23 involutions, Fischer group Fi 23 and the moonshine VOA Hiroshi Yamauchi Tokyo Woman’s Christian University Jointly with Thomas Creutzig and Ching Hung Lam Representation Theory XVI IUC Dubrovnik, Croatia June 27, 2019 H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 1 / 22
3-transposition groups Definition 1 p G , I q : 3-transposition group G : group, I : a set of involutions s.t. ð ñ I G “ I , G “ x I y , and | ab | ď 3 for @ a , b P I . Theorem 1 (Fischer’71, Cuypers-Hall’95) The list of almost simple 3-transposition groups is as follows. 1 G “ Sym n , I “ tp i j q | 1 ď i ă j ď n u 2 G “ O ˘ 2 n p 2 q , I “ transvections 3 G “ Sp 2 n p 2 q , I “ transvections 4 G “ O ˘ n p 3 q , I “ reflections 5 G “ SU n p 2 q , I “ transvections 6 G “ PΩ ` 8 p 2 or 3 q :Sym 3 , Fi 22 , 23 , 24 , I : unique H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 2 / 22
Basic sets Definition 2 Let p G , I q be a 3-transposition group and P a Sylow 2-subgroup. The intersection P X I is called a basic set of G and the size | P X I | is called the width of G . a , b P P X I ñ | ab | “ 2 ñ x P X I y : elementary abelian 2-group P X I : a maximal set of mutually commutative elements in I @ g P G , p P X I q g “ P g X I g “ P g X I ñ | P X I | : invariant of G Fi 22 Fi 23 Fi 24 Width 22 23 24 2 10 2 11 2 12 x P X I y 2 10 . M 22 2 11 . M 23 2 12 . M 24 Normalizer H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 3 / 22
CFSG Theorem 2 The complete list of finite simple group is given by: (0) Z { p Z ( p : prime) (1) Alt n ě 5 (2) Groups of Lie type (i.e. matrix groups over finite fields) (3) 26 sporadics (M 11 , 12 , 22 , 23 , 24 , Co 1 , 2 , 3 , Fi 22 , 23 , 24 , B , M , . . . ) Special symmetries in 24 dimension (cf. [FLM’88]): F 24 Z 24 c “ 24 VOA 2 V 6 “ V ` Λ ‘ V T ` G Λ ù ù Λ M 24 2 . Co 1 M Note that C M p 2B q “ 2 1 ` 24 . Co 1 and Aut p V ` Λ q “ 2 24 . Co 1 – p Λ { 2Λ q . Co 1 ` H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 4 / 22
Sunshine construction? Historically, Fi 24 B M ù ù 3-trans.gp. t 3 , 4 u -trans.gp. 6-trans.gp. Aim V 6 M — — — — 2 . B 3 . Fi 24 L p 1 { 2 , 0 q b VB W 3 p 4 { 5 q b VF — — — — ñ S 3 ˆ Fi 23 U 3A b Com V 6 p U 3A q —— —— ? S 3 ˆ 2 11 X r 23 s Note that C M p 2A q “ 2 . B and N M p 3A q “ 3 . Fi 24 (cf. S 3 “ 3:2) H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 5 / 22
Ising vectors Let V be a VOA of OZ-type, i.e. V “ ‘ n ě 0 V n , V 0 “ R 1 , V 1 “ 0. Then V 2 forms a commutative algebra with invariant bilinear form ab : “ a p 1 q b , p a | b q 1 “ a p 3 q b for a , b P V 2 . This algebra is called the Griess algebra of V . Lemma 3 (Miyamoto’96) e P V 2 : c “ c e Virasoro vector ð ñ ee “ 2 e and c e “ 2 p e | e q Definition 3 e P V 2 : Ising vector of σ -type ð ñ e : c “ 1 { 2 Virasoro vector s.t. x e y – L p 1 { 2 , 0 q (simple subVOA) There is no x e y -submodule isomorphic to L p 1 { 2 , 1 { 16 q in V H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 6 / 22
Miyamoto involutions Theorem 4 (Miyamoto’96) e : “ p´ 1 q 16o p e q P Aut p V q . If e P V is an Ising vector then τ e : “ p´ 1 q 2o p e q P Aut p V q . If τ e “ id V then σ Theorem 5 (Conway’85, Miyamoto’96, H¨ ohn’10) Ising vectors of V 6 2A-involutions of M 1:1 Ð Ý Ý Ñ P P e τ e By this correspondence, we can analyze 2A-involutions of M by considering corresponding Ising vectors of V 6 . We can generalize the above correspondence for the other groups. (cf. Lam-Y.’16 arXiv:1604.04989 ) H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 7 / 22
Dihedral subalgebras Theorem 6 (Sakuma’07, cf. Nina’s talk) e , f P V R : Ising vectors ñ | τ τ τ τ f | ď 6 (6-transposition property) τ e τ More precisely, there are 9 possible types of x e , f y : 3C ˝ | | | ˝´ ´ ´ ´ ´˝´ ´ ´ ´ ´˝´ ´ ´ ´ ´˝´ ´ ´ ´ ´ ˝´ ´ ´ ´ ´˝´ ´ ´ ´ ´˝´ ´ ´ ´ ´˝ 1A 2A 3A 4A 5A 6A 4B 2B x e , f y 1A 2A 3A 4A 5A 6A 4B 2B 3C 2 10 p e | f q 2 8 2 5 2 3 2 2 2 2 13 6 5 0 dim x e , f y 2 1 3 4 5 6 8 5 2 3 We will call U nX “ x e , f y the dihedral subalgebra of type nX . H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 8 / 22
Construction of 3-transposition groups I V : VOA of OZ-type D V : the set of Ising vectors of σ -type in V Set σ p D V q “ t σ e | e P D V u and G V “ x D V y . If e , f P D V then x e , f y is either 1A, 2A or 2B-type. Theorem 7 (Miyamoto’96, Matsuo’05, Cuipo-Lam-Y.’18) (1) p G V , σ p D V qq : 3-transposition group of symplectic type ( G V ď Sp 2 n p 2 q ) (2) If V “ x D V y then p V , G V q are classified ( V is a subVOA of V ` 2 R ) ? x e , f y : 2A-type ñ r τ e , τ f s “ 1 on V and | σ e σ f | “ 3 on V x τ e ,τ f y x e , f y : 2B-type ñ r τ e , τ f s “ 1 on V and | σ e σ f | ď 2 on V x τ e ,τ f y In order to realize Fischer groups, we need to use τ -involutions. H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 9 / 22
Construction of 3-transposition groups II V : VOA of OZ-type E V : the set of Ising vectors of V (including both σ and τ -types) Fix a , b P E V s.t. x a , b y : 3A-type ñ x τ a , τ b y – S 3 Set I a , b : “ t x P E V | p a | x q “ p b | y q “ 2 ´ 5 u (i.e. x a , x y – x b , x y – 2A-alg.) Theorem 8 (Lam-Y.’16) (1) x P I a , b ñ r τ x , τ a s “ r τ x , τ b s “ 1 (2) x , y P I a , b ñ x x , y y : 1A, 2A or 3A-types (3) x , y P I a , b ñ | τ x τ y | ď 3 in Aut p V q (4) G V : “ x τ x | x P I a , b y ñ G V : 3-transposition group in C Aut p V q x τ a , τ b y If we apply the theorem above to V 6 then we obtain G V 6 “ Fi 23 “ C M p S 3 q . Note that S 3 ˆ Fi 23 ă M whereas 3 . Fi 24 ă M but Fi 24 ă M . H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 10 / 22
Inductive structures p G , I q : 3-transposition group Pick a P I and set G r 1 s : “ G a “ x x P I | ax “ xa y{x a y ( G a ď C G p a q{x a y ) Similarly we define G r 2 s : “ G a , b “ p G a q b , G r 3 s : “ G a , b , c “ p G a , b q c , . . . Example G “ Fi 24 ñ G r 1 s “ Fi 23 , G r 2 s “ Fi 22 , G r 3 s “ Fi 21 “ PSU 6 p 2 q In the above process, a , b , c , . . . : mutually commutative elements in I Maximal collection : a basic set of p G , I q H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 11 / 22
Inductive subalgebras G V “ x τ x | x P I a , b y : 3-transposition group Let n be the width of G V . We define the inductive subalgebra X r n s : “ x a , b , x 1 , . . . , x n y Set X r 0 s : “ x a , b y and suppose we have defined X r i s : “ x a , b , x 1 , . . . , x i y . Then we choose x i ` 1 P I a , b s.t. x i ` 1 R X r i s and p x i ` 1 | x j q “ 2 ´ 5 , 1 ď j ď i and define X r i ` 1 s : “ x X r i s , x i ` 1 y as long as possible. Then t τ x i | 1 ď i ď n u gives a basic set of G V if G V is connected. H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 12 / 22
Inductive structures in VOA side Set D r 0 s : “ t τ x | x P I a , b u and D r i s : “ t τ y P D r i ´ 1 s | τ y τ x i “ τ x i τ y u . X r 0 s X r 1 s X r 2 s X r n s Ă Ă Ă ¨ ¨ ¨ Ă ù ù ù ù Com V X r 0 s Com V X r 1 s Com V X r 2 s Com V X r n s Ą Ą Ą ¨ ¨ ¨ Ą ö ö ö ö G r 0 s G r 1 s G r 2 s G r n s ¨ ¨ ¨ ↠ ↠ ↠ ↠ x D r 0 s y x D r 1 s y x D r 2 s y x D r n s y Ą Ą Ą ¨ ¨ ¨ Ą Theorem 9 (Lam-Y.’16) The Griess algebra of X r n s is uniquely determined and L p c 3 , 0 q b L p c 4 , 0 q b ¨ ¨ ¨ b L p c n ` 4 , 0 q Ă X r n s (full) where c i “ 1 ´ 6 {p i ` 2 qp i ` 3 q (unitary series). H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 13 / 22
Observation The central charge of X r n s is c 3 ` c 4 ` ¨ ¨ ¨ ` c n ` 4 “ p n ` 2 qp 5 n ` 29 q . 5 p n ` 7 q If n “ 23 : the central charge of X r 23 s is 24 On the other hand, S 3 ˆ Fi 23 ă M ñ X r 23 s Ă V 6 [Conway-Miyamoto] L p c 3 , 0 q b ¨ ¨ ¨ b L p c 27 , 0 q Ă X r 23 s Ă V 6 (cf. T.Creutzig @Dubrovnik 2017) finite extension V 6 M — — ——————— 2 . B 3 . Fi 24 — — ð ñ ? S 3 ˆ Fi 23 —— S 3 ˆ 2 11 X r 23 s H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 14 / 22
Construction of 2 ` 23 involutions F “ r ϵ 1 , . . . , ϵ n s Z “ Z ϵ 1 ‘ ¨ ¨ ¨ ‘ Z ϵ n , p ϵ i | ϵ j q “ 2 δ i , j (2-frame) ( n ” 8 p 16 q ) F ˚ “ 1 2 F : dual lattice of F π : F ˚ ↠ F ˚ { F – F n 2 Ą C : code ù L A p C q : “ π ´ 1 p C q : lattice L A p C q : even, unimodular iff C : doubly-even, self-dual F ˚ Q 1 “ 1 2 p ϵ 1 ` ¨ ¨ ¨ ` ϵ n q ������ p 11 ¨ ¨ ¨ 1 q P F n (all-one vector) 2 Ñ p x | 1 q P F 2 ù L B p C q : “ ν ´ 1 ν C : L A p C q Q x ÞÝ C p 0 q : sublattice ` 1 L Λ p C q : “ L B p C q \ p L B p C q ` ϵ 1 2 1 q looooomooooon “ ν ´ 1 C p 1 q n “ 24, C “ G ă F 24 ñ L A p G q “ N p A 24 1 q , L Λ p G q “ Λ 24 “ Λ 2 H. Yamauchi (TWCU) Fi 23 and the moonshine VOA Dubrovnik June 27, 2019 15 / 22
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