Constructing Majorana representations Madeleine Whybrow, University of Primorska Joint work with M. Pfeiffer
The Monster group ◮ Denoted M , the Monster group is the largest of the 26 sporadic groups in the classification of finite simple groups ◮ It was constructed by R. Griess in 1982 as Aut ( V M ) where V M is a 196 884 - dimensional, real, commutative, non-associative algebra known as the Griess or Monster algebra ◮ The Monster group contains two conjugacy classes of involutions - denoted 2A and 2B - and M = � 2 A � ◮ If t , s ∈ 2 A then ts is of order at most 6 and belongs to one of nine conjugacy classes: 1 A , 2 A , 2 B , 3 A , 3 C , 4 A , 4 B , 5 A , 6 A .
The Monster group ◮ In 1984, J. Conway showed that there exists a bijection ψ between the 2A involutions and certain idempotents in the Griess algebra called 2A-axes ◮ The 2A-axes generate the Griess algebra i.e. V M = �� ψ ( t ) : t ∈ 2 A �� ◮ If t , s ∈ 2 A then the algebra �� ψ ( t ) , ψ ( s ) �� is called a dihedral subalgebra of V M and has one of nine isomorphism types, depending on the conjugacy class of ts .
The Monster group Example Suppose that t , s ∈ 2 A such that ts ∈ 2 A as well. Then the algebra V := �� ψ ( t ) , ψ ( s ) �� is a 2 A dihedral algebra. The algebra V also contains the axis ψ ( ts ). In fact, it is of dimension 3: V = � ψ ( t ) , ψ ( s ) , ψ ( ts ) � R .
Monstrous Moonshine and VOAs ◮ In 1992, R. Borcherds famously proved Conway and Norton’s Monstrous Moonshine conjectures, which connect the Monster group to modular forms ◮ The central object in his proof is the Moonshine module, V # = � ∞ n =0 V # n . ◮ It belongs to a class of graded algebras know as vertex operator algebras, or VOA’s 2 ∼ ◮ In particular, we have Aut ( V # ) = M and V # = V M .
Majorana Theory We now let V be a real vector space equipped with a commutative algebra product · and an inner product ( , ) such that for all u , v , w ∈ V , we have: M1 ( u , v · w ) = ( u · v , w ); M2 ( u · u , v · v ) ≥ ( u · v , u · v ). Suppose that A ⊆ V such that V = �� A �� and such that for all a ∈ A we have: M3 ( a , a ) = 1 and a · a = a ; M4 V = V ( a ) ⊕ V ( a ) ⊕ V ( a ) 22 ⊕ V ( a ) where 1 0 1 1 25 V ( a ) = { v : v ∈ V , a · v = µ v } ; µ M5 V ( a ) = { λ a : λ ∈ R } . 1
Suppose furthermore that V obeys the Majorana fusion law. I.e. that for all v λ ∈ V a λ and v µ ∈ V a µ � V a v λ v µ ∈ ν ν ∈ λ ∗ µ where λ ∗ µ is a set given by 1 1 1 0 4 32 1 1 1 1 ∅ 4 32 1 1 0 ∅ 0 4 32 1 1 1 1 1 , 0 4 4 4 32 1 1 1 1 1 , 0 , 1 32 32 32 32 4 Then V is a Majorana algebra with Majorana axes A .
Suppose furthermore that V obeys the Majorana fusion law. I.e. that for all v λ ∈ V a λ and v µ ∈ V a µ � V a v λ v µ ∈ ν ν ∈ λ ∗ µ where λ ∗ µ is a set given by 1 1 1 0 4 32 1 1 1 1 ∅ 4 32 1 1 0 ∅ 0 4 32 1 1 1 1 1 , 0 4 4 4 32 1 1 1 1 1 , 0 , 1 32 32 32 32 4 Then V is a Majorana algebra with Majorana axes A .
Majorana Theory Let V be a Majorana algebra with Majorana axes A . For each a ∈ A , we can construct an involution τ ( a ) ∈ Aut ( V ) such that for u ∈ V ( a ) ⊕ V ( a ) ⊕ V ( a ) u 1 0 1 τ ( a )( u ) = 22 for u ∈ V ( a ) − u 1 25 called a Majorana involution. Given a group G and a normal set of involutions T such that G = � T � , if there exists a Majorana algebra V such that T = { τ ( a ) : a ∈ A } . then the tuple ( G , V , T ) is called a Majorana representation.
Majorana Theory Sakuma’s Theorem (A. A. Ivanov et al, 2010) Any Majorana algebra generated by two Majorana axes is isomorphic to a dihedral subalgebra of the Griess algebra.
The Algorithm In 2012, ´ Akos Seress announced the existence of an algorithm in GAP to construct the 2-closed Majorana representations of a given finite group. He never published his code or the full details of his algorithm and reproducing his work has been an important aim of the theory ever since.
The Algorithm Input: A finite group G and a normal set of involutions T such that G = � T � . Output: A spanning set C of V along with matrices indexed by the elements of C giving the inner and algebra products on V . If at any point in the algorithm a contradiction with the Majorana axioms is found, an appropriate error message is returned.
The Algorithm Step 0 - dihedral subalgebras. For every s , t ∈ T determine the isomorphism type of the algebra �� a t , a s �� . Step 1 - fusion law. Use the fusion laws to find additional eigenvectors. Step 2 - products from eigenvectors. Use eigenvectors to construct a system of linear equations whose unknowns are of the form a t · v for v ∈ C . Step 3 - the resurrection principle. Use a key result in Majorana theory to find a system of linear equations whose unknowns are of the from u · v for u , v ∈ C . Step 4 - rinse and repeat. Loop over steps 1 - 3 until all products are found.
The Algorithm Useful GAP features used: ◮ Data structures: lists and records; ◮ Linear algebra: sparse matrices (as part of the Gauss package). ◮ Optimisation and debugging: profiling package.
Results Three main types of results. 1. Construction of large examples - e.g. 286-dimensional representation of M 11 . 2. Classification results - e.g. Minimal 3-generated Majorana algebras (joint with A. Mamontov and A. Staroletov). 3. Important examples - e.g. An infinite family of Majorana algebras. Algorithm is published as part of a GAP Package MajoranaAlgebras Find our code at: https://github.com/MWhybrow92/MajoranaAlgebras .
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