28,... (not in OEIS) χ 3 (B n, Σ n ) = 1, 7, 21, 49, 100, 182, 361,... (not in OEIS) p even(n): The number of partitions of n with en even number of distinct blocks. p even(n) p odd(n) = 1, 0, 0, 1, 0, 1, 0, 0, 0,... (A010815) /home/vqd218/homedir/math/magma/subspaces.prg

16 s of subspace posets Subspaces of F n q ordered by inclusion (q is a prime power) L n (F q ) L n (F q ) obvious action Remove smallest and largest element: L n(f q ) = L n (F q ) {0, F n q} 0 χ(l n(f q )) = ( 1) n q (n 2) (Stanley) 1 χ 1 (L n(f q ), L n (F q )) = 0 2 χ 2 (L n(f q ), L n (F q )) = (q 1) (Thévenaz) 3 χ 3 (L n(f 2 ), L n (F 2 )) = 4, 12, 32, 80, 192,... (multiple entries in OEIS) /home/vqd218/homedir/math/magma/subspaces.prg

17 Summary There are global constraints on the p-local Euler characteristics χ(c p+ S (A)) defined for A-p-subgroups N (H)/H H χ 1 (C, A) = χ( C /A) is suprisingly often 1 χ 2 (C, A) may carry crucial information χ 3 (C, A) is bewildering χ r (C, A) for r > 3 is terra incognita