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 \({\bf F}_{p}\) or (usually) \({\bf F}_{p^2}\) for an odd prime \(p\). Their dimension is \(2^d\) with \(d = (n-1)/2,\, (n-2)/2\), or \((n-3)/2\) depending on \(n \bmod 6\). The method used is described in [Maas, 2009].

DoubleCoverSymmetricGroup(n: parameters): RngIntElt -> GrpMat

Sign          : MonStgElt                    Default: "-"
Characteristic: RngIntElt                    Default: 3

The double cover \(2S_n\) of the symmetric groups \(S_n\) of degree \(n\) for \(n \ge 4\) represented as a matrix group over the field \({\bf F}_{p}\) or (usually) \({\bf F}_{p^2}\), 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

The double cover \(2A_n\) of the symmetric groups \(A_n\) of degree \(n\) for \(n \ge 4\) represented as a matrix group over the field \({\bf F}_{p}\) or (usually) \({\bf F}_{p^2}\), where \(p\), which defaults to \(3\), can be specified as Characteristic.