Martin Wedel Jacobsen and Jesper Michael Møl ler. UAB Topology Seminar, December 17, 2010

Transcription

1 of matrices of fusion categories Martin Wedel Jacobsen and Jesper Michael Møl ler University of Copenhagen UAB Topology Seminar, December 17, 2010

2 of matrices of fusion categories Outline of talk G is a finite group and p is a prime number S G the poset of nonidentity p-subgroups of G T G (H, K ) = N G(H, K ) the transporter category of G L G (H, K ) = Op C G (H)\N G (H, K ) the linking category of G F G (H, K ) = C G(H)\N G (H, K ) the fusion category of G O G (H, K ) = N G(H, K )/K the orbit category of G F G (H, K ) = C G (H)\N G (H, K )/K the exterior quotient of the fusion category of G What are the of these finite categories?

3 of matrices of fusion categories Outline of talk 1 of matrices 2 of fusion categories 3 Categories of nonidentity subgroups Categories of centric subgroups Typical result χ(f G ) = [K ] µ(k ) (K ) F G

4 of matrices of fusion categories Definition (The ) The Euler characteristic of ζ = ( ζ(a, b) ) is the sum χ(ζ) = µ(a, b) a,b of the entries in the inverse µ = ζ 1 = k=0 ( 1)k (ζ E) k. Example ζ = ( g ) µ = ( g 1) χ(ζ) = g 1 ) ζ = ( µ = χ(ζ) = 1

5 of matrices of fusion categories Definition (The incidence matrix of a poset S) ζ(s) = ( ζ(a, b) ) a,b, ζ(a, b) = { 1 a b 0 otherwise Definition (The Euler characteristic of a finite poset S) χ(s) = χ(ζ(s)) Example (A poset with a terminal element) e a c d b ζ(s) = ( ) χ(s) = χ(ζ(s)) = 1

6 of matrices of fusion categories Definition (Simplices in a poset) A k-simplex, k 0, (from a to b) is a totally ordered subset of k + 1 points (with a as smallest and b as greatest element). Example ((ζ E) k counts k-simplices) 0-simplices 1-simplices 2-simplices ( ( ( ) ) ) = E = ζ(s) E = (ζ(s) E) 2 e c a b d

7 of matrices of fusion categories Lemma (Counting simplices in poset S) (ζ E) k (a, b) = #{k-simplices from a to b} (k 0) a,b (ζ E)k (a, b) = #{k-simplices in S} (k 0) Topological Euler characteristic of the realization S χ( S ) = ( 1) k #{k-simplices in S} = k=0 ( 1) k k=0 = a,b = a,b (ζ E) k (a, b) a,b S ( 1) k (ζ E) k (a, b) k=0 ζ 1 (a, b) = a,b x 1 = (1 + (x 1)) 1 = P k=0 ( 1) k (x 1) k µ(a, b) = ζ(s)

8 of matrices of fusion categories Assumptions G is a finite group and p is a prime number Definition (The posets S G and SG ) S G is the poset of all p-subgroups of G S G is the poset of nonidentity p-subgroups of G Proposition (Euler characteristic of the poset SG I) χ(sg ) = K Ob(SG ) µ(k ) = [K ] 1 µ(k ) G : N G(K ) Proof 1 = χ(s G ) = µ(h, K ) = µ(1, K ) + H,K 1 K = µ(1, 1) + 1 K µ(1, K ) + χ(s G ) 1 H K G µ(h, K )

9 of matrices of fusion categories Lemma (The Möbius function on S G [1, Lemme 4.1]) Let H and K be p-subgroups of G. Then µ(h, K ) = 0 unless H K with elementary abelian factor group where µ(h, K ) = ( 1) n p (n 2), K : H = p n In particular, µ(k ) = µ(1, K ) = 0 unless K is elementary abelian where µ(k ) = ( 1) n p (n 2), p n = K Example (Alternating groups at p = 2) n χ(sa n )

10 of matrices of fusion categories χ(s G ) = H Proposition ( II) ( 1 χ(s NG (H)/H )) Recursive calculations? Proof χ(sg ) = µ(h, K ) = µ(h, K ) = H,K H K H = µ(k /H) = H K [H,N G (H)] H = ( 1 χ(s NG (H)/H )) H K N G (H)/H K [H,N G (H)] µ( K ) µ(h, K )

11 of matrices of fusion categories What is known about χ(s G )? Theorem (Product formula) n 1 χ(sq n i=1 G ) = 1 χ(s i G i ) Proposition If G has a nonidentity normal p-subgroup then χ(sg ) = 1 Theorem (Brown 1975, Quillen 1978) 1 χ(sg ) is divisible by G p Example χ(ssl n (F ) = 1 + q) ( 1)n q (n 2) when q is a power of p i=1

12 of matrices of fusion categories What is unknown about χ(s G )! ( of Chevalley groups) Is 1 χ(s G n(q) ) = ( 1)n q #{positive roots} for G = A, B, C, D, E? ( of alternating groups) Describe the sequence n χ(s A n ) (Quillen conjecture 1978) O p (G) > 1 S G

13 of matrices of fusion categories The Euler characteristic of a matrix ζ = ( ζ(a, b) ) How do we define χ(ζ) when ζ is not invertible? Definition (Weightings and coweightings) A weighting for ζ is a column vector (k ) such that 1 ( ) ζ(a, b) (k b ) =. 1 A coweighting for ζ is a row vector (k ) such that (k a ) ( ζ(a, b) ) = ( 1 1 )

14 of matrices of fusion categories A matrix may have none or many (co)weightings If ( µ(a, b) ) is an inverse to ( ζ(a, b) ) then ( k a ) = ( µ(a, b) ) 1. = ( µ(a, b) ) (column sums) 1 b S ( ) ( ) ( ) ( kb = 1 1 µ(a, b) = µ(a, b) ) (row sums) a S are the unique weighting and coweighting for ζ χ(ζ) = 1 for ζ = ( ) (and ζ is not invertible) ζ = ( ) has no Euler characteristic

15 of matrices of fusion categories If ζ admits both a weighting k and a coweighting k then the sum of the values of the weighting k b = ( k a ζ(a, b)) ) k b = ( k a ζ(a, b)k b) = b b a a a b k a Definition (The Euler characteristic of a matrix (Leinster 2008)) χ(ζ) = b k b = a k a If ζ is invertible then χ(ζ) = a k a = a,b µ(a, b) as before.

16 of matrices of fusion categories Definition (The incidence matrix of a finite category C) ζ(c) = ( ζ(a, b) ) a,b ζ(a, b) = C(a, b) Definition (Euler characteristic of a category (Leinster 2008)) χ(c) = χ(ζ(c)) Proposition (Invarians under equivalence (Leinster 2008)) If there is an adjunction C D then χ(c) = χ(d) If C has an initial or terminal element then χ(c) = 1 If C and D are equivalent then χ(c) = χ(d)

17 of matrices of fusion categories Example (Skeletal subcategory of F A 6, p = 2) 6 4 e b 5 6 c a µ(c) = 1 d 2 1 1/2 1/2 1/2 1/2 0 1/ / /6 0 1/ /2 1/ / ζ(c) = χ(c) = µ(i, j) = 1/3

18 of matrices of fusion categories Theorem (A weighting, a coweighting, and the Euler characteristic of F G ) kk F = µ(k ) G : C G (K ) kf H = µ(h, K ) G : C K G (K ) = 1 ( 1 χ(s G )) CNG(H)(x)/H x C G (H) χ(fg ) = µ(k ) G : C K Ob(FG G (K ) = µ(k ) F ) [K ] π 0 (FG ) G (K ) In all cases k F K 0 K is elementary abelian If the Sylow p-subgroup P is normal in G then k H F 0 H = C P (g) for some g G

19 of matrices of fusion categories What is known about χ(f G )? Theorem (Product formula) Proposition n 1 χ(fq n i=1 G ) = 1 χ(f i G i ) If G has a nonidentity central p-subgroup then χ(f G ) = 1 G p χ(f G ) Z χ(f G ) = {ϕ F G (P) C P(ϕ) > 1} F G (P) when P, the Sylow p-subgroup, is abelian. i=1 χ(f G ) = χ(f a G ) and χ(f G ) = χ( F G )

20 of matrices of fusion categories Example (Alternating groups A n at p = 2) n χ(sa n ) χ(fa n ) 4 1 1/ / / / / /63 n χ(sa n ) χ(fa n ) / / / / / /405 Example (The smallest group with χ(fg ) > 1) There is a group G = C2 4 H, where H = (C 3 C 3 ) C 2, of order G = 288 with Euler characteristic χ(fg ) = 10/9 at p = 2.

21 of matrices of fusion categories What is unknown about χ(f G ) Are FG and F G a homotopy equivalent? Are FG and F G homotopy equivalent? Is χ(fg ) always positive when p divides the order of G? Can χ(fg ) get arbitrarily large? What is χ(fa n )? Does it converge for n? What is χ(f SL n (F q) )? Is there a G p -fold covering map E BFG where E is (homotopy) finite and χ(e) = G p χ(fg )?

22 of matrices of fusion categories Categories of nonidentity subgroups Categories of centric subgroups Theorem ( of nonidentity p-subgroup categories) C T G L G F G χ(c) µ(h) T [H] G (H) µ(h) L [H] G (H) µ(h) F [H] G (H) O G χ(t G ) + p 1 p [C] 1 O G (C)

23 of matrices of fusion categories Combinatorial identities Categories of nonidentity subgroups Categories of centric subgroups Corollary For any finite group G and any prime p, ( 1 χ(s NG (H)/H ) + µ(h)) = 0 H ( 1 χ(s CNG(H)(x)/H ) + µ(h)) = 0 H H x C G (H) ( H χ(s NG (H)/H ) H + µ(h)) = p 1 p C where H runs over the set of nonidentity p-subgroups of G and C over the set of nonidentity cyclic p-subgroups of G. C

24 of matrices of fusion categories Categories of nonidentity subgroups Categories of centric subgroups Definition The p-subgroup H G is p-centric if p C G (H): C H (H) Example ( of centric subgroup categories for alternating groups at p = 2) n A n χ(l c A n ) χ(sa c n ) χ(l c A n ) 1/12 1/24 13/ /4480 χ(fa c n ) 1/3 1/3 13/63 19/105 χ( F A c n ) 1/3 1/3 13/63 19/105 Conjecture χ(f c G ) = χ( F c G )

25 References of matrices of fusion categories Categories of nonidentity subgroups Categories of centric subgroups Charles Kratzer and Jacques Thévenaz, Type d homotopie des treillis et treillis des sous-groupes d un groupe fini, Comment. Math. Helv. 60 (1985), no. 1, MR (87b:06017)