The number of simple modules associated to Sol(q)

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1 The number of simple modules associated to Sol(q) Jason Semeraro Heilbronn Institute University of Leicester University of Bristol 8th August 2017

2 This is joint work with Justin Lynd.

3 Outline Blocks and fusion systems

4 Outline Blocks and fusion systems Alperin s weight conjecture

5 Outline Blocks and fusion systems Alperin s weight conjecture An example

6 Outline Blocks and fusion systems Alperin s weight conjecture An example p-local blocks and the gluing problem

7 Outline Blocks and fusion systems Alperin s weight conjecture An example p-local blocks and the gluing problem Malle Robinson Conjecture

8 Outline Blocks and fusion systems Alperin s weight conjecture An example p-local blocks and the gluing problem Malle Robinson Conjecture Sol(q)

9 Blocks G, a finite group

10 Blocks G, a finite group p, a prime

11 Blocks G, a finite group p, a prime k = k with char(k) = p

12 Blocks G, a finite group p, a prime k = k with char(k) = p b, a block (indecomposable summand) of kg

13 Blocks G, a finite group p, a prime k = k with char(k) = p b, a block (indecomposable summand) of kg M a simple kg-module = Mb = b, some b M b

14 Blocks G, a finite group p, a prime k = k with char(k) = p b, a block (indecomposable summand) of kg M a simple kg-module = Mb = b, some b M b How many such M in each b?

15 Fusion systems S, a Sylow p-subgroup of G

16 Fusion systems S, a Sylow p-subgroup of G F = F S (G) fusion system (category), objects: {P S}, morphisms: G-conjugation maps

17 Fusion systems S, a Sylow p-subgroup of G F = F S (G) fusion system (category), objects: {P S}, morphisms: G-conjugation maps P S is p-centric in G Z(P) Syl p (C G (P))

18 Fusion systems S, a Sylow p-subgroup of G F = F S (G) fusion system (category), objects: {P S}, morphisms: G-conjugation maps P S is p-centric in G Z(P) Syl p (C G (P)) P S is p-radical in G O p (N G (P)/PC G (P)) = 1

19 Fusion systems S, a Sylow p-subgroup of G F = F S (G) fusion system (category), objects: {P S}, morphisms: G-conjugation maps P S is p-centric in G Z(P) Syl p (C G (P)) P S is p-radical in G O p (N G (P)/PC G (P)) = 1 Both properties can be recovered from the category F

20 Fusion systems S, a Sylow p-subgroup of G F = F S (G) fusion system (category), objects: {P S}, morphisms: G-conjugation maps P S is p-centric in G Z(P) Syl p (C G (P)) P S is p-radical in G O p (N G (P)/PC G (P)) = 1 Both properties can be recovered from the category F F c := {P S P is p-centric}

21 Fusion systems S, a Sylow p-subgroup of G F = F S (G) fusion system (category), objects: {P S}, morphisms: G-conjugation maps P S is p-centric in G Z(P) Syl p (C G (P)) P S is p-radical in G O p (N G (P)/PC G (P)) = 1 Both properties can be recovered from the category F F c := {P S P is p-centric} F cr := {P S P is p-centric and p-radical}

22 Alperin s Weight Conjecture (for principal blocks) Suppose k b (principal block)

23 Alperin s Weight Conjecture (for principal blocks) Suppose k b (principal block) A, an algebra = z(a): number of projective simple A-modules

24 Alperin s Weight Conjecture (for principal blocks) Suppose k b (principal block) A, an algebra = z(a): number of projective simple A-modules AWC for b (Kessar): Conjecture (Alperin) The number of simples in b is l(f) := z(k(n G (P)/PC G (P)). P F cr /F Sum runs over a set of F-isomorphism class representatives

25 An example Suppose G = Sym(np), 3 p n < 2p, b principal block

26 An example Suppose G = Sym(np), 3 p n < 2p, b principal block p(n) = # partitions of n

27 An example Lemma Suppose G = Sym(np), 3 p n < 2p, b principal block p(n) = # partitions of n The number of simple b-modules is equal to p(n 1 )p(n 2 ) p(n p 1 ) (n) where the sum runs over (n) = (n 1,, n p 1 ) with n j 0 and p 1 j=1 n j = n. e.g., n = p = 3, (3) = (n 1, n 2 ) {(0, 3), (1, 2), (2, 1), (3, 0)} so we get = 10 simple b-modules

28 An example S Syl p (G) has an abelian subgroup A of index p and order p n

29 An example S Syl p (G) has an abelian subgroup A of index p and order p n F cr = {S, A} W S, W abelian of order p n p+2

30 An example S Syl p (G) has an abelian subgroup A of index p and order p n F cr = {S, A} W S, W abelian of order p n p+2 P = S = N G (P)/PC G (P) = C p 1 Sym(n p) (C p 1 ) 2

31 An example S Syl p (G) has an abelian subgroup A of index p and order p n F cr = {S, A} W S, W abelian of order p n p+2 P = S = N G (P)/PC G (P) = C p 1 Sym(n p) (C p 1 ) 2 P = W = N G (P)/PC G (P) = C p 1 Sym(n p) GL 2 (p)

32 An example S Syl p (G) has an abelian subgroup A of index p and order p n F cr = {S, A} W S, W abelian of order p n p+2 P = S = N G (P)/PC G (P) = C p 1 Sym(n p) (C p 1 ) 2 P = W = N G (P)/PC G (P) = C p 1 Sym(n p) GL 2 (p) P = A = N G (P)/PC G (P) = C p 1 Sym(n)

33 An example Lemma S Syl p (G) has an abelian subgroup A of index p and order p n F cr = {S, A} W S, W abelian of order p n p+2 P = S = N G (P)/PC G (P) = C p 1 Sym(n p) (C p 1 ) 2 P = W = N G (P)/PC G (P) = C p 1 Sym(n p) GL 2 (p) P = A = N G (P)/PC G (P) = C p 1 Sym(n) Let c p (n) denote the number of p-cores of size n If A = k(c p 1 Sym(n)) then z(a) = (n) c p (n 1 )... c p (n p 1 ) where the sum runs over (n) = (n 1,, n p 1 ) with n j 0 and p 1 j=1 n j = n.

34 An example e.g., n = p = 3:

35 An example e.g., n = p = 3: P = S = z(k(c 2 C 2 )) = 4

36 An example e.g., n = p = 3: P = S = z(k(c 2 C 2 )) = 4 P = W = z(kgl 2 (3)) = 2

37 An example e.g., n = p = 3: P = S = z(k(c 2 C 2 )) = 4 P = W = z(kgl 2 (3)) = 2 P = A = z(k(c 2 Sym(3)) = 4

38 An example e.g., n = p = 3: P = S = z(k(c 2 C 2 )) = 4 P = W = z(kgl 2 (3)) = 2 P = A = z(k(c 2 Sym(3)) = 4 So l(f 3 (Sym(9)) = = 10, as predicted by AWC!

39 An example e.g., n = p = 3: P = S = z(k(c 2 C 2 )) = 4 P = W = z(kgl 2 (3)) = 2 P = A = z(k(c 2 Sym(3)) = 4 So l(f 3 (Sym(9)) = = 10, as predicted by AWC! In general, combine the lemmas with the identity p(n) c p (n) = p(n p) p to see that AWC holds

40 p-local blocks What about a non-principal block, b?

41 p-local blocks What about a non-principal block, b? b = (F, α), F a saturated fusion system on the defect group of b and α lim [S(F c )] A2 F (Linckelmann) where,

42 p-local blocks What about a non-principal block, b? b = (F, α), F a saturated fusion system on the defect group of b and α lim [S(F c )] A2 F (Linckelmann) where, [S(F c )]: poset of F-iso classes of chains σ = (R 0 < R 1 < R n ) of elements in F c

43 p-local blocks What about a non-principal block, b? b = (F, α), F a saturated fusion system on the defect group of b and α lim [S(F c )] A2 F (Linckelmann) where, [S(F c )]: poset of F-iso classes of chains σ = (R 0 < R 1 < R n ) of elements in F c A 2 F : covariant functor A 2 F : σ H 2 (Aut F (σ), k )

44 p-local blocks What about a non-principal block, b? b = (F, α), F a saturated fusion system on the defect group of b and α lim [S(F c )] A2 F (Linckelmann) where, [S(F c )]: poset of F-iso classes of chains σ = (R 0 < R 1 < R n ) of elements in F c A 2 F : covariant functor A 2 F : σ H 2 (Aut F (σ), k ) Aut F (σ) Aut F (R n ): subgroup preserving R i e.g. b is principal = α = 0

45 p-local blocks Call (F, α) a p-local block

46 p-local blocks Call (F, α) a p-local block If (F, α) arises from a block b, l(f, α) := z(k αq Out F (Q)), Q F cr /F counts the number of b-weights

47 p-local blocks Call (F, α) a p-local block If (F, α) arises from a block b, l(f, α) := Q F cr /F counts the number of b-weights z(k αq Out F (Q)), k αq Out F (Q): algebra obtained from the group algebra k Out F (Q) by twisting with α Q

48 p-local blocks Call (F, α) a p-local block If (F, α) arises from a block b, l(f, α) := Q F cr /F counts the number of b-weights z(k αq Out F (Q)), k αq Out F (Q): algebra obtained from the group algebra k Out F (Q) by twisting with α Q What if (F, α) does not come from a block?

49 Gluing problem Always a natural map H 2 (F c, k ) lim [S(F c )] A2 F

50 Gluing problem Always a natural map H 2 (F c, k ) lim [S(F c )] A2 F Linckelmann asks the following in the case of blocks: Conjecture (Gluing problem) Is this map always surjective?

51 Gluing problem Always a natural map H 2 (F c, k ) lim [S(F c )] A2 F Linckelmann asks the following in the case of blocks: Conjecture (Gluing problem) Is this map always surjective? Libman: true for all fusion systems of S n, A n and GL n (q), q p

52 Gluing problem Always a natural map H 2 (F c, k ) lim [S(F c )] A2 F Linckelmann asks the following in the case of blocks: Conjecture (Gluing problem) Is this map always surjective? Libman: true for all fusion systems of S n, A n and GL n (q), q p Is there an exotic counterexample to the gluing problem?

53 Malle Robinson conjecture S a p-group s(s) = rank of largest elementary abelian subquotient (section)

54 Malle Robinson conjecture S a p-group s(s) = rank of largest elementary abelian subquotient (section) Conjecture (Malle Robinson) # simple b-modules p s(s)

55 Malle Robinson conjecture S a p-group s(s) = rank of largest elementary abelian subquotient (section) Conjecture (Malle Robinson) # simple b-modules p s(s) AWC suggests that the following generalized version should also hold Conjecture Let (F, α) be a p-local block with F is a saturated fusion system on S. Then l(f, α) p s(s). Work of Malle Robinson suggests that the conjecture holds for many non-exotic pairs (F, α)

56 Solomon Benson fusion systems Goal: gather more evidence for these conjectures

57 Solomon Benson fusion systems Goal: gather more evidence for these conjectures Many examples of exotic fusion systems, e.g. when p > 2 and S has an abelian maximal subgroup (Craven Oliver S)

58 Solomon Benson fusion systems Goal: gather more evidence for these conjectures Many examples of exotic fusion systems, e.g. when p > 2 and S has an abelian maximal subgroup (Craven Oliver S) when S is a Sylow 7-subgroup of the Monster (Parker S)

59 Solomon Benson fusion systems Goal: gather more evidence for these conjectures Many examples of exotic fusion systems, e.g. when p > 2 and S has an abelian maximal subgroup (Craven Oliver S) when S is a Sylow 7-subgroup of the Monster (Parker S) when p = 2: Solomon Benson fusion systems Sol(q), q odd (Aschbacher Chermak), (Levi Oliver), (Dwyer)

60 Solomon Benson fusion systems Goal: gather more evidence for these conjectures Many examples of exotic fusion systems, e.g. when p > 2 and S has an abelian maximal subgroup (Craven Oliver S) when S is a Sylow 7-subgroup of the Monster (Parker S) when p = 2: Solomon Benson fusion systems Sol(q), q odd (Aschbacher Chermak), (Levi Oliver), (Dwyer) Conjecturally the Solomon Benson systems are the only exotic simple 2-fusion systems

61 Solomon Benson fusion systems Goal: gather more evidence for these conjectures Many examples of exotic fusion systems, e.g. when p > 2 and S has an abelian maximal subgroup (Craven Oliver S) when S is a Sylow 7-subgroup of the Monster (Parker S) when p = 2: Solomon Benson fusion systems Sol(q), q odd (Aschbacher Chermak), (Levi Oliver), (Dwyer) Conjecturally the Solomon Benson systems are the only exotic simple 2-fusion systems Morally Sol(q) is a group of Lie-type in char. q, a close relative of Spin 7 (q)

62 Solomon Benson fusion systems Goal: gather more evidence for these conjectures Many examples of exotic fusion systems, e.g. when p > 2 and S has an abelian maximal subgroup (Craven Oliver S) when S is a Sylow 7-subgroup of the Monster (Parker S) when p = 2: Solomon Benson fusion systems Sol(q), q odd (Aschbacher Chermak), (Levi Oliver), (Dwyer) Conjecturally the Solomon Benson systems are the only exotic simple 2-fusion systems Morally Sol(q) is a group of Lie-type in char. q, a close relative of Spin 7 (q) Do the aforementioned representation-theoretic invariants reflect this?

63 ...yes We may assume q = 5 2k.

64 ...yes We may assume q = 5 2k. When k = 0, the elements of Sol(q) cr are essentially given by Chermak Oliver Shpectorov: P P Out F (P) S R 2 7 A 7 R 2 6 S 6 RR 2 9 S 3 Q 2 8 (C 3 ) 3 (C 2 S 3 ) QR 2 9 S 3 QR 2 9 (C 3 C 3 ) 1 C 2 C S (U) 2 9 S 3 E 2 4 GL 4 (2) C S (Ω 1 (T )) 2 7 GL 3 (2)

65 and when k > 0? Aschbacher Chermak essentially list elements of Sol(q) cr but: their classification contains some errors;

66 and when k > 0? Aschbacher Chermak essentially list elements of Sol(q) cr but: their classification contains some errors; Sol(q)-automorphism groups are not given explicitly

67 and when k > 0? Aschbacher Chermak essentially list elements of Sol(q) cr but: their classification contains some errors; Sol(q)-automorphism groups are not given explicitly the list of groups is quite different to the k = 0 case

68 and when k > 0? Aschbacher Chermak essentially list elements of Sol(q) cr but: their classification contains some errors; Sol(q)-automorphism groups are not given explicitly the list of groups is quite different to the k = 0 case Putting this all together we prove: Theorem (Lynd-S) Let F = Sol(q) be a Benson-Solomon system. Then Moreover, the natural map is an isomorphism in all cases. lim [S(F cr )] A2 F = 0. H 2 (F cr, k ) lim [S(F cr )] A2 F

69 Calculating l(sol(q), 0) We can calculate the number of weights for the unique p-local block (F, 0) supported by F: Theorem (Lynd-S) For all q > 2, we have l(sol(q), 0) = 12.

70 Calculating l(sol(q), 0) We can calculate the number of weights for the unique p-local block (F, 0) supported by F: Theorem (Lynd-S) For all q > 2, we have l(sol(q), 0) = 12. Note that: Sol( ) is behaving like a connected reductive integral group scheme G! (assuming AWC, l(f p (G(q)), 0) is independent of q)

71 Calculating l(sol(q), 0) We can calculate the number of weights for the unique p-local block (F, 0) supported by F: Theorem (Lynd-S) For all q > 2, we have l(sol(q), 0) = 12. Note that: Sol( ) is behaving like a connected reductive integral group scheme G! (assuming AWC, l(f p (G(q)), 0) is independent of q) The generalized Malle Robinson conjecture holds for Sol(q)

72 Calculating l(sol(q), 0) We can calculate the number of weights for the unique p-local block (F, 0) supported by F: Theorem (Lynd-S) For all q > 2, we have l(sol(q), 0) = 12. Note that: Sol( ) is behaving like a connected reductive integral group scheme G! (assuming AWC, l(f p (G(q)), 0) is independent of q) The generalized Malle Robinson conjecture holds for Sol(q) l(f 2 (Spin 7 (q)), 0) = 12. Is there a way to construct modules for Sol(q) from modules in the principal 2-block of Spin 7 (q)?

Fusion systems and self equivalences of p-completed classifying spaces of finite groups of Lie type

Fusion systems and self equivalences of p-completed classifying spaces of finite groups of Lie type Carles Broto Universitat Autònoma de Barcelona March 2012 Carles Broto (Universitat Autònoma de Barcelona)