A construction for the outer automorphism of S 6 Padraig Ó Catháin joint work with Neil Gillespie and Cheryl Praeger University of Queensland 5 August 2013 Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Automorphisms of finite groups G a finite group. ρ : G → Aut ( V ) a representation. Let σ ∈ Aut ( G ) . When can σ be realised as an element of Aut ( V ) ? I.e. when does there exist S ∈ Aut ( V ) such that, for all g ∈ G : ρ ( x σ ) = S − 1 ρ ( x ) S (Obvious) sufficient condition: σ is inner. (Obvious) necessary condition: χ ( x σ ) = χ ( x ) for all x ∈ G . Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Motivation: M 12 M 12 has two conjugacy classes of subgroups M 11 . Coset action on either class is 5-transitive on 12 points. The outer automorphism of M 12 swaps classes of M 11 s, and hence the two actions. These actions cannot be (linearly) equivalent: the traces are different, e.g. � 1 2 3 4 5 6 7 8 9 10 11 12 � P = 5 1 9 10 6 7 4 8 2 3 11 12 1 2 3 4 5 6 7 8 9 10 11 12 � � P σ = 10 1 6 2 7 5 3 9 8 4 12 11 Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Motivation: M 12 Theorem (M. Hall, 1962) Let H be a Hadamard matrix of order 12 . Modulo the centre of Aut ( H ) , the automorphisms of H are the Mathieu group M 12 of order 12 · 11 · 10 · 9 · 8 = 95040 . Here M 12 is represented as a quintuply transitive group of monomial permutations on the columns or rows of H. The row and column representations of H are isomorphic, but the correspondence given by P = HQH − 1 determines an outer automorphism of M 12 of order 2 . (Conway and Elkies also note that ( P , Q ) → ( Q , P ) exhibits the outer automorphism of M 12 .) Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Let H be Hadamard of order 12. Aut ( H ) = { ( P , Q ) | PHQ ⊤ = H , where P , Q are ± 1 -monomial } Consider the representations α : 2 . M 12 → P and β : 2 . M 12 → Q : α ( x ) = H β ( x ) H − 1 . By Hall: β ( x ) = α ( x σ ) for some outer automorphism σ of M 12 . So for every x ∈ 2 . M 12 , we have α ( x σ ) = H − 1 α ( x ) H , a linear representation of the outer automorphism of 2 . M 12 . Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Construction for the outer automorphism of M 12 While the outer automorphism of M 12 (acting on 12 points) cannot be realised linearly, it almost can. Given an element of M 12 , lift to 2 . M 12 1 conjugate by H 2 project back onto M 12 . 3 Sample automorphism: 1 2 3 4 5 6 7 8 9 10 11 12 � � π − 1 ( P ) = − 5 − 1 − 9 10 6 7 − 4 8 2 -3 11 -12 1 2 3 4 5 6 7 8 9 10 11 12 � � π − 1 ( P σ ) = 10 1 6 2 7 5 3 9 8 4 12 11 π − 1 ( P ) H = Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
The outer automorphism of S 6 Possibly the most famous outer automorphism of any group: ( 1 , 2 ) ( 1 , 2 )( 3 , 6 )( 4 , 5 ) → ( 1 , 2 , 3 ) ( 1 , 5 , 6 )( 2 , 3 , 4 ) → ( 1 , 2 , 3 , 4 , 5 , 6 ) ( 1 , 5 )( 2 , 3 , 6 ) → etc. Described by Sylvester: duads, synthemes, totals, etc. 1 Exhibited by the actions of Aut ( K 6 ) on points and a set of 2 one-factors Exhibited by action of S 6 on certain 2-colourings of K 5 3 Via the isomorphism S 6 ∼ = Sp 4 ( 2 ) 4 As a subgroup of M 12 , etc. 5 Can we find a matrix H which intertwines (lifts of) these representations of S 6 ? Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
A fragment from Moorhouse Moorhouse: classification of the complex Hadamard matrices with doubly transitive automorphism groups. A sporadic example: 1 1 1 1 1 1 1 1 ω ω ω ω 1 1 ω ω ω ω H = 1 1 ω ω ω ω 1 1 ω ω ω ω 1 1 ω ω ω ω which has automorphism group 3 . A 6 . Note: HH † = 6 I 6 . Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Aside: what we were actually doing... Attempting to construct codes with interesting automorphism groups from the rows of Hadamard matrices. The rows of the matrix (considered over F 4 ) generate ‘the hexacode’, used by Conway and Sloane to describe the Golay code. Considered over F 3 , we obtained a new nonlinear uniformly packed code, and a new frequency permutation array, etc. These ideas should extend to larger families of Hadamard matrices... Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
H intertwines two representations of 3 . A 6 . Restricting to A 6 we get inequivalent representations of A 6 . So every automorphism of H is of the form ρ ( x ) H ρ ( x σ ) . Now, ρ extends to a representation of 3 . S 6 . If we take y to be an odd permutation, then ρ ( y ) H ρ ( y σ ) = H † . But H is not Hermitian... In any case, H could never intertwine representations of 3 . S 6 : involutions of shape 1 4 2 have non-zero trace over C , but are mapped to elements of shape 2 3 . (Cf. elements of order 3.) Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
The split-quaternions, Ξ Discovered by Hamilton Cockle in 1849 4-dimensional over R : generated by { 1 , i , τ, i τ } , where i 2 = − 1 τ 2 = 1 τ i τ = − i . Isomorphic to M 2 ( R ) : � 0 � 1 − 1 0 � � i �→ τ �→ 1 0 0 − 1 Not a division algebra: ( i + τ i ) 2 = 0 (Cockle’s "impossibles"). Contains C ∼ = � 1 , i � as a subalgebra. Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Take S = τ I 6 , then, for any odd permutation y ∈ S 6 , we have: S ρ ( y ) H ρ ( y σ ) S = SH † S = H So odd permutations can be made into Ξ -linear automorphisms of H . Note that τ and τ i have trace 0: so all odd permutations lift to elements of equal trace. Even permutations are defined over C , odd permutations involve τ . We have 3 . S 6 acting on H with the row and column actions differing by an outer automorphism of S 6 . To compute the image of σ ∈ S 6 under an outer automorphism: lift σ to 3 . S 6 , if odd, multiply by S , conjugate by H , restrict to S 6 . Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Conclusion A construction of the outer automorphism for S 6 which does not depend on finding two non-conjugate subgroups of index 6. Could be realised as a C -linear 12-dimensional representation. Question: Which automorphisms of which representations of which groups can be realised linearly in this sense? Question: Can we find other interesting combinatorial objects hiding in the rings of intertwiners of representations? Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Thank you! Padraig Ó Catháin Outer automorphism of S 6 5 August 2013
Recommend
More recommend
