Double Covers of the Alternating and Symmetric Groups

These functions construct representations of the groups 2A_n and 2S_n as matrix groups over a field GF(p) or (usually) GF(p²) for an odd prime p. Their dimension is 2^d with d = (n - 1)/2, (n - 2)/2, or (n - 3)/2 depending on n mod 6. The method used is described in [Maa09].

DoubleCoverSymmetricGroup(n: parameters) : RngIntElt -> GrpMat

    Sign: MonStgElt                     Default: "-"

    Characteristic: RngIntElt           Default: 3

n ≥4 represented as a matrix group over the field GF(p) or (usually) GF(p²), where p, which defaults to 3, can be specified as Characteristic.

There are two isomorphism classes of groups with this structure, which are distinguished by the optional parameter Sign, which can be positive, (Sign = "+" or "plus"), or (default) negative (Sign = "-" or "minus"). The inverse images of the Coxeter generators of S_n have order 2 when Sign is "+" and 4 when Sign is "-".

DoubleCoverAlternatingGroup(n: parameters) : RngIntElt -> GrpMat

    Characteristic: RngIntElt           Default: 3

n ≥4 represented as a matrix group over the field GF(p) or (usually) GF(p²), where p, which defaults to 3, can be specified as Characteristic.