Twisted Tori

The functionality presented here deals with the computation of the twisted tori of a finite group of Lie type.

Note that for a given group \(G(k)\), the twisted tori are returned as subgroups of the standard torus of \(G(K)\) for the smallest field extension \(K\) of \(k\), where this is possible.

For finite fields and an untwisted group of Lie type \(G(k)\), a twisted torus \(T_w(k)\) of \(G(k)\) has the form

\[T_w(k) = \{ t\in T(K) | t^{\sigma w} = t \},\]

where \(T(K)\) is the standard \(K\)-split torus of \(G(K)\), \(\sigma\) is the generator of the Galois group \({\operatorname{Gal}}(K:k)\) and \(w\) is an element of the Weyl group of \(G(k)\).

TwistedTorusOrder(R, w): RootDtm, GrpPermElt → SeqEnum

Given the root datum \(R\) and a Weyl group element \(w\), computes the orders of the cyclic components of the twisted torus \(T_w(k)\subset G(R,k)\) as sequence of polynomials in \(q\), the order of the field \(k\).

TwistedToriOrders(G): GrpLie → SeqEnum

TwistedToriOrders(R): RootDtm → SeqEnum

Given a group of Lie type \(G\) or a root datum \(R\), takes for every conjugacy class of the Weyl group of \(G\) a representative \(w\), and computes TwistedTorusOrder(R, w). Returns the sequence of the lists consisting of TwistedTorusOrder(R, w) and \(w\) for every conjugacy class.

TwistedTorus(G, w): GrpLie, GrpPermElt → List

Computes the twisted torus \(T_w(k)\) of the group of Lie type \(G\) for the given element \(w\) of the Weyl group of \(G\). Returned is the list consisting of three elements, first of them being the sequence of orders of cyclic parts of the torus, the second being the sequence of generators of the respective orders and the third being \(w\). See [Haller, 2005] for the algorithm used.

TwistedTori(G): GrpLie → SeqEnum

Computes one twisted torus \(T_w(k)\) of the group of Lie type \(G\) for each conjugacy class \(w^W\) of the Weyl group \(W\) of \(G\). A sequence of them is returned. See [Haller, 2005] for the algorithm used.

Example: Grp Lie Tori

We compute all twisted tori of \(A_1(5)\): As we may notice, the second one is contained in the group over the quadratic field extension:

> G := GroupOfLieType("A1", 5);
> TwistedToriOrders(G);
[ [*
    [
        q - 1
    ],
    Id($)
*], [*
    [
        q + 1
    ],
    (1, 2)
*] ]
> TwistedTori(G);
[ [*
    [ 4 ],
    [ (2)  ],
    Id($)
*], [*
    [ 6 ],
    [ ( k.1^4)  ],
    (1, 2)
*] ]
> Universe($1[2][2]);
$: Group of Lie type A1 over Finite field of size 5^2

Example: Grp Lie Tori2

These are the orders of the decompositions of all (up to conjugacy) maximal tori of the group \(G_2(q)\) as polynomials in \(q\):

> R := RootDatum("G2");
> [ t[1] : t in TwistedToriOrders(R) ];
[
    [  q - 1,  q - 1  ],
    [  q + 1,  q + 1  ],
    [  q^2 - 1        ],
    [  q^2 - 1        ],
    [  q^2 + q + 1    ],
    [  q^2 - q + 1    ]
]