Double Covers of the Alternating and Symmetric Groups

These functions construct representations of the groups 2An and 2Sn as matrix groups over a field GF(p) or (usually) GF(p2) for an odd prime p. Their dimension is 2d 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 Sn 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.
V2.29, 21 October 2025